Use mixed Horner scheme in Math.exp/expm1 to improve instruction parallelization (AssemblyScript#311)

Author: MaxGraey

std/assembly/math.ts

@@ -422,8 +422,10 @@ export namespace NativeMath {
     } else if (hx > 0x3E300000) {
       hi = x;
     } else return 1.0 + x;
-    var xx = x * x;
-    var c = x - xx * (P1 + xx * (P2 + xx * (P3 + xx * (P4 + xx * P5))));
+    var xs = x * x;
+    // var c = x - xp2 * (P1 + xp2 * (P2 + xp2 * (P3 + xp2 * (P4 + xp2 * P5))));
+    var xq = xs * xs;
+    var c = x - (xs * P1 + xq * ((P2 + xs * P3) + xq * (P4 + xs * P5)));
     var y = 1.0 + (x * c / (2 - c) - lo + hi);
     if (k == 0) return y;
     return scalbn(y, k);
@@ -464,7 +466,9 @@ export namespace NativeMath {
     } else if (hx < 0x3C900000) return x;
     var hfx = 0.5 * x;
     var hxs = x * hfx;
-    var r1 = 1.0 + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5))));
+    // var r1 = 1.0 + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5))));
+    var hxq = hxs * hxs;
+    var r1 = (1.0 + hxs * Q1) + hxq * ((Q2 + hxs * Q3) + hxq * (Q4 + hxs * Q5));
     t = 3.0 - r1 * hfx;
     var e = hxs * ((r1 - t) / (6.0 - x * t));
     if (k == 0) return x - (x * e - hxs);

tests/compiler/std/array.optimized.wat

@@ -3457,7 +3457,7 @@
   if
    i32.const 0
    i32.const 552
-   i32.const 955
+   i32.const 959
    i32.const 4
    call $~lib/env/abort
    unreachable
@@ -5173,7 +5173,7 @@
   if
    i32.const 0
    i32.const 552
-   i32.const 964
+   i32.const 968
    i32.const 24
    call $~lib/env/abort
    unreachable

tests/compiler/std/array.untouched.wat

@@ -4512,7 +4512,7 @@
   if
    i32.const 0
    i32.const 552
-   i32.const 955
+   i32.const 959
    i32.const 4
    call $~lib/env/abort
    unreachable
@@ -7609,7 +7609,7 @@
   if
    i32.const 0
    i32.const 552
-   i32.const 964
+   i32.const 968
    i32.const 24
    call $~lib/env/abort
    unreachable

tests/compiler/std/libm.optimized.wat

@@ -1695,20 +1695,29 @@
     return
    end
   end
-  f64.const 1
   get_local $0
   f64.const 0.5
   get_local $0
   f64.mul
   tee_local $8
   f64.mul
   tee_local $2
-  f64.const -0.03333333333333313
   get_local $2
+  f64.mul
+  set_local $1
+  f64.const 3
+  f64.const 1
+  get_local $2
+  f64.const -0.03333333333333313
+  f64.mul
+  f64.add
+  get_local $1
   f64.const 1.5873015872548146e-03
   get_local $2
   f64.const -7.93650757867488e-05
-  get_local $2
+  f64.mul
+  f64.add
+  get_local $1
   f64.const 4.008217827329362e-06
   get_local $2
   f64.const -2.0109921818362437e-07
@@ -1718,20 +1727,14 @@
   f64.add
   f64.mul
   f64.add
-  f64.mul
-  f64.add
-  f64.mul
-  f64.add
   tee_local $9
+  get_local $8
+  f64.mul
+  f64.sub
   set_local $1
   get_local $2
   get_local $9
-  f64.const 3
   get_local $1
-  get_local $8
-  f64.mul
-  f64.sub
-  tee_local $1
   f64.sub
   f64.const 6
   get_local $0
@@ -1954,10 +1957,12 @@
  )
  (func $~lib/math/NativeMath.exp (; 25 ;) (type $FF) (param $0 f64) (result f64)
   (local $1 i32)
-  (local $2 i32)
-  (local $3 f64)
-  (local $4 i32)
+  (local $2 f64)
+  (local $3 i32)
+  (local $4 f64)
   (local $5 f64)
+  (local $6 i32)
+  (local $7 f64)
   get_local $0
   i64.reinterpret/f64
   i64.const 32
@@ -1966,7 +1971,7 @@
   tee_local $1
   i32.const 31
   i32.shr_u
-  set_local $4
+  set_local $6
   get_local $1
   i32.const 2147483647
   i32.and
@@ -2017,22 +2022,22 @@
     i32.trunc_s/f64
    else    
     i32.const 1
-    get_local $4
+    get_local $6
     i32.const 1
     i32.shl
     i32.sub
    end
-   tee_local $2
+   tee_local $3
    f64.convert_s/i32
    tee_local $0
    f64.const 0.6931471803691238
    f64.mul
    f64.sub
-   tee_local $3
+   tee_local $4
    get_local $0
    f64.const 1.9082149292705877e-10
    f64.mul
-   tee_local $5
+   tee_local $7
    f64.sub
    set_local $0
   else   
@@ -2046,54 +2051,58 @@
     return
    end
    get_local $0
-   set_local $3
+   set_local $4
   end
-  f64.const 1
   get_local $0
   get_local $0
+  f64.mul
+  tee_local $2
+  get_local $2
+  f64.mul
+  set_local $5
+  f64.const 1
   get_local $0
   get_local $0
-  f64.mul
-  tee_local $0
+  get_local $2
   f64.const 0.16666666666666602
-  get_local $0
+  f64.mul
+  get_local $5
   f64.const -2.7777777777015593e-03
-  get_local $0
+  get_local $2
   f64.const 6.613756321437934e-05
-  get_local $0
-  f64.const -1.6533902205465252e-06
-  get_local $0
-  f64.const 4.1381367970572385e-08
   f64.mul
   f64.add
+  get_local $5
+  f64.const -1.6533902205465252e-06
+  get_local $2
+  f64.const 4.1381367970572385e-08
   f64.mul
   f64.add
   f64.mul
   f64.add
   f64.mul
   f64.add
-  f64.mul
   f64.sub
   tee_local $0
   f64.mul
   f64.const 2
   get_local $0
   f64.sub
   f64.div
-  get_local $5
+  get_local $7
   f64.sub
-  get_local $3
+  get_local $4
   f64.add
   f64.add
   set_local $0
-  get_local $2
+  get_local $3
   i32.eqz
   if
    get_local $0
    return
   end
   get_local $0
-  get_local $2
+  get_local $3
   call $~lib/math/NativeMath.scalbn
  )
  (func $~lib/math/NativeMath.cosh (; 26 ;) (type $FF) (param $0 f64) (result f64)