diff --git a/std/src/f128/tests.rs b/std/src/f128/tests.rs
index b64c7f856a15f..bd7a921c502a7 100644
--- a/std/src/f128/tests.rs
+++ b/std/src/f128/tests.rs
@@ -1,29 +1,31 @@
-#![allow(dead_code)] // FIXME(f16_f128): remove once constants are used
+#![cfg(not(bootstrap))]
+// FIXME(f16_f128): only tested on platforms that have symbols and aren't buggy
+#![cfg(reliable_f128)]
+
+use crate::f128::consts;
+use crate::num::*;
 
 /// Smallest number
 const TINY_BITS: u128 = 0x1;
+
 /// Next smallest number
 const TINY_UP_BITS: u128 = 0x2;
+
 /// Exponent = 0b11...10, Sifnificand 0b1111..10. Min val > 0
-const MAX_DOWN_BITS: u128 = 0x7ffeffffffffffffffffffffffffffff;
+const MAX_DOWN_BITS: u128 = 0x7ffefffffffffffffffffffffffffffe;
+
 /// Zeroed exponent, full significant
 const LARGEST_SUBNORMAL_BITS: u128 = 0x0000ffffffffffffffffffffffffffff;
+
 /// Exponent = 0b1, zeroed significand
 const SMALLEST_NORMAL_BITS: u128 = 0x00010000000000000000000000000000;
+
 /// First pattern over the mantissa
 const NAN_MASK1: u128 = 0x0000aaaaaaaaaaaaaaaaaaaaaaaaaaaa;
+
 /// Second pattern over the mantissa
 const NAN_MASK2: u128 = 0x00005555555555555555555555555555;
 
-/// Compare by value
-#[allow(unused_macros)]
-macro_rules! assert_f128_eq {
-    ($a:expr, $b:expr) => {
-        let (l, r): (&f128, &f128) = (&$a, &$b);
-        assert_eq!(*l, *r, "\na: {:#0130x}\nb: {:#0130x}", l.to_bits(), r.to_bits())
-    };
-}
-
 /// Compare by representation
 #[allow(unused_macros)]
 macro_rules! assert_f128_biteq {
@@ -31,10 +33,503 @@ macro_rules! assert_f128_biteq {
         let (l, r): (&f128, &f128) = (&$a, &$b);
         let lb = l.to_bits();
         let rb = r.to_bits();
-        assert_eq!(
-            lb, rb,
-            "float {:?} is not bitequal to {:?}.\na: {:#0130x}\nb: {:#0130x}",
-            *l, *r, lb, rb
-        );
+        assert_eq!(lb, rb, "float {l:?} is not bitequal to {r:?}.\na: {lb:#034x}\nb: {rb:#034x}");
     };
 }
+
+#[test]
+fn test_num_f128() {
+    test_num(10f128, 2f128);
+}
+
+// FIXME(f16_f128): add min and max tests when available
+
+#[test]
+fn test_nan() {
+    let nan: f128 = f128::NAN;
+    assert!(nan.is_nan());
+    assert!(!nan.is_infinite());
+    assert!(!nan.is_finite());
+    assert!(nan.is_sign_positive());
+    assert!(!nan.is_sign_negative());
+    // FIXME(f16_f128): classify
+    // assert!(!nan.is_normal());
+    // assert_eq!(Fp::Nan, nan.classify());
+}
+
+#[test]
+fn test_infinity() {
+    let inf: f128 = f128::INFINITY;
+    assert!(inf.is_infinite());
+    assert!(!inf.is_finite());
+    assert!(inf.is_sign_positive());
+    assert!(!inf.is_sign_negative());
+    assert!(!inf.is_nan());
+    // FIXME(f16_f128): classify
+    // assert!(!inf.is_normal());
+    // assert_eq!(Fp::Infinite, inf.classify());
+}
+
+#[test]
+fn test_neg_infinity() {
+    let neg_inf: f128 = f128::NEG_INFINITY;
+    assert!(neg_inf.is_infinite());
+    assert!(!neg_inf.is_finite());
+    assert!(!neg_inf.is_sign_positive());
+    assert!(neg_inf.is_sign_negative());
+    assert!(!neg_inf.is_nan());
+    // FIXME(f16_f128): classify
+    // assert!(!neg_inf.is_normal());
+    // assert_eq!(Fp::Infinite, neg_inf.classify());
+}
+
+#[test]
+fn test_zero() {
+    let zero: f128 = 0.0f128;
+    assert_eq!(0.0, zero);
+    assert!(!zero.is_infinite());
+    assert!(zero.is_finite());
+    assert!(zero.is_sign_positive());
+    assert!(!zero.is_sign_negative());
+    assert!(!zero.is_nan());
+    // FIXME(f16_f128): classify
+    // assert!(!zero.is_normal());
+    // assert_eq!(Fp::Zero, zero.classify());
+}
+
+#[test]
+fn test_neg_zero() {
+    let neg_zero: f128 = -0.0;
+    assert_eq!(0.0, neg_zero);
+    assert!(!neg_zero.is_infinite());
+    assert!(neg_zero.is_finite());
+    assert!(!neg_zero.is_sign_positive());
+    assert!(neg_zero.is_sign_negative());
+    assert!(!neg_zero.is_nan());
+    // FIXME(f16_f128): classify
+    // assert!(!neg_zero.is_normal());
+    // assert_eq!(Fp::Zero, neg_zero.classify());
+}
+
+#[test]
+fn test_one() {
+    let one: f128 = 1.0f128;
+    assert_eq!(1.0, one);
+    assert!(!one.is_infinite());
+    assert!(one.is_finite());
+    assert!(one.is_sign_positive());
+    assert!(!one.is_sign_negative());
+    assert!(!one.is_nan());
+    // FIXME(f16_f128): classify
+    // assert!(one.is_normal());
+    // assert_eq!(Fp::Normal, one.classify());
+}
+
+#[test]
+fn test_is_nan() {
+    let nan: f128 = f128::NAN;
+    let inf: f128 = f128::INFINITY;
+    let neg_inf: f128 = f128::NEG_INFINITY;
+    assert!(nan.is_nan());
+    assert!(!0.0f128.is_nan());
+    assert!(!5.3f128.is_nan());
+    assert!(!(-10.732f128).is_nan());
+    assert!(!inf.is_nan());
+    assert!(!neg_inf.is_nan());
+}
+
+#[test]
+fn test_is_infinite() {
+    let nan: f128 = f128::NAN;
+    let inf: f128 = f128::INFINITY;
+    let neg_inf: f128 = f128::NEG_INFINITY;
+    assert!(!nan.is_infinite());
+    assert!(inf.is_infinite());
+    assert!(neg_inf.is_infinite());
+    assert!(!0.0f128.is_infinite());
+    assert!(!42.8f128.is_infinite());
+    assert!(!(-109.2f128).is_infinite());
+}
+
+#[test]
+fn test_is_finite() {
+    let nan: f128 = f128::NAN;
+    let inf: f128 = f128::INFINITY;
+    let neg_inf: f128 = f128::NEG_INFINITY;
+    assert!(!nan.is_finite());
+    assert!(!inf.is_finite());
+    assert!(!neg_inf.is_finite());
+    assert!(0.0f128.is_finite());
+    assert!(42.8f128.is_finite());
+    assert!((-109.2f128).is_finite());
+}
+
+// FIXME(f16_f128): add `test_is_normal` and `test_classify` when classify is working
+// FIXME(f16_f128): add missing math functions when available
+
+#[test]
+fn test_abs() {
+    assert_eq!(f128::INFINITY.abs(), f128::INFINITY);
+    assert_eq!(1f128.abs(), 1f128);
+    assert_eq!(0f128.abs(), 0f128);
+    assert_eq!((-0f128).abs(), 0f128);
+    assert_eq!((-1f128).abs(), 1f128);
+    assert_eq!(f128::NEG_INFINITY.abs(), f128::INFINITY);
+    assert_eq!((1f128 / f128::NEG_INFINITY).abs(), 0f128);
+    assert!(f128::NAN.abs().is_nan());
+}
+
+#[test]
+fn test_is_sign_positive() {
+    assert!(f128::INFINITY.is_sign_positive());
+    assert!(1f128.is_sign_positive());
+    assert!(0f128.is_sign_positive());
+    assert!(!(-0f128).is_sign_positive());
+    assert!(!(-1f128).is_sign_positive());
+    assert!(!f128::NEG_INFINITY.is_sign_positive());
+    assert!(!(1f128 / f128::NEG_INFINITY).is_sign_positive());
+    assert!(f128::NAN.is_sign_positive());
+    assert!(!(-f128::NAN).is_sign_positive());
+}
+
+#[test]
+fn test_is_sign_negative() {
+    assert!(!f128::INFINITY.is_sign_negative());
+    assert!(!1f128.is_sign_negative());
+    assert!(!0f128.is_sign_negative());
+    assert!((-0f128).is_sign_negative());
+    assert!((-1f128).is_sign_negative());
+    assert!(f128::NEG_INFINITY.is_sign_negative());
+    assert!((1f128 / f128::NEG_INFINITY).is_sign_negative());
+    assert!(!f128::NAN.is_sign_negative());
+    assert!((-f128::NAN).is_sign_negative());
+}
+
+#[test]
+fn test_next_up() {
+    let tiny = f128::from_bits(TINY_BITS);
+    let tiny_up = f128::from_bits(TINY_UP_BITS);
+    let max_down = f128::from_bits(MAX_DOWN_BITS);
+    let largest_subnormal = f128::from_bits(LARGEST_SUBNORMAL_BITS);
+    let smallest_normal = f128::from_bits(SMALLEST_NORMAL_BITS);
+    assert_f128_biteq!(f128::NEG_INFINITY.next_up(), f128::MIN);
+    assert_f128_biteq!(f128::MIN.next_up(), -max_down);
+    assert_f128_biteq!((-1.0 - f128::EPSILON).next_up(), -1.0);
+    assert_f128_biteq!((-smallest_normal).next_up(), -largest_subnormal);
+    assert_f128_biteq!((-tiny_up).next_up(), -tiny);
+    assert_f128_biteq!((-tiny).next_up(), -0.0f128);
+    assert_f128_biteq!((-0.0f128).next_up(), tiny);
+    assert_f128_biteq!(0.0f128.next_up(), tiny);
+    assert_f128_biteq!(tiny.next_up(), tiny_up);
+    assert_f128_biteq!(largest_subnormal.next_up(), smallest_normal);
+    assert_f128_biteq!(1.0f128.next_up(), 1.0 + f128::EPSILON);
+    assert_f128_biteq!(f128::MAX.next_up(), f128::INFINITY);
+    assert_f128_biteq!(f128::INFINITY.next_up(), f128::INFINITY);
+
+    // Check that NaNs roundtrip.
+    let nan0 = f128::NAN;
+    let nan1 = f128::from_bits(f128::NAN.to_bits() ^ 0x002a_aaaa);
+    let nan2 = f128::from_bits(f128::NAN.to_bits() ^ 0x0055_5555);
+    assert_f128_biteq!(nan0.next_up(), nan0);
+    assert_f128_biteq!(nan1.next_up(), nan1);
+    assert_f128_biteq!(nan2.next_up(), nan2);
+}
+
+#[test]
+fn test_next_down() {
+    let tiny = f128::from_bits(TINY_BITS);
+    let tiny_up = f128::from_bits(TINY_UP_BITS);
+    let max_down = f128::from_bits(MAX_DOWN_BITS);
+    let largest_subnormal = f128::from_bits(LARGEST_SUBNORMAL_BITS);
+    let smallest_normal = f128::from_bits(SMALLEST_NORMAL_BITS);
+    assert_f128_biteq!(f128::NEG_INFINITY.next_down(), f128::NEG_INFINITY);
+    assert_f128_biteq!(f128::MIN.next_down(), f128::NEG_INFINITY);
+    assert_f128_biteq!((-max_down).next_down(), f128::MIN);
+    assert_f128_biteq!((-1.0f128).next_down(), -1.0 - f128::EPSILON);
+    assert_f128_biteq!((-largest_subnormal).next_down(), -smallest_normal);
+    assert_f128_biteq!((-tiny).next_down(), -tiny_up);
+    assert_f128_biteq!((-0.0f128).next_down(), -tiny);
+    assert_f128_biteq!((0.0f128).next_down(), -tiny);
+    assert_f128_biteq!(tiny.next_down(), 0.0f128);
+    assert_f128_biteq!(tiny_up.next_down(), tiny);
+    assert_f128_biteq!(smallest_normal.next_down(), largest_subnormal);
+    assert_f128_biteq!((1.0 + f128::EPSILON).next_down(), 1.0f128);
+    assert_f128_biteq!(f128::MAX.next_down(), max_down);
+    assert_f128_biteq!(f128::INFINITY.next_down(), f128::MAX);
+
+    // Check that NaNs roundtrip.
+    let nan0 = f128::NAN;
+    let nan1 = f128::from_bits(f128::NAN.to_bits() ^ 0x002a_aaaa);
+    let nan2 = f128::from_bits(f128::NAN.to_bits() ^ 0x0055_5555);
+    assert_f128_biteq!(nan0.next_down(), nan0);
+    assert_f128_biteq!(nan1.next_down(), nan1);
+    assert_f128_biteq!(nan2.next_down(), nan2);
+}
+
+#[test]
+fn test_recip() {
+    let nan: f128 = f128::NAN;
+    let inf: f128 = f128::INFINITY;
+    let neg_inf: f128 = f128::NEG_INFINITY;
+    assert_eq!(1.0f128.recip(), 1.0);
+    assert_eq!(2.0f128.recip(), 0.5);
+    assert_eq!((-0.4f128).recip(), -2.5);
+    assert_eq!(0.0f128.recip(), inf);
+    assert!(nan.recip().is_nan());
+    assert_eq!(inf.recip(), 0.0);
+    assert_eq!(neg_inf.recip(), 0.0);
+}
+
+#[test]
+fn test_to_degrees() {
+    let pi: f128 = consts::PI;
+    let nan: f128 = f128::NAN;
+    let inf: f128 = f128::INFINITY;
+    let neg_inf: f128 = f128::NEG_INFINITY;
+    assert_eq!(0.0f128.to_degrees(), 0.0);
+    assert_approx_eq!((-5.8f128).to_degrees(), -332.315521);
+    assert_eq!(pi.to_degrees(), 180.0);
+    assert!(nan.to_degrees().is_nan());
+    assert_eq!(inf.to_degrees(), inf);
+    assert_eq!(neg_inf.to_degrees(), neg_inf);
+    assert_eq!(1_f128.to_degrees(), 57.2957795130823208767981548141051703);
+}
+
+#[test]
+fn test_to_radians() {
+    let pi: f128 = consts::PI;
+    let nan: f128 = f128::NAN;
+    let inf: f128 = f128::INFINITY;
+    let neg_inf: f128 = f128::NEG_INFINITY;
+    assert_eq!(0.0f128.to_radians(), 0.0);
+    assert_approx_eq!(154.6f128.to_radians(), 2.698279);
+    assert_approx_eq!((-332.31f128).to_radians(), -5.799903);
+    // check approx rather than exact because round trip for pi doesn't fall on an exactly
+    // representable value (unlike `f32` and `f64`).
+    assert_approx_eq!(180.0f128.to_radians(), pi);
+    assert!(nan.to_radians().is_nan());
+    assert_eq!(inf.to_radians(), inf);
+    assert_eq!(neg_inf.to_radians(), neg_inf);
+}
+
+#[test]
+fn test_real_consts() {
+    // FIXME(f16_f128): add math tests when available
+    use super::consts;
+
+    let pi: f128 = consts::PI;
+    let frac_pi_2: f128 = consts::FRAC_PI_2;
+    let frac_pi_3: f128 = consts::FRAC_PI_3;
+    let frac_pi_4: f128 = consts::FRAC_PI_4;
+    let frac_pi_6: f128 = consts::FRAC_PI_6;
+    let frac_pi_8: f128 = consts::FRAC_PI_8;
+    let frac_1_pi: f128 = consts::FRAC_1_PI;
+    let frac_2_pi: f128 = consts::FRAC_2_PI;
+    // let frac_2_sqrtpi: f128 = consts::FRAC_2_SQRT_PI;
+    // let sqrt2: f128 = consts::SQRT_2;
+    // let frac_1_sqrt2: f128 = consts::FRAC_1_SQRT_2;
+    // let e: f128 = consts::E;
+    // let log2_e: f128 = consts::LOG2_E;
+    // let log10_e: f128 = consts::LOG10_E;
+    // let ln_2: f128 = consts::LN_2;
+    // let ln_10: f128 = consts::LN_10;
+
+    assert_approx_eq!(frac_pi_2, pi / 2f128);
+    assert_approx_eq!(frac_pi_3, pi / 3f128);
+    assert_approx_eq!(frac_pi_4, pi / 4f128);
+    assert_approx_eq!(frac_pi_6, pi / 6f128);
+    assert_approx_eq!(frac_pi_8, pi / 8f128);
+    assert_approx_eq!(frac_1_pi, 1f128 / pi);
+    assert_approx_eq!(frac_2_pi, 2f128 / pi);
+    // assert_approx_eq!(frac_2_sqrtpi, 2f128 / pi.sqrt());
+    // assert_approx_eq!(sqrt2, 2f128.sqrt());
+    // assert_approx_eq!(frac_1_sqrt2, 1f128 / 2f128.sqrt());
+    // assert_approx_eq!(log2_e, e.log2());
+    // assert_approx_eq!(log10_e, e.log10());
+    // assert_approx_eq!(ln_2, 2f128.ln());
+    // assert_approx_eq!(ln_10, 10f128.ln());
+}
+
+#[test]
+fn test_float_bits_conv() {
+    assert_eq!((1f128).to_bits(), 0x3fff0000000000000000000000000000);
+    assert_eq!((12.5f128).to_bits(), 0x40029000000000000000000000000000);
+    assert_eq!((1337f128).to_bits(), 0x40094e40000000000000000000000000);
+    assert_eq!((-14.25f128).to_bits(), 0xc002c800000000000000000000000000);
+    assert_approx_eq!(f128::from_bits(0x3fff0000000000000000000000000000), 1.0);
+    assert_approx_eq!(f128::from_bits(0x40029000000000000000000000000000), 12.5);
+    assert_approx_eq!(f128::from_bits(0x40094e40000000000000000000000000), 1337.0);
+    assert_approx_eq!(f128::from_bits(0xc002c800000000000000000000000000), -14.25);
+
+    // Check that NaNs roundtrip their bits regardless of signaling-ness
+    // 0xA is 0b1010; 0x5 is 0b0101 -- so these two together clobbers all the mantissa bits
+    let masked_nan1 = f128::NAN.to_bits() ^ NAN_MASK1;
+    let masked_nan2 = f128::NAN.to_bits() ^ NAN_MASK2;
+    assert!(f128::from_bits(masked_nan1).is_nan());
+    assert!(f128::from_bits(masked_nan2).is_nan());
+
+    assert_eq!(f128::from_bits(masked_nan1).to_bits(), masked_nan1);
+    assert_eq!(f128::from_bits(masked_nan2).to_bits(), masked_nan2);
+}
+
+#[test]
+#[should_panic]
+fn test_clamp_min_greater_than_max() {
+    let _ = 1.0f128.clamp(3.0, 1.0);
+}
+
+#[test]
+#[should_panic]
+fn test_clamp_min_is_nan() {
+    let _ = 1.0f128.clamp(f128::NAN, 1.0);
+}
+
+#[test]
+#[should_panic]
+fn test_clamp_max_is_nan() {
+    let _ = 1.0f128.clamp(3.0, f128::NAN);
+}
+
+#[test]
+fn test_total_cmp() {
+    use core::cmp::Ordering;
+
+    fn quiet_bit_mask() -> u128 {
+        1 << (f128::MANTISSA_DIGITS - 2)
+    }
+
+    // FIXME(f16_f128): test subnormals when powf is available
+    // fn min_subnorm() -> f128 {
+    //     f128::MIN_POSITIVE / f128::powf(2.0, f128::MANTISSA_DIGITS as f128 - 1.0)
+    // }
+
+    // fn max_subnorm() -> f128 {
+    //     f128::MIN_POSITIVE - min_subnorm()
+    // }
+
+    fn q_nan() -> f128 {
+        f128::from_bits(f128::NAN.to_bits() | quiet_bit_mask())
+    }
+
+    fn s_nan() -> f128 {
+        f128::from_bits((f128::NAN.to_bits() & !quiet_bit_mask()) + 42)
+    }
+
+    assert_eq!(Ordering::Equal, (-q_nan()).total_cmp(&-q_nan()));
+    assert_eq!(Ordering::Equal, (-s_nan()).total_cmp(&-s_nan()));
+    assert_eq!(Ordering::Equal, (-f128::INFINITY).total_cmp(&-f128::INFINITY));
+    assert_eq!(Ordering::Equal, (-f128::MAX).total_cmp(&-f128::MAX));
+    assert_eq!(Ordering::Equal, (-2.5_f128).total_cmp(&-2.5));
+    assert_eq!(Ordering::Equal, (-1.0_f128).total_cmp(&-1.0));
+    assert_eq!(Ordering::Equal, (-1.5_f128).total_cmp(&-1.5));
+    assert_eq!(Ordering::Equal, (-0.5_f128).total_cmp(&-0.5));
+    assert_eq!(Ordering::Equal, (-f128::MIN_POSITIVE).total_cmp(&-f128::MIN_POSITIVE));
+    // assert_eq!(Ordering::Equal, (-max_subnorm()).total_cmp(&-max_subnorm()));
+    // assert_eq!(Ordering::Equal, (-min_subnorm()).total_cmp(&-min_subnorm()));
+    assert_eq!(Ordering::Equal, (-0.0_f128).total_cmp(&-0.0));
+    assert_eq!(Ordering::Equal, 0.0_f128.total_cmp(&0.0));
+    // assert_eq!(Ordering::Equal, min_subnorm().total_cmp(&min_subnorm()));
+    // assert_eq!(Ordering::Equal, max_subnorm().total_cmp(&max_subnorm()));
+    assert_eq!(Ordering::Equal, f128::MIN_POSITIVE.total_cmp(&f128::MIN_POSITIVE));
+    assert_eq!(Ordering::Equal, 0.5_f128.total_cmp(&0.5));
+    assert_eq!(Ordering::Equal, 1.0_f128.total_cmp(&1.0));
+    assert_eq!(Ordering::Equal, 1.5_f128.total_cmp(&1.5));
+    assert_eq!(Ordering::Equal, 2.5_f128.total_cmp(&2.5));
+    assert_eq!(Ordering::Equal, f128::MAX.total_cmp(&f128::MAX));
+    assert_eq!(Ordering::Equal, f128::INFINITY.total_cmp(&f128::INFINITY));
+    assert_eq!(Ordering::Equal, s_nan().total_cmp(&s_nan()));
+    assert_eq!(Ordering::Equal, q_nan().total_cmp(&q_nan()));
+
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-s_nan()));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-f128::INFINITY));
+    assert_eq!(Ordering::Less, (-f128::INFINITY).total_cmp(&-f128::MAX));
+    assert_eq!(Ordering::Less, (-f128::MAX).total_cmp(&-2.5));
+    assert_eq!(Ordering::Less, (-2.5_f128).total_cmp(&-1.5));
+    assert_eq!(Ordering::Less, (-1.5_f128).total_cmp(&-1.0));
+    assert_eq!(Ordering::Less, (-1.0_f128).total_cmp(&-0.5));
+    assert_eq!(Ordering::Less, (-0.5_f128).total_cmp(&-f128::MIN_POSITIVE));
+    // assert_eq!(Ordering::Less, (-f128::MIN_POSITIVE).total_cmp(&-max_subnorm()));
+    // assert_eq!(Ordering::Less, (-max_subnorm()).total_cmp(&-min_subnorm()));
+    // assert_eq!(Ordering::Less, (-min_subnorm()).total_cmp(&-0.0));
+    assert_eq!(Ordering::Less, (-0.0_f128).total_cmp(&0.0));
+    // assert_eq!(Ordering::Less, 0.0_f128.total_cmp(&min_subnorm()));
+    // assert_eq!(Ordering::Less, min_subnorm().total_cmp(&max_subnorm()));
+    // assert_eq!(Ordering::Less, max_subnorm().total_cmp(&f128::MIN_POSITIVE));
+    assert_eq!(Ordering::Less, f128::MIN_POSITIVE.total_cmp(&0.5));
+    assert_eq!(Ordering::Less, 0.5_f128.total_cmp(&1.0));
+    assert_eq!(Ordering::Less, 1.0_f128.total_cmp(&1.5));
+    assert_eq!(Ordering::Less, 1.5_f128.total_cmp(&2.5));
+    assert_eq!(Ordering::Less, 2.5_f128.total_cmp(&f128::MAX));
+    assert_eq!(Ordering::Less, f128::MAX.total_cmp(&f128::INFINITY));
+    assert_eq!(Ordering::Less, f128::INFINITY.total_cmp(&s_nan()));
+    assert_eq!(Ordering::Less, s_nan().total_cmp(&q_nan()));
+
+    assert_eq!(Ordering::Greater, (-s_nan()).total_cmp(&-q_nan()));
+    assert_eq!(Ordering::Greater, (-f128::INFINITY).total_cmp(&-s_nan()));
+    assert_eq!(Ordering::Greater, (-f128::MAX).total_cmp(&-f128::INFINITY));
+    assert_eq!(Ordering::Greater, (-2.5_f128).total_cmp(&-f128::MAX));
+    assert_eq!(Ordering::Greater, (-1.5_f128).total_cmp(&-2.5));
+    assert_eq!(Ordering::Greater, (-1.0_f128).total_cmp(&-1.5));
+    assert_eq!(Ordering::Greater, (-0.5_f128).total_cmp(&-1.0));
+    assert_eq!(Ordering::Greater, (-f128::MIN_POSITIVE).total_cmp(&-0.5));
+    // assert_eq!(Ordering::Greater, (-max_subnorm()).total_cmp(&-f128::MIN_POSITIVE));
+    // assert_eq!(Ordering::Greater, (-min_subnorm()).total_cmp(&-max_subnorm()));
+    // assert_eq!(Ordering::Greater, (-0.0_f128).total_cmp(&-min_subnorm()));
+    assert_eq!(Ordering::Greater, 0.0_f128.total_cmp(&-0.0));
+    // assert_eq!(Ordering::Greater, min_subnorm().total_cmp(&0.0));
+    // assert_eq!(Ordering::Greater, max_subnorm().total_cmp(&min_subnorm()));
+    // assert_eq!(Ordering::Greater, f128::MIN_POSITIVE.total_cmp(&max_subnorm()));
+    assert_eq!(Ordering::Greater, 0.5_f128.total_cmp(&f128::MIN_POSITIVE));
+    assert_eq!(Ordering::Greater, 1.0_f128.total_cmp(&0.5));
+    assert_eq!(Ordering::Greater, 1.5_f128.total_cmp(&1.0));
+    assert_eq!(Ordering::Greater, 2.5_f128.total_cmp(&1.5));
+    assert_eq!(Ordering::Greater, f128::MAX.total_cmp(&2.5));
+    assert_eq!(Ordering::Greater, f128::INFINITY.total_cmp(&f128::MAX));
+    assert_eq!(Ordering::Greater, s_nan().total_cmp(&f128::INFINITY));
+    assert_eq!(Ordering::Greater, q_nan().total_cmp(&s_nan()));
+
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-s_nan()));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-f128::INFINITY));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-f128::MAX));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-2.5));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-1.5));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-1.0));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-0.5));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-f128::MIN_POSITIVE));
+    // assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-max_subnorm()));
+    // assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-min_subnorm()));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-0.0));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&0.0));
+    // assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&min_subnorm()));
+    // assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&max_subnorm()));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&f128::MIN_POSITIVE));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&0.5));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&1.0));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&1.5));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&2.5));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&f128::MAX));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&f128::INFINITY));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&s_nan()));
+
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-f128::INFINITY));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-f128::MAX));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-2.5));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-1.5));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-1.0));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-0.5));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-f128::MIN_POSITIVE));
+    // assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-max_subnorm()));
+    // assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-min_subnorm()));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-0.0));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&0.0));
+    // assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&min_subnorm()));
+    // assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&max_subnorm()));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&f128::MIN_POSITIVE));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&0.5));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&1.0));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&1.5));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&2.5));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&f128::MAX));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&f128::INFINITY));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&s_nan()));
+}
diff --git a/std/src/f16/tests.rs b/std/src/f16/tests.rs
index d65c43eca4bb8..bb6a811529e17 100644
--- a/std/src/f16/tests.rs
+++ b/std/src/f16/tests.rs
@@ -1,35 +1,37 @@
-#![allow(dead_code)] // FIXME(f16_f128): remove once constants are used
+#![cfg(not(bootstrap))]
+// FIXME(f16_f128): only tested on platforms that have symbols and aren't buggy
+#![cfg(reliable_f16)]
+
+use crate::f16::consts;
+use crate::num::*;
 
 // We run out of precision pretty quickly with f16
-const F16_APPROX_L1: f16 = 0.001;
+// const F16_APPROX_L1: f16 = 0.001;
 const F16_APPROX_L2: f16 = 0.01;
-const F16_APPROX_L3: f16 = 0.1;
+// const F16_APPROX_L3: f16 = 0.1;
 const F16_APPROX_L4: f16 = 0.5;
 
 /// Smallest number
 const TINY_BITS: u16 = 0x1;
+
 /// Next smallest number
 const TINY_UP_BITS: u16 = 0x2;
+
 /// Exponent = 0b11...10, Sifnificand 0b1111..10. Min val > 0
 const MAX_DOWN_BITS: u16 = 0x7bfe;
+
 /// Zeroed exponent, full significant
 const LARGEST_SUBNORMAL_BITS: u16 = 0x03ff;
+
 /// Exponent = 0b1, zeroed significand
 const SMALLEST_NORMAL_BITS: u16 = 0x0400;
+
 /// First pattern over the mantissa
 const NAN_MASK1: u16 = 0x02aa;
+
 /// Second pattern over the mantissa
 const NAN_MASK2: u16 = 0x0155;
 
-/// Compare by value
-#[allow(unused_macros)]
-macro_rules! assert_f16_eq {
-    ($a:expr, $b:expr) => {
-        let (l, r): (&f16, &f16) = (&$a, &$b);
-        assert_eq!(*l, *r, "\na: {:#018x}\nb: {:#018x}", l.to_bits(), r.to_bits())
-    };
-}
-
 /// Compare by representation
 #[allow(unused_macros)]
 macro_rules! assert_f16_biteq {
@@ -37,10 +39,500 @@ macro_rules! assert_f16_biteq {
         let (l, r): (&f16, &f16) = (&$a, &$b);
         let lb = l.to_bits();
         let rb = r.to_bits();
-        assert_eq!(
-            lb, rb,
-            "float {:?} is not bitequal to {:?}.\na: {:#018x}\nb: {:#018x}",
-            *l, *r, lb, rb
-        );
+        assert_eq!(lb, rb, "float {l:?} ({lb:#04x}) is not bitequal to {r:?} ({rb:#04x})");
     };
 }
+
+#[test]
+fn test_num_f16() {
+    test_num(10f16, 2f16);
+}
+
+// FIXME(f16_f128): add min and max tests when available
+
+#[test]
+fn test_nan() {
+    let nan: f16 = f16::NAN;
+    assert!(nan.is_nan());
+    assert!(!nan.is_infinite());
+    assert!(!nan.is_finite());
+    assert!(nan.is_sign_positive());
+    assert!(!nan.is_sign_negative());
+    // FIXME(f16_f128): classify
+    // assert!(!nan.is_normal());
+    // assert_eq!(Fp::Nan, nan.classify());
+}
+
+#[test]
+fn test_infinity() {
+    let inf: f16 = f16::INFINITY;
+    assert!(inf.is_infinite());
+    assert!(!inf.is_finite());
+    assert!(inf.is_sign_positive());
+    assert!(!inf.is_sign_negative());
+    assert!(!inf.is_nan());
+    // FIXME(f16_f128): classify
+    // assert!(!inf.is_normal());
+    // assert_eq!(Fp::Infinite, inf.classify());
+}
+
+#[test]
+fn test_neg_infinity() {
+    let neg_inf: f16 = f16::NEG_INFINITY;
+    assert!(neg_inf.is_infinite());
+    assert!(!neg_inf.is_finite());
+    assert!(!neg_inf.is_sign_positive());
+    assert!(neg_inf.is_sign_negative());
+    assert!(!neg_inf.is_nan());
+    // FIXME(f16_f128): classify
+    // assert!(!neg_inf.is_normal());
+    // assert_eq!(Fp::Infinite, neg_inf.classify());
+}
+
+#[test]
+fn test_zero() {
+    let zero: f16 = 0.0f16;
+    assert_eq!(0.0, zero);
+    assert!(!zero.is_infinite());
+    assert!(zero.is_finite());
+    assert!(zero.is_sign_positive());
+    assert!(!zero.is_sign_negative());
+    assert!(!zero.is_nan());
+    // FIXME(f16_f128): classify
+    // assert!(!zero.is_normal());
+    // assert_eq!(Fp::Zero, zero.classify());
+}
+
+#[test]
+fn test_neg_zero() {
+    let neg_zero: f16 = -0.0;
+    assert_eq!(0.0, neg_zero);
+    assert!(!neg_zero.is_infinite());
+    assert!(neg_zero.is_finite());
+    assert!(!neg_zero.is_sign_positive());
+    assert!(neg_zero.is_sign_negative());
+    assert!(!neg_zero.is_nan());
+    // FIXME(f16_f128): classify
+    // assert!(!neg_zero.is_normal());
+    // assert_eq!(Fp::Zero, neg_zero.classify());
+}
+
+#[test]
+fn test_one() {
+    let one: f16 = 1.0f16;
+    assert_eq!(1.0, one);
+    assert!(!one.is_infinite());
+    assert!(one.is_finite());
+    assert!(one.is_sign_positive());
+    assert!(!one.is_sign_negative());
+    assert!(!one.is_nan());
+    // FIXME(f16_f128): classify
+    // assert!(one.is_normal());
+    // assert_eq!(Fp::Normal, one.classify());
+}
+
+#[test]
+fn test_is_nan() {
+    let nan: f16 = f16::NAN;
+    let inf: f16 = f16::INFINITY;
+    let neg_inf: f16 = f16::NEG_INFINITY;
+    assert!(nan.is_nan());
+    assert!(!0.0f16.is_nan());
+    assert!(!5.3f16.is_nan());
+    assert!(!(-10.732f16).is_nan());
+    assert!(!inf.is_nan());
+    assert!(!neg_inf.is_nan());
+}
+
+#[test]
+fn test_is_infinite() {
+    let nan: f16 = f16::NAN;
+    let inf: f16 = f16::INFINITY;
+    let neg_inf: f16 = f16::NEG_INFINITY;
+    assert!(!nan.is_infinite());
+    assert!(inf.is_infinite());
+    assert!(neg_inf.is_infinite());
+    assert!(!0.0f16.is_infinite());
+    assert!(!42.8f16.is_infinite());
+    assert!(!(-109.2f16).is_infinite());
+}
+
+#[test]
+fn test_is_finite() {
+    let nan: f16 = f16::NAN;
+    let inf: f16 = f16::INFINITY;
+    let neg_inf: f16 = f16::NEG_INFINITY;
+    assert!(!nan.is_finite());
+    assert!(!inf.is_finite());
+    assert!(!neg_inf.is_finite());
+    assert!(0.0f16.is_finite());
+    assert!(42.8f16.is_finite());
+    assert!((-109.2f16).is_finite());
+}
+
+// FIXME(f16_f128): add `test_is_normal` and `test_classify` when classify is working
+// FIXME(f16_f128): add missing math functions when available
+
+#[test]
+fn test_abs() {
+    assert_eq!(f16::INFINITY.abs(), f16::INFINITY);
+    assert_eq!(1f16.abs(), 1f16);
+    assert_eq!(0f16.abs(), 0f16);
+    assert_eq!((-0f16).abs(), 0f16);
+    assert_eq!((-1f16).abs(), 1f16);
+    assert_eq!(f16::NEG_INFINITY.abs(), f16::INFINITY);
+    assert_eq!((1f16 / f16::NEG_INFINITY).abs(), 0f16);
+    assert!(f16::NAN.abs().is_nan());
+}
+
+#[test]
+fn test_is_sign_positive() {
+    assert!(f16::INFINITY.is_sign_positive());
+    assert!(1f16.is_sign_positive());
+    assert!(0f16.is_sign_positive());
+    assert!(!(-0f16).is_sign_positive());
+    assert!(!(-1f16).is_sign_positive());
+    assert!(!f16::NEG_INFINITY.is_sign_positive());
+    assert!(!(1f16 / f16::NEG_INFINITY).is_sign_positive());
+    assert!(f16::NAN.is_sign_positive());
+    assert!(!(-f16::NAN).is_sign_positive());
+}
+
+#[test]
+fn test_is_sign_negative() {
+    assert!(!f16::INFINITY.is_sign_negative());
+    assert!(!1f16.is_sign_negative());
+    assert!(!0f16.is_sign_negative());
+    assert!((-0f16).is_sign_negative());
+    assert!((-1f16).is_sign_negative());
+    assert!(f16::NEG_INFINITY.is_sign_negative());
+    assert!((1f16 / f16::NEG_INFINITY).is_sign_negative());
+    assert!(!f16::NAN.is_sign_negative());
+    assert!((-f16::NAN).is_sign_negative());
+}
+
+#[test]
+fn test_next_up() {
+    let tiny = f16::from_bits(TINY_BITS);
+    let tiny_up = f16::from_bits(TINY_UP_BITS);
+    let max_down = f16::from_bits(MAX_DOWN_BITS);
+    let largest_subnormal = f16::from_bits(LARGEST_SUBNORMAL_BITS);
+    let smallest_normal = f16::from_bits(SMALLEST_NORMAL_BITS);
+    assert_f16_biteq!(f16::NEG_INFINITY.next_up(), f16::MIN);
+    assert_f16_biteq!(f16::MIN.next_up(), -max_down);
+    assert_f16_biteq!((-1.0 - f16::EPSILON).next_up(), -1.0);
+    assert_f16_biteq!((-smallest_normal).next_up(), -largest_subnormal);
+    assert_f16_biteq!((-tiny_up).next_up(), -tiny);
+    assert_f16_biteq!((-tiny).next_up(), -0.0f16);
+    assert_f16_biteq!((-0.0f16).next_up(), tiny);
+    assert_f16_biteq!(0.0f16.next_up(), tiny);
+    assert_f16_biteq!(tiny.next_up(), tiny_up);
+    assert_f16_biteq!(largest_subnormal.next_up(), smallest_normal);
+    assert_f16_biteq!(1.0f16.next_up(), 1.0 + f16::EPSILON);
+    assert_f16_biteq!(f16::MAX.next_up(), f16::INFINITY);
+    assert_f16_biteq!(f16::INFINITY.next_up(), f16::INFINITY);
+
+    // Check that NaNs roundtrip.
+    let nan0 = f16::NAN;
+    let nan1 = f16::from_bits(f16::NAN.to_bits() ^ NAN_MASK1);
+    let nan2 = f16::from_bits(f16::NAN.to_bits() ^ NAN_MASK2);
+    assert_f16_biteq!(nan0.next_up(), nan0);
+    assert_f16_biteq!(nan1.next_up(), nan1);
+    assert_f16_biteq!(nan2.next_up(), nan2);
+}
+
+#[test]
+fn test_next_down() {
+    let tiny = f16::from_bits(TINY_BITS);
+    let tiny_up = f16::from_bits(TINY_UP_BITS);
+    let max_down = f16::from_bits(MAX_DOWN_BITS);
+    let largest_subnormal = f16::from_bits(LARGEST_SUBNORMAL_BITS);
+    let smallest_normal = f16::from_bits(SMALLEST_NORMAL_BITS);
+    assert_f16_biteq!(f16::NEG_INFINITY.next_down(), f16::NEG_INFINITY);
+    assert_f16_biteq!(f16::MIN.next_down(), f16::NEG_INFINITY);
+    assert_f16_biteq!((-max_down).next_down(), f16::MIN);
+    assert_f16_biteq!((-1.0f16).next_down(), -1.0 - f16::EPSILON);
+    assert_f16_biteq!((-largest_subnormal).next_down(), -smallest_normal);
+    assert_f16_biteq!((-tiny).next_down(), -tiny_up);
+    assert_f16_biteq!((-0.0f16).next_down(), -tiny);
+    assert_f16_biteq!((0.0f16).next_down(), -tiny);
+    assert_f16_biteq!(tiny.next_down(), 0.0f16);
+    assert_f16_biteq!(tiny_up.next_down(), tiny);
+    assert_f16_biteq!(smallest_normal.next_down(), largest_subnormal);
+    assert_f16_biteq!((1.0 + f16::EPSILON).next_down(), 1.0f16);
+    assert_f16_biteq!(f16::MAX.next_down(), max_down);
+    assert_f16_biteq!(f16::INFINITY.next_down(), f16::MAX);
+
+    // Check that NaNs roundtrip.
+    let nan0 = f16::NAN;
+    let nan1 = f16::from_bits(f16::NAN.to_bits() ^ NAN_MASK1);
+    let nan2 = f16::from_bits(f16::NAN.to_bits() ^ NAN_MASK2);
+    assert_f16_biteq!(nan0.next_down(), nan0);
+    assert_f16_biteq!(nan1.next_down(), nan1);
+    assert_f16_biteq!(nan2.next_down(), nan2);
+}
+
+#[test]
+fn test_recip() {
+    let nan: f16 = f16::NAN;
+    let inf: f16 = f16::INFINITY;
+    let neg_inf: f16 = f16::NEG_INFINITY;
+    assert_eq!(1.0f16.recip(), 1.0);
+    assert_eq!(2.0f16.recip(), 0.5);
+    assert_eq!((-0.4f16).recip(), -2.5);
+    assert_eq!(0.0f16.recip(), inf);
+    assert!(nan.recip().is_nan());
+    assert_eq!(inf.recip(), 0.0);
+    assert_eq!(neg_inf.recip(), 0.0);
+}
+
+#[test]
+fn test_to_degrees() {
+    let pi: f16 = consts::PI;
+    let nan: f16 = f16::NAN;
+    let inf: f16 = f16::INFINITY;
+    let neg_inf: f16 = f16::NEG_INFINITY;
+    assert_eq!(0.0f16.to_degrees(), 0.0);
+    assert_approx_eq!((-5.8f16).to_degrees(), -332.315521);
+    assert_approx_eq!(pi.to_degrees(), 180.0, F16_APPROX_L4);
+    assert!(nan.to_degrees().is_nan());
+    assert_eq!(inf.to_degrees(), inf);
+    assert_eq!(neg_inf.to_degrees(), neg_inf);
+    assert_eq!(1_f16.to_degrees(), 57.2957795130823208767981548141051703);
+}
+
+#[test]
+fn test_to_radians() {
+    let pi: f16 = consts::PI;
+    let nan: f16 = f16::NAN;
+    let inf: f16 = f16::INFINITY;
+    let neg_inf: f16 = f16::NEG_INFINITY;
+    assert_eq!(0.0f16.to_radians(), 0.0);
+    assert_approx_eq!(154.6f16.to_radians(), 2.698279);
+    assert_approx_eq!((-332.31f16).to_radians(), -5.799903);
+    assert_approx_eq!(180.0f16.to_radians(), pi, F16_APPROX_L2);
+    assert!(nan.to_radians().is_nan());
+    assert_eq!(inf.to_radians(), inf);
+    assert_eq!(neg_inf.to_radians(), neg_inf);
+}
+
+#[test]
+fn test_real_consts() {
+    // FIXME(f16_f128): add math tests when available
+    use super::consts;
+
+    let pi: f16 = consts::PI;
+    let frac_pi_2: f16 = consts::FRAC_PI_2;
+    let frac_pi_3: f16 = consts::FRAC_PI_3;
+    let frac_pi_4: f16 = consts::FRAC_PI_4;
+    let frac_pi_6: f16 = consts::FRAC_PI_6;
+    let frac_pi_8: f16 = consts::FRAC_PI_8;
+    let frac_1_pi: f16 = consts::FRAC_1_PI;
+    let frac_2_pi: f16 = consts::FRAC_2_PI;
+    // let frac_2_sqrtpi: f16 = consts::FRAC_2_SQRT_PI;
+    // let sqrt2: f16 = consts::SQRT_2;
+    // let frac_1_sqrt2: f16 = consts::FRAC_1_SQRT_2;
+    // let e: f16 = consts::E;
+    // let log2_e: f16 = consts::LOG2_E;
+    // let log10_e: f16 = consts::LOG10_E;
+    // let ln_2: f16 = consts::LN_2;
+    // let ln_10: f16 = consts::LN_10;
+
+    assert_approx_eq!(frac_pi_2, pi / 2f16);
+    assert_approx_eq!(frac_pi_3, pi / 3f16);
+    assert_approx_eq!(frac_pi_4, pi / 4f16);
+    assert_approx_eq!(frac_pi_6, pi / 6f16);
+    assert_approx_eq!(frac_pi_8, pi / 8f16);
+    assert_approx_eq!(frac_1_pi, 1f16 / pi);
+    assert_approx_eq!(frac_2_pi, 2f16 / pi);
+    // assert_approx_eq!(frac_2_sqrtpi, 2f16 / pi.sqrt());
+    // assert_approx_eq!(sqrt2, 2f16.sqrt());
+    // assert_approx_eq!(frac_1_sqrt2, 1f16 / 2f16.sqrt());
+    // assert_approx_eq!(log2_e, e.log2());
+    // assert_approx_eq!(log10_e, e.log10());
+    // assert_approx_eq!(ln_2, 2f16.ln());
+    // assert_approx_eq!(ln_10, 10f16.ln());
+}
+
+#[test]
+fn test_float_bits_conv() {
+    assert_eq!((1f16).to_bits(), 0x3c00);
+    assert_eq!((12.5f16).to_bits(), 0x4a40);
+    assert_eq!((1337f16).to_bits(), 0x6539);
+    assert_eq!((-14.25f16).to_bits(), 0xcb20);
+    assert_approx_eq!(f16::from_bits(0x3c00), 1.0);
+    assert_approx_eq!(f16::from_bits(0x4a40), 12.5);
+    assert_approx_eq!(f16::from_bits(0x6539), 1337.0);
+    assert_approx_eq!(f16::from_bits(0xcb20), -14.25);
+
+    // Check that NaNs roundtrip their bits regardless of signaling-ness
+    let masked_nan1 = f16::NAN.to_bits() ^ NAN_MASK1;
+    let masked_nan2 = f16::NAN.to_bits() ^ NAN_MASK2;
+    assert!(f16::from_bits(masked_nan1).is_nan());
+    assert!(f16::from_bits(masked_nan2).is_nan());
+
+    assert_eq!(f16::from_bits(masked_nan1).to_bits(), masked_nan1);
+    assert_eq!(f16::from_bits(masked_nan2).to_bits(), masked_nan2);
+}
+
+#[test]
+#[should_panic]
+fn test_clamp_min_greater_than_max() {
+    let _ = 1.0f16.clamp(3.0, 1.0);
+}
+
+#[test]
+#[should_panic]
+fn test_clamp_min_is_nan() {
+    let _ = 1.0f16.clamp(f16::NAN, 1.0);
+}
+
+#[test]
+#[should_panic]
+fn test_clamp_max_is_nan() {
+    let _ = 1.0f16.clamp(3.0, f16::NAN);
+}
+
+#[test]
+fn test_total_cmp() {
+    use core::cmp::Ordering;
+
+    fn quiet_bit_mask() -> u16 {
+        1 << (f16::MANTISSA_DIGITS - 2)
+    }
+
+    // FIXME(f16_f128): test subnormals when powf is available
+    // fn min_subnorm() -> f16 {
+    //     f16::MIN_POSITIVE / f16::powf(2.0, f16::MANTISSA_DIGITS as f16 - 1.0)
+    // }
+
+    // fn max_subnorm() -> f16 {
+    //     f16::MIN_POSITIVE - min_subnorm()
+    // }
+
+    fn q_nan() -> f16 {
+        f16::from_bits(f16::NAN.to_bits() | quiet_bit_mask())
+    }
+
+    fn s_nan() -> f16 {
+        f16::from_bits((f16::NAN.to_bits() & !quiet_bit_mask()) + 42)
+    }
+
+    assert_eq!(Ordering::Equal, (-q_nan()).total_cmp(&-q_nan()));
+    assert_eq!(Ordering::Equal, (-s_nan()).total_cmp(&-s_nan()));
+    assert_eq!(Ordering::Equal, (-f16::INFINITY).total_cmp(&-f16::INFINITY));
+    assert_eq!(Ordering::Equal, (-f16::MAX).total_cmp(&-f16::MAX));
+    assert_eq!(Ordering::Equal, (-2.5_f16).total_cmp(&-2.5));
+    assert_eq!(Ordering::Equal, (-1.0_f16).total_cmp(&-1.0));
+    assert_eq!(Ordering::Equal, (-1.5_f16).total_cmp(&-1.5));
+    assert_eq!(Ordering::Equal, (-0.5_f16).total_cmp(&-0.5));
+    assert_eq!(Ordering::Equal, (-f16::MIN_POSITIVE).total_cmp(&-f16::MIN_POSITIVE));
+    // assert_eq!(Ordering::Equal, (-max_subnorm()).total_cmp(&-max_subnorm()));
+    // assert_eq!(Ordering::Equal, (-min_subnorm()).total_cmp(&-min_subnorm()));
+    assert_eq!(Ordering::Equal, (-0.0_f16).total_cmp(&-0.0));
+    assert_eq!(Ordering::Equal, 0.0_f16.total_cmp(&0.0));
+    // assert_eq!(Ordering::Equal, min_subnorm().total_cmp(&min_subnorm()));
+    // assert_eq!(Ordering::Equal, max_subnorm().total_cmp(&max_subnorm()));
+    assert_eq!(Ordering::Equal, f16::MIN_POSITIVE.total_cmp(&f16::MIN_POSITIVE));
+    assert_eq!(Ordering::Equal, 0.5_f16.total_cmp(&0.5));
+    assert_eq!(Ordering::Equal, 1.0_f16.total_cmp(&1.0));
+    assert_eq!(Ordering::Equal, 1.5_f16.total_cmp(&1.5));
+    assert_eq!(Ordering::Equal, 2.5_f16.total_cmp(&2.5));
+    assert_eq!(Ordering::Equal, f16::MAX.total_cmp(&f16::MAX));
+    assert_eq!(Ordering::Equal, f16::INFINITY.total_cmp(&f16::INFINITY));
+    assert_eq!(Ordering::Equal, s_nan().total_cmp(&s_nan()));
+    assert_eq!(Ordering::Equal, q_nan().total_cmp(&q_nan()));
+
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-s_nan()));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-f16::INFINITY));
+    assert_eq!(Ordering::Less, (-f16::INFINITY).total_cmp(&-f16::MAX));
+    assert_eq!(Ordering::Less, (-f16::MAX).total_cmp(&-2.5));
+    assert_eq!(Ordering::Less, (-2.5_f16).total_cmp(&-1.5));
+    assert_eq!(Ordering::Less, (-1.5_f16).total_cmp(&-1.0));
+    assert_eq!(Ordering::Less, (-1.0_f16).total_cmp(&-0.5));
+    assert_eq!(Ordering::Less, (-0.5_f16).total_cmp(&-f16::MIN_POSITIVE));
+    // assert_eq!(Ordering::Less, (-f16::MIN_POSITIVE).total_cmp(&-max_subnorm()));
+    // assert_eq!(Ordering::Less, (-max_subnorm()).total_cmp(&-min_subnorm()));
+    // assert_eq!(Ordering::Less, (-min_subnorm()).total_cmp(&-0.0));
+    assert_eq!(Ordering::Less, (-0.0_f16).total_cmp(&0.0));
+    // assert_eq!(Ordering::Less, 0.0_f16.total_cmp(&min_subnorm()));
+    // assert_eq!(Ordering::Less, min_subnorm().total_cmp(&max_subnorm()));
+    // assert_eq!(Ordering::Less, max_subnorm().total_cmp(&f16::MIN_POSITIVE));
+    assert_eq!(Ordering::Less, f16::MIN_POSITIVE.total_cmp(&0.5));
+    assert_eq!(Ordering::Less, 0.5_f16.total_cmp(&1.0));
+    assert_eq!(Ordering::Less, 1.0_f16.total_cmp(&1.5));
+    assert_eq!(Ordering::Less, 1.5_f16.total_cmp(&2.5));
+    assert_eq!(Ordering::Less, 2.5_f16.total_cmp(&f16::MAX));
+    assert_eq!(Ordering::Less, f16::MAX.total_cmp(&f16::INFINITY));
+    assert_eq!(Ordering::Less, f16::INFINITY.total_cmp(&s_nan()));
+    assert_eq!(Ordering::Less, s_nan().total_cmp(&q_nan()));
+
+    assert_eq!(Ordering::Greater, (-s_nan()).total_cmp(&-q_nan()));
+    assert_eq!(Ordering::Greater, (-f16::INFINITY).total_cmp(&-s_nan()));
+    assert_eq!(Ordering::Greater, (-f16::MAX).total_cmp(&-f16::INFINITY));
+    assert_eq!(Ordering::Greater, (-2.5_f16).total_cmp(&-f16::MAX));
+    assert_eq!(Ordering::Greater, (-1.5_f16).total_cmp(&-2.5));
+    assert_eq!(Ordering::Greater, (-1.0_f16).total_cmp(&-1.5));
+    assert_eq!(Ordering::Greater, (-0.5_f16).total_cmp(&-1.0));
+    assert_eq!(Ordering::Greater, (-f16::MIN_POSITIVE).total_cmp(&-0.5));
+    // assert_eq!(Ordering::Greater, (-max_subnorm()).total_cmp(&-f16::MIN_POSITIVE));
+    // assert_eq!(Ordering::Greater, (-min_subnorm()).total_cmp(&-max_subnorm()));
+    // assert_eq!(Ordering::Greater, (-0.0_f16).total_cmp(&-min_subnorm()));
+    assert_eq!(Ordering::Greater, 0.0_f16.total_cmp(&-0.0));
+    // assert_eq!(Ordering::Greater, min_subnorm().total_cmp(&0.0));
+    // assert_eq!(Ordering::Greater, max_subnorm().total_cmp(&min_subnorm()));
+    // assert_eq!(Ordering::Greater, f16::MIN_POSITIVE.total_cmp(&max_subnorm()));
+    assert_eq!(Ordering::Greater, 0.5_f16.total_cmp(&f16::MIN_POSITIVE));
+    assert_eq!(Ordering::Greater, 1.0_f16.total_cmp(&0.5));
+    assert_eq!(Ordering::Greater, 1.5_f16.total_cmp(&1.0));
+    assert_eq!(Ordering::Greater, 2.5_f16.total_cmp(&1.5));
+    assert_eq!(Ordering::Greater, f16::MAX.total_cmp(&2.5));
+    assert_eq!(Ordering::Greater, f16::INFINITY.total_cmp(&f16::MAX));
+    assert_eq!(Ordering::Greater, s_nan().total_cmp(&f16::INFINITY));
+    assert_eq!(Ordering::Greater, q_nan().total_cmp(&s_nan()));
+
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-s_nan()));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-f16::INFINITY));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-f16::MAX));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-2.5));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-1.5));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-1.0));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-0.5));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-f16::MIN_POSITIVE));
+    // assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-max_subnorm()));
+    // assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-min_subnorm()));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&-0.0));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&0.0));
+    // assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&min_subnorm()));
+    // assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&max_subnorm()));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&f16::MIN_POSITIVE));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&0.5));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&1.0));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&1.5));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&2.5));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&f16::MAX));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&f16::INFINITY));
+    assert_eq!(Ordering::Less, (-q_nan()).total_cmp(&s_nan()));
+
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-f16::INFINITY));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-f16::MAX));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-2.5));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-1.5));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-1.0));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-0.5));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-f16::MIN_POSITIVE));
+    // assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-max_subnorm()));
+    // assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-min_subnorm()));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&-0.0));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&0.0));
+    // assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&min_subnorm()));
+    // assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&max_subnorm()));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&f16::MIN_POSITIVE));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&0.5));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&1.0));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&1.5));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&2.5));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&f16::MAX));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&f16::INFINITY));
+    assert_eq!(Ordering::Less, (-s_nan()).total_cmp(&s_nan()));
+}
diff --git a/std/src/macros.rs b/std/src/macros.rs
index 58df83bd79d23..972b6015932db 100644
--- a/std/src/macros.rs
+++ b/std/src/macros.rs
@@ -373,10 +373,17 @@ macro_rules! dbg {
     };
 }
 
+/// Verify that floats are within a tolerance of each other, 1.0e-6 by default.
 #[cfg(test)]
 macro_rules! assert_approx_eq {
-    ($a:expr, $b:expr) => {{
+    ($a:expr, $b:expr) => {{ assert_approx_eq!($a, $b, 1.0e-6) }};
+    ($a:expr, $b:expr, $lim:expr) => {{
         let (a, b) = (&$a, &$b);
-        assert!((*a - *b).abs() < 1.0e-6, "{} is not approximately equal to {}", *a, *b);
+        let diff = (*a - *b).abs();
+        assert!(
+            diff < $lim,
+            "{a:?} is not approximately equal to {b:?} (threshold {lim:?}, actual {diff:?})",
+            lim = $lim
+        );
     }};
 }