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

Percentage Accurate: 27.0% → 31.4%

Time: 20.7s

Alternatives: 8

Speedup: 225.0×

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: 27.0% 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: 31.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot t_1 \leq 5 \cdot 10^{+281}:\\ \;\;\;\;t_1 \cdot \left(x \cdot \cos \left(\frac{{\left(\sqrt[3]{\mathsf{fma}\left(2, y, 1\right)} \cdot \sqrt[3]{z \cdot t}\right)}^{3}}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))))
   (if (<= (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) t_1) 5e+281)
     (*
      t_1
      (* x (cos (/ (pow (* (cbrt (fma 2.0 y 1.0)) (cbrt (* z t))) 3.0) 16.0))))
     x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 5e+281) {
		tmp = t_1 * (x * cos((pow((cbrt(fma(2.0, y, 1.0)) * cbrt((z * t))), 3.0) / 16.0)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 5e+281)
		tmp = Float64(t_1 * Float64(x * cos(Float64((Float64(cbrt(fma(2.0, y, 1.0)) * cbrt(Float64(z * t))) ^ 3.0) / 16.0))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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] * t$95$1), $MachinePrecision], 5e+281], N[(t$95$1 * N[(x * N[Cos[N[(N[Power[N[(N[Power[N[(2.0 * y + 1.0), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(z * t), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / 16.0), $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 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 5.00000000000000016e281

   1. Initial program 48.2%

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

      1. add-cube-cbrt 49.3%

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(\sqrt[3]{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t} \cdot \sqrt[3]{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}\right) \cdot \sqrt[3]{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      2. pow3 49.2%

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{{\left(\sqrt[3]{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}\right)}^{3}}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      3. associate-*l* 49.2%

         \[\leadsto \left(x \cdot \cos \left(\frac{{\left(\sqrt[3]{\color{blue}{\left(y \cdot 2 + 1\right) \cdot \left(z \cdot t\right)}}\right)}^{3}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      4. *-commutative 49.2%

         \[\leadsto \left(x \cdot \cos \left(\frac{{\left(\sqrt[3]{\left(\color{blue}{2 \cdot y} + 1\right) \cdot \left(z \cdot t\right)}\right)}^{3}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      5. fma-def 49.2%

         \[\leadsto \left(x \cdot \cos \left(\frac{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(2, y, 1\right)} \cdot \left(z \cdot t\right)}\right)}^{3}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   3. Applied egg-rr 49.2%

      \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(2, y, 1\right) \cdot \left(z \cdot t\right)}\right)}^{3}}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   4. Step-by-step derivation

      1. cbrt-prod 49.4%

         \[\leadsto \left(x \cdot \cos \left(\frac{{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(2, y, 1\right)} \cdot \sqrt[3]{z \cdot t}\right)}}^{3}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   5. Applied egg-rr 49.4%

      \[\leadsto \left(x \cdot \cos \left(\frac{{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(2, y, 1\right)} \cdot \sqrt[3]{z \cdot t}\right)}}^{3}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   if 5.00000000000000016e281 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-*l/ 0.0%

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

      5. fma-def 0.0%

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

      6. associate-*l/ 0.0%

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

      7. *-commutative 0.0%

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

      8. fma-def 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in b around 0 3.2%

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

   5. Taylor expanded in z around 0 11.2%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \cdot \left(x \cdot \cos \left(\frac{{\left(\sqrt[3]{\mathsf{fma}\left(2, y, 1\right)} \cdot \sqrt[3]{z \cdot t}\right)}^{3}}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2: 31.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot t_1 \leq 5 \cdot 10^{+281}:\\ \;\;\;\;t_1 \cdot \left(x \cdot \cos \left(\frac{{\left(\sqrt[3]{\mathsf{fma}\left(2, y, 1\right) \cdot \left(z \cdot t\right)}\right)}^{3}}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))))
   (if (<= (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) t_1) 5e+281)
     (* t_1 (* x (cos (/ (pow (cbrt (* (fma 2.0 y 1.0) (* z t))) 3.0) 16.0))))
     x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 5e+281) {
		tmp = t_1 * (x * cos((pow(cbrt((fma(2.0, y, 1.0) * (z * t))), 3.0) / 16.0)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 5e+281)
		tmp = Float64(t_1 * Float64(x * cos(Float64((cbrt(Float64(fma(2.0, y, 1.0) * Float64(z * t))) ^ 3.0) / 16.0))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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] * t$95$1), $MachinePrecision], 5e+281], N[(t$95$1 * N[(x * N[Cos[N[(N[Power[N[Power[N[(N[(2.0 * y + 1.0), $MachinePrecision] * N[(z * t), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] / 16.0), $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 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 5.00000000000000016e281

   1. Initial program 48.2%

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

      1. add-cube-cbrt 49.3%

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(\sqrt[3]{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t} \cdot \sqrt[3]{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}\right) \cdot \sqrt[3]{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      2. pow3 49.2%

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{{\left(\sqrt[3]{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}\right)}^{3}}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      3. associate-*l* 49.2%

         \[\leadsto \left(x \cdot \cos \left(\frac{{\left(\sqrt[3]{\color{blue}{\left(y \cdot 2 + 1\right) \cdot \left(z \cdot t\right)}}\right)}^{3}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      4. *-commutative 49.2%

         \[\leadsto \left(x \cdot \cos \left(\frac{{\left(\sqrt[3]{\left(\color{blue}{2 \cdot y} + 1\right) \cdot \left(z \cdot t\right)}\right)}^{3}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      5. fma-def 49.2%

         \[\leadsto \left(x \cdot \cos \left(\frac{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(2, y, 1\right)} \cdot \left(z \cdot t\right)}\right)}^{3}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   3. Applied egg-rr 49.2%

      \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(2, y, 1\right) \cdot \left(z \cdot t\right)}\right)}^{3}}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   if 5.00000000000000016e281 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-*l/ 0.0%

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

      5. fma-def 0.0%

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

      6. associate-*l/ 0.0%

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

      7. *-commutative 0.0%

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

      8. fma-def 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in b around 0 3.2%

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

   5. Taylor expanded in z around 0 11.2%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \cdot \left(x \cdot \cos \left(\frac{{\left(\sqrt[3]{\mathsf{fma}\left(2, y, 1\right) \cdot \left(z \cdot t\right)}\right)}^{3}}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 31.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+281}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{1}{\frac{\frac{\frac{16}{t}}{z}}{\mathsf{fma}\left(2, y, 1\right)}}\right) \cdot \cos \left(t \cdot \frac{b \cdot \mathsf{fma}\left(2, a, 1\right)}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
      5e+281)
   (*
    x
    (*
     (cos (/ 1.0 (/ (/ (/ 16.0 t) z) (fma 2.0 y 1.0))))
     (cos (* t (/ (* b (fma 2.0 a 1.0)) 16.0)))))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 5e+281) {
		tmp = x * (cos((1.0 / (((16.0 / t) / z) / fma(2.0, y, 1.0)))) * cos((t * ((b * fma(2.0, a, 1.0)) / 16.0))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 5e+281)
		tmp = Float64(x * Float64(cos(Float64(1.0 / Float64(Float64(Float64(16.0 / t) / z) / fma(2.0, y, 1.0)))) * cos(Float64(t * Float64(Float64(b * fma(2.0, a, 1.0)) / 16.0)))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+281], N[(x * N[(N[Cos[N[(1.0 / N[(N[(N[(16.0 / t), $MachinePrecision] / z), $MachinePrecision] / N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(t * N[(N[(b * N[(2.0 * a + 1.0), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]), $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 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 5.00000000000000016e281

   1. Initial program 48.2%

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

      1. associate-*l* 48.2%

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

      2. *-commutative 48.2%

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

      3. *-commutative 48.2%

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

      4. associate-*l/ 48.2%

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

      5. fma-def 48.2%

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

      6. associate-*l/ 48.2%

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

      7. *-commutative 48.2%

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

      8. fma-def 48.2%

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

   3. Simplified 48.2%

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

      1. associate-/r/ 48.2%

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

      2. fma-def 48.2%

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

      3. associate-/l* 48.7%

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

      4. clear-num 49.2%

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

      5. *-commutative 49.2%

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

      6. fma-def 49.2%

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

   5. Applied egg-rr 49.2%

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

   if 5.00000000000000016e281 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-*l/ 0.0%

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

      5. fma-def 0.0%

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

      6. associate-*l/ 0.0%

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

      7. *-commutative 0.0%

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

      8. fma-def 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in b around 0 3.2%

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

   5. Taylor expanded in z around 0 11.2%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+281}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{1}{\frac{\frac{\frac{16}{t}}{z}}{\mathsf{fma}\left(2, y, 1\right)}}\right) \cdot \cos \left(t \cdot \frac{b \cdot \mathsf{fma}\left(2, a, 1\right)}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 31.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+281}:\\ \;\;\;\;x \cdot \left(\cos \left(t \cdot \frac{b \cdot \mathsf{fma}\left(2, a, 1\right)}{16}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, y, 1\right)}{\frac{\frac{16}{t}}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
      5e+281)
   (*
    x
    (*
     (cos (* t (/ (* b (fma 2.0 a 1.0)) 16.0)))
     (cos (/ (fma 2.0 y 1.0) (/ (/ 16.0 t) z)))))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 5e+281) {
		tmp = x * (cos((t * ((b * fma(2.0, a, 1.0)) / 16.0))) * cos((fma(2.0, y, 1.0) / ((16.0 / t) / z))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 5e+281)
		tmp = Float64(x * Float64(cos(Float64(t * Float64(Float64(b * fma(2.0, a, 1.0)) / 16.0))) * cos(Float64(fma(2.0, y, 1.0) / Float64(Float64(16.0 / t) / z)))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+281], N[(x * N[(N[Cos[N[(t * N[(N[(b * N[(2.0 * a + 1.0), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(2.0 * y + 1.0), $MachinePrecision] / N[(N[(16.0 / t), $MachinePrecision] / z), $MachinePrecision]), $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 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 5.00000000000000016e281

   1. Initial program 48.2%

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

      1. associate-*l* 48.2%

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

      2. *-commutative 48.2%

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

      3. *-commutative 48.2%

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

      4. associate-*l/ 48.2%

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

      5. fma-def 48.2%

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

      6. associate-*l/ 48.2%

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

      7. *-commutative 48.2%

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

      8. fma-def 48.2%

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

   3. Simplified 48.2%

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

      1. associate-/r/ 48.2%

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

      2. fma-def 48.2%

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

      3. associate-/l* 48.7%

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

      4. *-commutative 48.7%

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

      5. fma-def 48.7%

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

   5. Applied egg-rr 48.7%

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

   if 5.00000000000000016e281 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-*l/ 0.0%

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

      5. fma-def 0.0%

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

      6. associate-*l/ 0.0%

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

      7. *-commutative 0.0%

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

      8. fma-def 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in b around 0 3.2%

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

   5. Taylor expanded in z around 0 11.2%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+281}:\\ \;\;\;\;x \cdot \left(\cos \left(t \cdot \frac{b \cdot \mathsf{fma}\left(2, a, 1\right)}{16}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, y, 1\right)}{\frac{\frac{16}{t}}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 31.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot t_1 \leq 5 \cdot 10^{+281}:\\ \;\;\;\;t_1 \cdot \left(x \cdot \cos \left(\frac{z \cdot t}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))))
   (if (<= (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) t_1) 5e+281)
     (* t_1 (* x (cos (/ (* z t) 16.0))))
     x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 5e+281) {
		tmp = t_1 * (x * cos(((z * t) / 16.0)));
	} else {
		tmp = x;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(((t * ((1.0d0 + (2.0d0 * a)) * b)) / 16.0d0))
    if (((x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * t_1) <= 5d+281) then
        tmp = t_1 * (x * cos(((z * t) / 16.0d0)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (((x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 5e+281) {
		tmp = t_1 * (x * Math.cos(((z * t) / 16.0)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))
	tmp = 0
	if ((x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 5e+281:
		tmp = t_1 * (x * math.cos(((z * t) / 16.0)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 5e+281)
		tmp = Float64(t_1 * Float64(x * cos(Float64(Float64(z * t) / 16.0))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	tmp = 0.0;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 5e+281)
		tmp = t_1 * (x * cos(((z * t) / 16.0)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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] * t$95$1), $MachinePrecision], 5e+281], N[(t$95$1 * N[(x * N[Cos[N[(N[(z * t), $MachinePrecision] / 16.0), $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 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 5.00000000000000016e281

   1. Initial program 48.2%

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

   2. Taylor expanded in y around 0 48.5%

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

   if 5.00000000000000016e281 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-*l/ 0.0%

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

      5. fma-def 0.0%

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

      6. associate-*l/ 0.0%

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

      7. *-commutative 0.0%

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

      8. fma-def 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in b around 0 3.2%

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

   5. Taylor expanded in z around 0 11.2%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \cdot \left(x \cdot \cos \left(\frac{z \cdot t}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 29.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{+101}:\\ \;\;\;\;x \cdot \left(\cos \left(t \cdot \frac{z}{16}\right) \cdot \cos \left(\frac{1}{\frac{\frac{16}{b}}{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 3.7e+101)
   (* x (* (cos (* t (/ z 16.0))) (cos (/ 1.0 (/ (/ 16.0 b) t)))))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 3.7e+101) {
		tmp = x * (cos((t * (z / 16.0))) * cos((1.0 / ((16.0 / b) / t))));
	} else {
		tmp = x;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (t <= 3.7d+101) then
        tmp = x * (cos((t * (z / 16.0d0))) * cos((1.0d0 / ((16.0d0 / b) / t))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 3.7e+101) {
		tmp = x * (Math.cos((t * (z / 16.0))) * Math.cos((1.0 / ((16.0 / b) / t))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 3.7e+101:
		tmp = x * (math.cos((t * (z / 16.0))) * math.cos((1.0 / ((16.0 / b) / t))))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 3.7e+101)
		tmp = Float64(x * Float64(cos(Float64(t * Float64(z / 16.0))) * cos(Float64(1.0 / Float64(Float64(16.0 / b) / t)))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 3.7e+101)
		tmp = x * (cos((t * (z / 16.0))) * cos((1.0 / ((16.0 / b) / t))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 3.7e+101], N[(x * N[(N[Cos[N[(t * N[(z / 16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(1.0 / N[(N[(16.0 / b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < 3.6999999999999997e101

   1. Initial program 32.1%

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

      1. associate-*l* 32.1%

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

      2. *-commutative 32.1%

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

      3. *-commutative 32.1%

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

      4. associate-*l/ 32.1%

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

      5. fma-def 32.1%

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

      6. associate-*l/ 32.1%

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

      7. *-commutative 32.1%

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

      8. fma-def 32.1%

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

   3. Simplified 32.1%

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

      1. associate-*l/ 32.1%

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

      2. fma-udef 32.1%

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

      3. *-commutative 32.1%

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

      4. clear-num 32.2%

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

      5. associate-*l* 32.7%

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

      6. *-commutative 32.7%

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

      7. fma-udef 32.7%

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

   5. Applied egg-rr 32.7%

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

   6. Taylor expanded in y around 0 34.0%

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

   7. Taylor expanded in a around 0 34.4%

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

      1. *-commutative 34.4%

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

   9. Simplified 34.4%

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

   10. Taylor expanded in t around 0 34.4%

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

       1. associate-/r* 34.5%

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

   12. Simplified 34.5%

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

   if 3.6999999999999997e101 < t

   1. Initial program 6.1%

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

      1. associate-*l* 6.1%

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

      2. *-commutative 6.1%

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

      3. *-commutative 6.1%

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

      4. associate-*l/ 6.1%

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

      5. fma-def 6.1%

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

      6. associate-*l/ 6.1%

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

      7. *-commutative 6.1%

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

      8. fma-def 6.1%

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

   3. Simplified 6.1%

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

   4. Taylor expanded in b around 0 9.0%

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

   5. Taylor expanded in z around 0 13.8%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{+101}:\\ \;\;\;\;x \cdot \left(\cos \left(t \cdot \frac{z}{16}\right) \cdot \cos \left(\frac{1}{\frac{\frac{16}{b}}{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 29.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \left(\cos \left(t \cdot \frac{z}{16}\right) \cdot \cos \left(t \cdot \left(b \cdot 0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 5.5e+98)
   (* x (* (cos (* t (/ z 16.0))) (cos (* t (* b 0.0625)))))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 5.5e+98) {
		tmp = x * (cos((t * (z / 16.0))) * cos((t * (b * 0.0625))));
	} else {
		tmp = x;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (t <= 5.5d+98) then
        tmp = x * (cos((t * (z / 16.0d0))) * cos((t * (b * 0.0625d0))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 5.5e+98) {
		tmp = x * (Math.cos((t * (z / 16.0))) * Math.cos((t * (b * 0.0625))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 5.5e+98:
		tmp = x * (math.cos((t * (z / 16.0))) * math.cos((t * (b * 0.0625))))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 5.5e+98)
		tmp = Float64(x * Float64(cos(Float64(t * Float64(z / 16.0))) * cos(Float64(t * Float64(b * 0.0625)))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 5.5e+98)
		tmp = x * (cos((t * (z / 16.0))) * cos((t * (b * 0.0625))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 5.5e+98], N[(x * N[(N[Cos[N[(t * N[(z / 16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(t * N[(b * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < 5.49999999999999946e98

   1. Initial program 32.1%

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

      1. associate-*l* 32.1%

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

      2. *-commutative 32.1%

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

      3. *-commutative 32.1%

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

      4. associate-*l/ 32.1%

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

      5. fma-def 32.1%

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

      6. associate-*l/ 32.1%

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

      7. *-commutative 32.1%

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

      8. fma-def 32.1%

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

   3. Simplified 32.1%

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

      1. associate-*l/ 32.1%

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

      2. fma-udef 32.1%

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

      3. *-commutative 32.1%

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

      4. clear-num 32.2%

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

      5. associate-*l* 32.7%

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

      6. *-commutative 32.7%

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

      7. fma-udef 32.7%

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

   5. Applied egg-rr 32.7%

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

   6. Taylor expanded in y around 0 34.0%

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

   7. Taylor expanded in a around 0 34.3%

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

      1. associate-*r* 34.3%

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

   9. Simplified 34.3%

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

   if 5.49999999999999946e98 < t

   1. Initial program 6.1%

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

      1. associate-*l* 6.1%

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

      2. *-commutative 6.1%

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

      3. *-commutative 6.1%

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

      4. associate-*l/ 6.1%

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

      5. fma-def 6.1%

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

      6. associate-*l/ 6.1%

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

      7. *-commutative 6.1%

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

      8. fma-def 6.1%

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

   3. Simplified 6.1%

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

   4. Taylor expanded in b around 0 9.0%

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

   5. Taylor expanded in z around 0 13.8%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \left(\cos \left(t \cdot \frac{z}{16}\right) \cdot \cos \left(t \cdot \left(b \cdot 0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 30.4% accurate, 225.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 x)

C

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

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
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Python

def code(x, y, z, t, a, b):
	return x

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 28.4%

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

   1. associate-*l* 28.4%

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

   2. *-commutative 28.4%

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

   3. *-commutative 28.4%

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

   4. associate-*l/ 28.4%

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

   5. fma-def 28.4%

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

   6. associate-*l/ 28.4%

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

   7. *-commutative 28.4%

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

   8. fma-def 28.4%

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

3. Simplified 28.4%

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

4. Taylor expanded in b around 0 28.5%

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

5. Taylor expanded in z around 0 31.0%

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

6. Final simplification 31.0%

   \[\leadsto x \]

Developer target: 29.8% accurate, 1.0× 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 2023290 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2.0))))))

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