Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1

Percentage Accurate: 27.8% → 32.3%

Time: 12.6s

Alternatives: 10

Speedup: 269.0×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (*
  (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
  (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t) / 16.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

Python

def code(x, y, z, t, a, b):
	return (x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 27.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (*
  (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
  (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t) / 16.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

Python

def code(x, y, z, t, a, b):
	return (x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 32.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot t\_1 \leq 10^{+303}:\\ \;\;\;\;\left(\left(-x\right) \cdot \cos \left(\mathsf{fma}\left(\frac{z \cdot \mathsf{fma}\left(2, y, 1\right)}{16}, t\_m, \mathsf{PI}\left(\right)\right)\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0))))
   (if (<= (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0))) t_1) 1e+303)
     (* (* (- x) (cos (fma (/ (* z (fma 2.0 y 1.0)) 16.0) t_m (PI)))) t_1)
     x)))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1e303

   1. Initial program 49.3%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \left(x \cdot \color{blue}{\sin \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      3. sin-sum N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\sin \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \left(\sin \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \color{blue}{\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      5. sin-PI/2 N/A

         \[\leadsto \left(x \cdot \left(\sin \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot \color{blue}{1}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\sin \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) - \left(\mathsf{neg}\left(\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right)\right) \cdot 1\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\sin \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) - \left(\mathsf{neg}\left(\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right)\right) \cdot 1\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   4. Applied rewrites 49.5%

      \[\leadsto \left(x \cdot \color{blue}{\left(\sin \left(\frac{t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)}{16}\right) \cdot 0 - \cos \left(\mathsf{fma}\left(\frac{z \cdot \mathsf{fma}\left(2, y, 1\right)}{16}, t, \mathsf{PI}\left(\right)\right)\right) \cdot 1\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   if 1e303 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 0.1%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{\mathsf{neg}\left(16\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{t \cdot \color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(t \cdot \left(a \cdot 2 + 1\right)\right) \cdot b}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(t \cdot \left(a \cdot 2 + 1\right)\right) \cdot \frac{b}{\mathsf{neg}\left(16\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot t\right)} \cdot \frac{b}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\left(a \cdot 2 + 1\right) \cdot t, \frac{b}{\mathsf{neg}\left(16\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   4. Applied rewrites 0.8%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t, \frac{b}{-16}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   6. Applied rewrites 0.1%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\left(-\cos \left(\mathsf{fma}\left(\frac{b}{-16} \cdot t, \mathsf{fma}\left(a, 2, 1\right), \mathsf{PI}\left(\right)\right)\right)\right)} \]

   7. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \cos \mathsf{PI}\left(\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \cos \mathsf{PI}\left(\right)\right)} \]

      2. cos-PI N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{-1}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot x}\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot x \]

      6. lower-*.f64 11.4

         \[\leadsto \color{blue}{1 \cdot x} \]

   9. Applied rewrites 11.4%

      \[\leadsto \color{blue}{1 \cdot x} \]
   10. Step-by-step derivation

   11. Applied rewrites 11.4%

       \[\leadsto x \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 10^{+303}:\\ \;\;\;\;\left(\left(-x\right) \cdot \cos \left(\mathsf{fma}\left(\frac{z \cdot \mathsf{fma}\left(2, y, 1\right)}{16}, t, \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 32.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 10^{+303}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(\left({y}^{-1} + 2\right) \cdot y\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\_m\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
      1e+303)
   (*
    (* x (cos (/ (* (* (* (+ (pow y -1.0) 2.0) y) z) t_m) 16.0)))
    (cos (* (* b t_m) 0.0625)))
   x))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+303) {
		tmp = (x * cos((((((pow(y, -1.0) + 2.0) * y) * z) * t_m) / 16.0))) * cos(((b * t_m) * 0.0625));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((x * cos((((((y * 2.0d0) + 1.0d0) * z) * t_m) / 16.0d0))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t_m) / 16.0d0))) <= 1d+303) then
        tmp = (x * cos(((((((y ** (-1.0d0)) + 2.0d0) * y) * z) * t_m) / 16.0d0))) * cos(((b * t_m) * 0.0625d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (((x * Math.cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+303) {
		tmp = (x * Math.cos((((((Math.pow(y, -1.0) + 2.0) * y) * z) * t_m) / 16.0))) * Math.cos(((b * t_m) * 0.0625));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m, a, b):
	tmp = 0
	if ((x * math.cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * math.cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+303:
		tmp = (x * math.cos((((((math.pow(y, -1.0) + 2.0) * y) * z) * t_m) / 16.0))) * math.cos(((b * t_m) * 0.0625))
	else:
		tmp = x
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+303)
		tmp = Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64((y ^ -1.0) + 2.0) * y) * z) * t_m) / 16.0))) * cos(Float64(Float64(b * t_m) * 0.0625)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m, a, b)
	tmp = 0.0;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+303)
		tmp = (x * cos(((((((y ^ -1.0) + 2.0) * y) * z) * t_m) / 16.0))) * cos(((b * t_m) * 0.0625));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+303], N[(N[(x * N[Cos[N[(N[(N[(N[(N[(N[Power[y, -1.0], $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(b * t$95$m), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1e303

   1. Initial program 49.3%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

      3. lower-*.f64 49.2

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\color{blue}{\left(b \cdot t\right)} \cdot 0.0625\right) \]

   5. Applied rewrites 49.2%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot 0.0625\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\color{blue}{\left(y \cdot \left(2 + \frac{1}{y}\right)\right)} \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\color{blue}{\left(\left(2 + \frac{1}{y}\right) \cdot y\right)} \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\color{blue}{\left(\left(2 + \frac{1}{y}\right) \cdot y\right)} \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(\color{blue}{\left(\frac{1}{y} + 2\right)} \cdot y\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(\color{blue}{\left(\frac{1}{y} + 2\right)} \cdot y\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      5. lower-/.f64 49.4

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(\left(\color{blue}{\frac{1}{y}} + 2\right) \cdot y\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \]

   8. Applied rewrites 49.4%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\color{blue}{\left(\left(\frac{1}{y} + 2\right) \cdot y\right)} \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \]

   if 1e303 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 0.1%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{\mathsf{neg}\left(16\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{t \cdot \color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(t \cdot \left(a \cdot 2 + 1\right)\right) \cdot b}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(t \cdot \left(a \cdot 2 + 1\right)\right) \cdot \frac{b}{\mathsf{neg}\left(16\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot t\right)} \cdot \frac{b}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\left(a \cdot 2 + 1\right) \cdot t, \frac{b}{\mathsf{neg}\left(16\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   4. Applied rewrites 0.8%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t, \frac{b}{-16}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   6. Applied rewrites 0.1%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\left(-\cos \left(\mathsf{fma}\left(\frac{b}{-16} \cdot t, \mathsf{fma}\left(a, 2, 1\right), \mathsf{PI}\left(\right)\right)\right)\right)} \]

   7. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \cos \mathsf{PI}\left(\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \cos \mathsf{PI}\left(\right)\right)} \]

      2. cos-PI N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{-1}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot x}\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot x \]

      6. lower-*.f64 11.4

         \[\leadsto \color{blue}{1 \cdot x} \]

   9. Applied rewrites 11.4%

      \[\leadsto \color{blue}{1 \cdot x} \]
   10. Step-by-step derivation

   11. Applied rewrites 11.4%

       \[\leadsto x \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 10^{+303}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(\left({y}^{-1} + 2\right) \cdot y\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 10^{+303}:\\ \;\;\;\;\left(x \cdot \sin \left(\frac{t\_m \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)}{-16} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\left(b \cdot t\_m\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
      1e+303)
   (*
    (* x (sin (+ (/ (* t_m (* z (fma 2.0 y 1.0))) -16.0) (/ (PI) 2.0))))
    (cos (* (* b t_m) 0.0625)))
   x))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1e303

   1. Initial program 49.3%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

      3. lower-*.f64 49.2

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\color{blue}{\left(b \cdot t\right)} \cdot 0.0625\right) \]

   5. Applied rewrites 49.2%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot 0.0625\right)} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)}\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \left(x \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right)}\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(x \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(x \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

   7. Applied rewrites 48.7%

      \[\leadsto \left(x \cdot \color{blue}{\sin \left(\frac{t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)}{-16} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \]

   if 1e303 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 0.1%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{\mathsf{neg}\left(16\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{t \cdot \color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(t \cdot \left(a \cdot 2 + 1\right)\right) \cdot b}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(t \cdot \left(a \cdot 2 + 1\right)\right) \cdot \frac{b}{\mathsf{neg}\left(16\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot t\right)} \cdot \frac{b}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\left(a \cdot 2 + 1\right) \cdot t, \frac{b}{\mathsf{neg}\left(16\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   4. Applied rewrites 0.8%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t, \frac{b}{-16}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   6. Applied rewrites 0.1%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\left(-\cos \left(\mathsf{fma}\left(\frac{b}{-16} \cdot t, \mathsf{fma}\left(a, 2, 1\right), \mathsf{PI}\left(\right)\right)\right)\right)} \]

   7. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \cos \mathsf{PI}\left(\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \cos \mathsf{PI}\left(\right)\right)} \]

      2. cos-PI N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{-1}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot x}\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot x \]

      6. lower-*.f64 11.4

         \[\leadsto \color{blue}{1 \cdot x} \]

   9. Applied rewrites 11.4%

      \[\leadsto \color{blue}{1 \cdot x} \]
   10. Step-by-step derivation

   11. Applied rewrites 11.4%

       \[\leadsto x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 10^{+303}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\mathsf{fma}\left(2, y, 1\right) \cdot \left(t\_m \cdot z\right)}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\_m\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
      1e+303)
   (*
    (* x (cos (/ (* (fma 2.0 y 1.0) (* t_m z)) 16.0)))
    (cos (* (* b t_m) 0.0625)))
   x))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+303) {
		tmp = (x * cos(((fma(2.0, y, 1.0) * (t_m * z)) / 16.0))) * cos(((b * t_m) * 0.0625));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+303)
		tmp = Float64(Float64(x * cos(Float64(Float64(fma(2.0, y, 1.0) * Float64(t_m * z)) / 16.0))) * cos(Float64(Float64(b * t_m) * 0.0625)));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+303], N[(N[(x * N[Cos[N[(N[(N[(2.0 * y + 1.0), $MachinePrecision] * N[(t$95$m * z), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(b * t$95$m), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1e303

   1. Initial program 49.3%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

      3. lower-*.f64 49.2

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\color{blue}{\left(b \cdot t\right)} \cdot 0.0625\right) \]

   5. Applied rewrites 49.2%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot 0.0625\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(\left(y \cdot 2 + 1\right) \cdot z\right)} \cdot t}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(y \cdot 2 + 1\right) \cdot \left(z \cdot t\right)}}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(y \cdot 2 + 1\right) \cdot \color{blue}{\left(t \cdot z\right)}}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(y \cdot 2 + 1\right) \cdot \color{blue}{\left(t \cdot z\right)}}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      6. lower-*.f64 49.2

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(y \cdot 2 + 1\right) \cdot \left(t \cdot z\right)}}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(y \cdot 2 + 1\right)} \cdot \left(t \cdot z\right)}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\color{blue}{y \cdot 2} + 1\right) \cdot \left(t \cdot z\right)}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      9. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\color{blue}{2 \cdot y} + 1\right) \cdot \left(t \cdot z\right)}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      10. lower-fma.f64 49.2

          \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\mathsf{fma}\left(2, y, 1\right)} \cdot \left(t \cdot z\right)}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \]

   7. Applied rewrites 49.2%

      \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\mathsf{fma}\left(2, y, 1\right) \cdot \left(t \cdot z\right)}}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \]

   if 1e303 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 0.1%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{\mathsf{neg}\left(16\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{t \cdot \color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(t \cdot \left(a \cdot 2 + 1\right)\right) \cdot b}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(t \cdot \left(a \cdot 2 + 1\right)\right) \cdot \frac{b}{\mathsf{neg}\left(16\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot t\right)} \cdot \frac{b}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\left(a \cdot 2 + 1\right) \cdot t, \frac{b}{\mathsf{neg}\left(16\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   4. Applied rewrites 0.8%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t, \frac{b}{-16}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   6. Applied rewrites 0.1%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\left(-\cos \left(\mathsf{fma}\left(\frac{b}{-16} \cdot t, \mathsf{fma}\left(a, 2, 1\right), \mathsf{PI}\left(\right)\right)\right)\right)} \]

   7. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \cos \mathsf{PI}\left(\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \cos \mathsf{PI}\left(\right)\right)} \]

      2. cos-PI N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{-1}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot x}\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot x \]

      6. lower-*.f64 11.4

         \[\leadsto \color{blue}{1 \cdot x} \]

   9. Applied rewrites 11.4%

      \[\leadsto \color{blue}{1 \cdot x} \]
   10. Step-by-step derivation

   11. Applied rewrites 11.4%

       \[\leadsto x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 10^{+303}:\\ \;\;\;\;\left(x \cdot \cos \left(\left(b \cdot t\_m\right) \cdot -0.0625\right)\right) \cdot \cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
      1e+303)
   (*
    (* x (cos (* (* b t_m) -0.0625)))
    (cos (* -0.0625 (* (* (fma 2.0 y 1.0) z) t_m))))
   x))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+303) {
		tmp = (x * cos(((b * t_m) * -0.0625))) * cos((-0.0625 * ((fma(2.0, y, 1.0) * z) * t_m)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+303)
		tmp = Float64(Float64(x * cos(Float64(Float64(b * t_m) * -0.0625))) * cos(Float64(-0.0625 * Float64(Float64(fma(2.0, y, 1.0) * z) * t_m))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+303], N[(N[(x * N[Cos[N[(N[(b * t$95$m), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.0625 * N[(N[(N[(2.0 * y + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1e303

   1. Initial program 49.3%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

      3. lower-*.f64 49.2

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\color{blue}{\left(b \cdot t\right)} \cdot 0.0625\right) \]

   5. Applied rewrites 49.2%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot 0.0625\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right)} \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

      4. cos-neg-rev N/A

         \[\leadsto \left(x \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right)}\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot t\right)\right)}\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(x \cdot \cos \left(\color{blue}{\frac{-1}{16}} \cdot \left(b \cdot t\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\cos \left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right)}\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{-1}{16}\right)}\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{-1}{16}\right)}\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(b \cdot t\right)} \cdot \frac{-1}{16}\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

      11. cos-neg-rev N/A

          \[\leadsto \left(x \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{-1}{16}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{-1}{16}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \left(x \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{-1}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \left(x \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{-1}{16}\right)\right) \cdot \cos \left(\color{blue}{\frac{-1}{16}} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{-1}{16}\right)\right) \cdot \cos \color{blue}{\left(\frac{-1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)} \]

      16. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{-1}{16}\right)\right) \cdot \cos \left(\frac{-1}{16} \cdot \color{blue}{\left(\left(z \cdot \left(1 + 2 \cdot y\right)\right) \cdot t\right)}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{-1}{16}\right)\right) \cdot \cos \left(\frac{-1}{16} \cdot \color{blue}{\left(\left(z \cdot \left(1 + 2 \cdot y\right)\right) \cdot t\right)}\right) \]

   8. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\left(x \cdot \cos \left(\left(b \cdot t\right) \cdot -0.0625\right)\right) \cdot \cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right)\right)} \]

   if 1e303 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 0.1%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{\mathsf{neg}\left(16\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{t \cdot \color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(t \cdot \left(a \cdot 2 + 1\right)\right) \cdot b}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(t \cdot \left(a \cdot 2 + 1\right)\right) \cdot \frac{b}{\mathsf{neg}\left(16\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot t\right)} \cdot \frac{b}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\left(a \cdot 2 + 1\right) \cdot t, \frac{b}{\mathsf{neg}\left(16\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   4. Applied rewrites 0.8%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t, \frac{b}{-16}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   6. Applied rewrites 0.1%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\left(-\cos \left(\mathsf{fma}\left(\frac{b}{-16} \cdot t, \mathsf{fma}\left(a, 2, 1\right), \mathsf{PI}\left(\right)\right)\right)\right)} \]

   7. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \cos \mathsf{PI}\left(\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \cos \mathsf{PI}\left(\right)\right)} \]

      2. cos-PI N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{-1}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot x}\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot x \]

      6. lower-*.f64 11.4

         \[\leadsto \color{blue}{1 \cdot x} \]

   9. Applied rewrites 11.4%

      \[\leadsto \color{blue}{1 \cdot x} \]
   10. Step-by-step derivation

   11. Applied rewrites 11.4%

       \[\leadsto x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 31.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 10^{+303}:\\ \;\;\;\;\left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\_m\right) \cdot 0.125\right)\right) \cdot \cos \left(\left(b \cdot t\_m\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
      1e+303)
   (* (* x (cos (* (* (* z y) t_m) 0.125))) (cos (* (* b t_m) 0.0625)))
   x))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+303) {
		tmp = (x * cos((((z * y) * t_m) * 0.125))) * cos(((b * t_m) * 0.0625));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((x * cos((((((y * 2.0d0) + 1.0d0) * z) * t_m) / 16.0d0))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t_m) / 16.0d0))) <= 1d+303) then
        tmp = (x * cos((((z * y) * t_m) * 0.125d0))) * cos(((b * t_m) * 0.0625d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (((x * Math.cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+303) {
		tmp = (x * Math.cos((((z * y) * t_m) * 0.125))) * Math.cos(((b * t_m) * 0.0625));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m, a, b):
	tmp = 0
	if ((x * math.cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * math.cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+303:
		tmp = (x * math.cos((((z * y) * t_m) * 0.125))) * math.cos(((b * t_m) * 0.0625))
	else:
		tmp = x
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+303)
		tmp = Float64(Float64(x * cos(Float64(Float64(Float64(z * y) * t_m) * 0.125))) * cos(Float64(Float64(b * t_m) * 0.0625)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m, a, b)
	tmp = 0.0;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+303)
		tmp = (x * cos((((z * y) * t_m) * 0.125))) * cos(((b * t_m) * 0.0625));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+303], N[(N[(x * N[Cos[N[(N[(N[(z * y), $MachinePrecision] * t$95$m), $MachinePrecision] * 0.125), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(b * t$95$m), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1e303

   1. Initial program 49.3%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

      3. lower-*.f64 49.2

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\color{blue}{\left(b \cdot t\right)} \cdot 0.0625\right) \]

   5. Applied rewrites 49.2%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot 0.0625\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\color{blue}{\left(2 \cdot y\right)} \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 48.9

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\color{blue}{\left(2 \cdot y\right)} \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \]

   8. Applied rewrites 48.9%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\color{blue}{\left(2 \cdot y\right)} \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{1}{8} \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)}\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot \left(y \cdot z\right)\right) \cdot \frac{1}{8}\right)}\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot \left(y \cdot z\right)\right) \cdot \frac{1}{8}\right)}\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

       3. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \cdot \frac{1}{8}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(\left(y \cdot z\right) \cdot t\right)} \cdot \frac{1}{8}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

       5. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\left(\color{blue}{\left(z \cdot y\right)} \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

       6. lower-*.f64 48.9

          \[\leadsto \left(x \cdot \cos \left(\left(\color{blue}{\left(z \cdot y\right)} \cdot t\right) \cdot 0.125\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \]

   11. Applied rewrites 48.9%

       \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(\left(z \cdot y\right) \cdot t\right) \cdot 0.125\right)}\right) \cdot \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \]

   if 1e303 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 0.1%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{\mathsf{neg}\left(16\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{t \cdot \color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(t \cdot \left(a \cdot 2 + 1\right)\right) \cdot b}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(t \cdot \left(a \cdot 2 + 1\right)\right) \cdot \frac{b}{\mathsf{neg}\left(16\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot t\right)} \cdot \frac{b}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\left(a \cdot 2 + 1\right) \cdot t, \frac{b}{\mathsf{neg}\left(16\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   4. Applied rewrites 0.8%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t, \frac{b}{-16}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   6. Applied rewrites 0.1%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\left(-\cos \left(\mathsf{fma}\left(\frac{b}{-16} \cdot t, \mathsf{fma}\left(a, 2, 1\right), \mathsf{PI}\left(\right)\right)\right)\right)} \]

   7. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \cos \mathsf{PI}\left(\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \cos \mathsf{PI}\left(\right)\right)} \]

      2. cos-PI N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{-1}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot x}\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot x \]

      6. lower-*.f64 11.4

         \[\leadsto \color{blue}{1 \cdot x} \]

   9. Applied rewrites 11.4%

      \[\leadsto \color{blue}{1 \cdot x} \]
   10. Step-by-step derivation

   11. Applied rewrites 11.4%

       \[\leadsto x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 31.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 10^{+303}:\\ \;\;\;\;\cos \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot \left(t\_m \cdot -0.0625\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
      1e+303)
   (* (cos (* (* (fma a 2.0 1.0) b) (* t_m -0.0625))) x)
   x))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+303) {
		tmp = cos(((fma(a, 2.0, 1.0) * b) * (t_m * -0.0625))) * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+303)
		tmp = Float64(cos(Float64(Float64(fma(a, 2.0, 1.0) * b) * Float64(t_m * -0.0625))) * x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+303], N[(N[Cos[N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[(t$95$m * -0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1e303

   1. Initial program 49.3%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

      3. lower-*.f64 49.2

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\color{blue}{\left(b \cdot t\right)} \cdot 0.0625\right) \]

   5. Applied rewrites 49.2%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot 0.0625\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x} \]

      3. cos-neg-rev N/A

         \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right)} \cdot x \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \cdot x \]

      5. metadata-eval N/A

         \[\leadsto \cos \left(\color{blue}{\frac{-1}{16}} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x \]

      6. lower-cos.f64 N/A

         \[\leadsto \color{blue}{\cos \left(\frac{-1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \cdot x \]

      7. lower-*.f64 N/A

         \[\leadsto \cos \color{blue}{\left(\frac{-1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \cdot x \]

      8. *-commutative N/A

         \[\leadsto \cos \left(\frac{-1}{16} \cdot \color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)}\right) \cdot x \]

      9. lower-*.f64 N/A

         \[\leadsto \cos \left(\frac{-1}{16} \cdot \color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)}\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right)\right) \cdot x \]

      11. lower-*.f64 N/A

          \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right)\right) \cdot x \]

      12. +-commutative N/A

          \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(\left(\color{blue}{\left(2 \cdot a + 1\right)} \cdot t\right) \cdot b\right)\right) \cdot x \]

      13. lower-fma.f64 46.9

          \[\leadsto \cos \left(-0.0625 \cdot \left(\left(\color{blue}{\mathsf{fma}\left(2, a, 1\right)} \cdot t\right) \cdot b\right)\right) \cdot x \]

   8. Applied rewrites 46.9%

      \[\leadsto \color{blue}{\cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(2, a, 1\right) \cdot t\right) \cdot b\right)\right) \cdot x} \]
   9. Step-by-step derivation

   10. Applied rewrites 47.5%

       \[\leadsto \cos \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot \left(t \cdot -0.0625\right)\right) \cdot x \]

   if 1e303 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 0.1%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{\mathsf{neg}\left(16\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{t \cdot \color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(t \cdot \left(a \cdot 2 + 1\right)\right) \cdot b}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(t \cdot \left(a \cdot 2 + 1\right)\right) \cdot \frac{b}{\mathsf{neg}\left(16\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot t\right)} \cdot \frac{b}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\left(a \cdot 2 + 1\right) \cdot t, \frac{b}{\mathsf{neg}\left(16\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   4. Applied rewrites 0.8%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t, \frac{b}{-16}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   6. Applied rewrites 0.1%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\left(-\cos \left(\mathsf{fma}\left(\frac{b}{-16} \cdot t, \mathsf{fma}\left(a, 2, 1\right), \mathsf{PI}\left(\right)\right)\right)\right)} \]

   7. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \cos \mathsf{PI}\left(\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \cos \mathsf{PI}\left(\right)\right)} \]

      2. cos-PI N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{-1}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot x}\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot x \]

      6. lower-*.f64 11.4

         \[\leadsto \color{blue}{1 \cdot x} \]

   9. Applied rewrites 11.4%

      \[\leadsto \color{blue}{1 \cdot x} \]
   10. Step-by-step derivation

   11. Applied rewrites 11.4%

       \[\leadsto x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 29.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \sin \left(\mathsf{fma}\left(b \cdot t\_m, -0.0625, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot x \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (* (sin (fma (* b t_m) -0.0625 (/ (PI) 2.0))) x))

Derivation

1. Initial program 29.3%

   \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

   3. lower-*.f64 30.1

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\color{blue}{\left(b \cdot t\right)} \cdot 0.0625\right) \]

5. Applied rewrites 30.1%

   \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot 0.0625\right)} \]

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x} \]

   3. cos-neg-rev N/A

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right)} \cdot x \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \cdot x \]

   5. metadata-eval N/A

      \[\leadsto \cos \left(\color{blue}{\frac{-1}{16}} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x \]

   6. lower-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \left(\frac{-1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \cdot x \]

   7. lower-*.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\frac{-1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \cdot x \]

   8. *-commutative N/A

      \[\leadsto \cos \left(\frac{-1}{16} \cdot \color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)}\right) \cdot x \]

   9. lower-*.f64 N/A

      \[\leadsto \cos \left(\frac{-1}{16} \cdot \color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)}\right) \cdot x \]

   10. *-commutative N/A

       \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right)\right) \cdot x \]

   11. lower-*.f64 N/A

       \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right)\right) \cdot x \]

   12. +-commutative N/A

       \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(\left(\color{blue}{\left(2 \cdot a + 1\right)} \cdot t\right) \cdot b\right)\right) \cdot x \]

   13. lower-fma.f64 30.3

       \[\leadsto \cos \left(-0.0625 \cdot \left(\left(\color{blue}{\mathsf{fma}\left(2, a, 1\right)} \cdot t\right) \cdot b\right)\right) \cdot x \]

8. Applied rewrites 30.3%

   \[\leadsto \color{blue}{\cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(2, a, 1\right) \cdot t\right) \cdot b\right)\right) \cdot x} \]

9. Taylor expanded in a around 0

   \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]
10. Step-by-step derivation

11. Applied rewrites 30.9%

    \[\leadsto \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right) \cdot x \]
12. Step-by-step derivation

13. Applied rewrites 31.7%

    \[\leadsto \sin \left(\mathsf{fma}\left(b \cdot t, -0.0625, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot x \]
14. Add Preprocessing

Alternative 9: 29.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \sin \left(\mathsf{fma}\left(0.0625, b \cdot t\_m, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot x \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (* (sin (fma 0.0625 (* b t_m) (/ (PI) 2.0))) x))

Derivation

1. Initial program 29.3%

   \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

   3. lower-*.f64 30.1

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\color{blue}{\left(b \cdot t\right)} \cdot 0.0625\right) \]

5. Applied rewrites 30.1%

   \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot 0.0625\right)} \]

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x} \]

   3. cos-neg-rev N/A

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right)} \cdot x \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \cdot x \]

   5. metadata-eval N/A

      \[\leadsto \cos \left(\color{blue}{\frac{-1}{16}} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x \]

   6. lower-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \left(\frac{-1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \cdot x \]

   7. lower-*.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\frac{-1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \cdot x \]

   8. *-commutative N/A

      \[\leadsto \cos \left(\frac{-1}{16} \cdot \color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)}\right) \cdot x \]

   9. lower-*.f64 N/A

      \[\leadsto \cos \left(\frac{-1}{16} \cdot \color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)}\right) \cdot x \]

   10. *-commutative N/A

       \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right)\right) \cdot x \]

   11. lower-*.f64 N/A

       \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right)\right) \cdot x \]

   12. +-commutative N/A

       \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(\left(\color{blue}{\left(2 \cdot a + 1\right)} \cdot t\right) \cdot b\right)\right) \cdot x \]

   13. lower-fma.f64 30.3

       \[\leadsto \cos \left(-0.0625 \cdot \left(\left(\color{blue}{\mathsf{fma}\left(2, a, 1\right)} \cdot t\right) \cdot b\right)\right) \cdot x \]

8. Applied rewrites 30.3%

   \[\leadsto \color{blue}{\cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(2, a, 1\right) \cdot t\right) \cdot b\right)\right) \cdot x} \]

9. Taylor expanded in a around 0

   \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]
10. Step-by-step derivation

11. Applied rewrites 30.9%

    \[\leadsto \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right) \cdot x \]
12. Step-by-step derivation

13. Applied rewrites 31.6%

    \[\leadsto \sin \left(\mathsf{fma}\left(0.0625, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot x \]
14. Add Preprocessing

Alternative 10: 30.9% accurate, 269.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ x \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b) :precision binary64 x)

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	return x;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m, double a, double b) {
	return x;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m, a, b):
	return x

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	return x
end

MATLAB

t_m = abs(t);
function tmp = code(x, y, z, t_m, a, b)
	tmp = x;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := x

Derivation

1. Initial program 29.3%

   \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

   2. cos-neg-rev N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]

   3. sin-+PI/2-rev N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lower-sin.f64 N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   5. lift-/.f64 N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{\mathsf{neg}\left(16\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{t \cdot \color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   10. associate-*r* N/A

       \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(t \cdot \left(a \cdot 2 + 1\right)\right) \cdot b}}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   11. associate-/l* N/A

       \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(t \cdot \left(a \cdot 2 + 1\right)\right) \cdot \frac{b}{\mathsf{neg}\left(16\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   12. *-commutative N/A

       \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot t\right)} \cdot \frac{b}{\mathsf{neg}\left(16\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\left(a \cdot 2 + 1\right) \cdot t, \frac{b}{\mathsf{neg}\left(16\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

4. Applied rewrites 29.2%

   \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t, \frac{b}{-16}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

6. Applied rewrites 29.2%

   \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\left(-\cos \left(\mathsf{fma}\left(\frac{b}{-16} \cdot t, \mathsf{fma}\left(a, 2, 1\right), \mathsf{PI}\left(\right)\right)\right)\right)} \]

7. Taylor expanded in t around 0

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \cos \mathsf{PI}\left(\right)\right)} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \cos \mathsf{PI}\left(\right)\right)} \]

   2. cos-PI N/A

      \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{-1}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot x}\right) \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

   5. metadata-eval N/A

      \[\leadsto \color{blue}{1} \cdot x \]

   6. lower-*.f64 31.2

      \[\leadsto \color{blue}{1 \cdot x} \]

9. Applied rewrites 31.2%

   \[\leadsto \color{blue}{1 \cdot x} \]
10. Step-by-step derivation

11. Applied rewrites 31.2%

    \[\leadsto x \]
12. Add Preprocessing

Developer Target 1: 30.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * cos(((b / 16.0) * (t / ((1.0 - (a * 2.0)) + pow((a * 2.0), 2.0)))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * cos(((b / 16.0d0) * (t / ((1.0d0 - (a * 2.0d0)) + ((a * 2.0d0) ** 2.0d0)))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.cos(((b / 16.0) * (t / ((1.0 - (a * 2.0)) + Math.pow((a * 2.0), 2.0)))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.cos(((b / 16.0) * (t / ((1.0 - (a * 2.0)) + math.pow((a * 2.0), 2.0)))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * cos(Float64(Float64(b / 16.0) * Float64(t / Float64(Float64(1.0 - Float64(a * 2.0)) + (Float64(a * 2.0) ^ 2.0))))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * cos(((b / 16.0) * (t / ((1.0 - (a * 2.0)) + ((a * 2.0) ^ 2.0)))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Cos[N[(N[(b / 16.0), $MachinePrecision] * N[(t / N[(N[(1.0 - N[(a * 2.0), $MachinePrecision]), $MachinePrecision] + N[Power[N[(a * 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024320 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (* x (cos (* (/ b 16) (/ t (+ (- 1 (* a 2)) (pow (* a 2) 2)))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))