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

Percentage Accurate: 27.3% → 31.2%

Time: 26.7s

Alternatives: 5

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: 27.3% 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: 31.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{-105}:\\ \;\;\;\;x\_m \cdot \left(\cos \left({\left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)}\right)}^{3}\right) \cdot \cos \left(\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot \frac{t}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
        (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
       2e-105)
    (*
     x_m
     (*
      (cos (pow (cbrt (* (fma y 2.0 1.0) (* z (* t 0.0625)))) 3.0))
      (cos (* (* b (fma a 2.0 1.0)) (/ t 16.0)))))
    x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 2e-105) {
		tmp = x_m * (cos(pow(cbrt((fma(y, 2.0, 1.0) * (z * (t * 0.0625)))), 3.0)) * cos(((b * fma(a, 2.0, 1.0)) * (t / 16.0))));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x_m * 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))) <= 2e-105)
		tmp = Float64(x_m * Float64(cos((cbrt(Float64(fma(y, 2.0, 1.0) * Float64(z * Float64(t * 0.0625)))) ^ 3.0)) * cos(Float64(Float64(b * fma(a, 2.0, 1.0)) * Float64(t / 16.0)))));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * 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], 2e-105], N[(x$95$m * N[(N[Cos[N[Power[N[Power[N[(N[(y * 2.0 + 1.0), $MachinePrecision] * N[(z * N[(t * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(b * N[(a * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(t / 16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

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)))) < 1.99999999999999993e-105

   1. Initial program 48.5%

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

      1. associate-*l* 48.5%

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

      2. *-commutative 48.5%

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

      3. *-commutative 48.5%

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

      4. associate-/l* 48.5%

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

      5. fma-define 48.5%

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

      6. associate-/l* 48.5%

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

      7. fma-define 48.5%

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

   3. Simplified 48.5%

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

      1. associate-*r* 48.8%

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

      2. add-cube-cbrt 49.4%

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

      3. pow3 49.7%

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

      4. div-inv 49.7%

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

      5. metadata-eval 49.7%

         \[\leadsto x \cdot \left(\cos \left({\left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot \color{blue}{0.0625}\right)\right)}\right)}^{3}\right) \cdot \cos \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot \frac{t}{16}\right)\right) \]

   6. Applied egg-rr 49.7%

      \[\leadsto x \cdot \left(\cos \color{blue}{\left({\left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)}\right)}^{3}\right)} \cdot \cos \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot \frac{t}{16}\right)\right) \]

   if 1.99999999999999993e-105 < (*.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 18.0%

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

   3. Simplified 19.7%

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

   5. Taylor expanded in a around 0 20.7%

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

      1. *-commutative 20.7%

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

      2. *-commutative 20.7%

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

   7. Simplified 20.7%

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

   8. Taylor expanded in t around 0 24.1%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{-105}:\\ \;\;\;\;x \cdot \left(\cos \left({\left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)}\right)}^{3}\right) \cdot \cos \left(\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot \frac{t}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 31.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{-105}:\\ \;\;\;\;\cos \left({\left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)}\right)}^{3}\right) \cdot \left(x\_m \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(a \cdot -2 + -1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
        (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
       2e-105)
    (*
     (cos (pow (cbrt (* (fma y 2.0 1.0) (* z (* t 0.0625)))) 3.0))
     (* x_m (cos (* 0.0625 (* b (* t (+ (* a -2.0) -1.0)))))))
    x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 2e-105) {
		tmp = cos(pow(cbrt((fma(y, 2.0, 1.0) * (z * (t * 0.0625)))), 3.0)) * (x_m * cos((0.0625 * (b * (t * ((a * -2.0) + -1.0))))));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x_m * 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))) <= 2e-105)
		tmp = Float64(cos((cbrt(Float64(fma(y, 2.0, 1.0) * Float64(z * Float64(t * 0.0625)))) ^ 3.0)) * Float64(x_m * cos(Float64(0.0625 * Float64(b * Float64(t * Float64(Float64(a * -2.0) + -1.0)))))));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * 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], 2e-105], N[(N[Cos[N[Power[N[Power[N[(N[(y * 2.0 + 1.0), $MachinePrecision] * N[(z * N[(t * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] * N[(x$95$m * N[Cos[N[(0.0625 * N[(b * N[(t * N[(N[(a * -2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

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)))) < 1.99999999999999993e-105

   1. Initial program 48.5%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in b around inf 48.6%

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

      1. fma-define 48.6%

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

      2. metadata-eval 48.6%

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

      3. div-inv 48.6%

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

      4. associate-*r* 48.3%

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

      5. associate-/l* 48.3%

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

      6. add-cube-cbrt 49.1%

         \[\leadsto \cos \color{blue}{\left(\left(\sqrt[3]{\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}} \cdot \sqrt[3]{\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}}\right) \cdot \sqrt[3]{\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}}\right)} \cdot \left(x \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(-2 \cdot a - 1\right)\right)\right)\right)\right) \]

      7. pow3 49.6%

         \[\leadsto \cos \color{blue}{\left({\left(\sqrt[3]{\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}}\right)}^{3}\right)} \cdot \left(x \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(-2 \cdot a - 1\right)\right)\right)\right)\right) \]

      8. associate-/l* 49.6%

         \[\leadsto \cos \left({\left(\sqrt[3]{\color{blue}{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot \frac{t}{16}}}\right)}^{3}\right) \cdot \left(x \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(-2 \cdot a - 1\right)\right)\right)\right)\right) \]

      9. associate-*r* 49.7%

         \[\leadsto \cos \left({\left(\sqrt[3]{\color{blue}{\left(y \cdot 2 + 1\right) \cdot \left(z \cdot \frac{t}{16}\right)}}\right)}^{3}\right) \cdot \left(x \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(-2 \cdot a - 1\right)\right)\right)\right)\right) \]

      10. div-inv 49.7%

          \[\leadsto \cos \left({\left(\sqrt[3]{\left(y \cdot 2 + 1\right) \cdot \left(z \cdot \color{blue}{\left(t \cdot \frac{1}{16}\right)}\right)}\right)}^{3}\right) \cdot \left(x \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(-2 \cdot a - 1\right)\right)\right)\right)\right) \]

      11. metadata-eval 49.7%

          \[\leadsto \cos \left({\left(\sqrt[3]{\left(y \cdot 2 + 1\right) \cdot \left(z \cdot \left(t \cdot \color{blue}{0.0625}\right)\right)}\right)}^{3}\right) \cdot \left(x \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(-2 \cdot a - 1\right)\right)\right)\right)\right) \]

      12. *-commutative 49.7%

          \[\leadsto \cos \left({\left(\sqrt[3]{\color{blue}{\left(z \cdot \left(t \cdot 0.0625\right)\right) \cdot \left(y \cdot 2 + 1\right)}}\right)}^{3}\right) \cdot \left(x \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(-2 \cdot a - 1\right)\right)\right)\right)\right) \]

      13. fma-define 49.7%

          \[\leadsto \cos \left({\left(\sqrt[3]{\left(z \cdot \left(t \cdot 0.0625\right)\right) \cdot \color{blue}{\mathsf{fma}\left(y, 2, 1\right)}}\right)}^{3}\right) \cdot \left(x \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(-2 \cdot a - 1\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 49.7%

      \[\leadsto \cos \color{blue}{\left({\left(\sqrt[3]{\left(z \cdot \left(t \cdot 0.0625\right)\right) \cdot \mathsf{fma}\left(y, 2, 1\right)}\right)}^{3}\right)} \cdot \left(x \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(-2 \cdot a - 1\right)\right)\right)\right)\right) \]

   if 1.99999999999999993e-105 < (*.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 18.0%

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

   3. Simplified 19.7%

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

   5. Taylor expanded in a around 0 20.7%

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

      1. *-commutative 20.7%

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

      2. *-commutative 20.7%

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

   7. Simplified 20.7%

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

   8. Taylor expanded in t around 0 24.1%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{-105}:\\ \;\;\;\;\cos \left({\left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)}\right)}^{3}\right) \cdot \left(x \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(a \cdot -2 + -1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 29.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{+34}:\\ \;\;\;\;x\_m \cdot \left(\cos \left(z \cdot \left(t \cdot 0.0625\right)\right) \cdot \log \left(e^{\cos \left(t \cdot \left(b \cdot -0.0625\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<= t 2.05e+34)
    (* x_m (* (cos (* z (* t 0.0625))) (log (exp (cos (* t (* b -0.0625)))))))
    x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 2.05e+34) {
		tmp = x_m * (cos((z * (t * 0.0625))) * log(exp(cos((t * (b * -0.0625))))));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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 <= 2.05d+34) then
        tmp = x_m * (cos((z * (t * 0.0625d0))) * log(exp(cos((t * (b * (-0.0625d0)))))))
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 2.05e+34) {
		tmp = x_m * (Math.cos((z * (t * 0.0625))) * Math.log(Math.exp(Math.cos((t * (b * -0.0625))))));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if t <= 2.05e+34:
		tmp = x_m * (math.cos((z * (t * 0.0625))) * math.log(math.exp(math.cos((t * (b * -0.0625))))))
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (t <= 2.05e+34)
		tmp = Float64(x_m * Float64(cos(Float64(z * Float64(t * 0.0625))) * log(exp(cos(Float64(t * Float64(b * -0.0625)))))));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 2.05e+34)
		tmp = x_m * (cos((z * (t * 0.0625))) * log(exp(cos((t * (b * -0.0625))))));
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[t, 2.05e+34], N[(x$95$m * N[(N[Cos[N[(z * N[(t * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Log[N[Exp[N[Cos[N[(t * N[(b * -0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.0499999999999999e34

   1. Initial program 35.7%

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

   3. Simplified 37.2%

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

   5. Taylor expanded in a around 0 37.3%

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

      1. *-commutative 37.3%

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

      2. *-commutative 37.3%

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

   7. Simplified 37.3%

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

      1. add-cbrt-cube 16.7%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right) \cdot \left(x \cdot \cos \left(\left(t \cdot b\right) \cdot -0.0625\right)\right)\right) \cdot \left(\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right) \cdot \left(x \cdot \cos \left(\left(t \cdot b\right) \cdot -0.0625\right)\right)\right)\right) \cdot \left(\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right) \cdot \left(x \cdot \cos \left(\left(t \cdot b\right) \cdot -0.0625\right)\right)\right)}} \]

      2. pow1/3 9.0%

         \[\leadsto \color{blue}{{\left(\left(\left(\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right) \cdot \left(x \cdot \cos \left(\left(t \cdot b\right) \cdot -0.0625\right)\right)\right) \cdot \left(\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right) \cdot \left(x \cdot \cos \left(\left(t \cdot b\right) \cdot -0.0625\right)\right)\right)\right) \cdot \left(\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right) \cdot \left(x \cdot \cos \left(\left(t \cdot b\right) \cdot -0.0625\right)\right)\right)\right)}^{0.3333333333333333}} \]

   9. Applied egg-rr 9.0%

      \[\leadsto \color{blue}{{\left({\left(\cos \left(\left(z \cdot \left(t \cdot 0.0625\right)\right) \cdot \mathsf{fma}\left(y, 2, 1\right)\right) \cdot \left(\cos \left(t \cdot \left(b \cdot -0.0625\right)\right) \cdot x\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

   10. Taylor expanded in y around 0 38.3%

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

       1. *-commutative 38.3%

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

       2. *-commutative 38.3%

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

       3. *-commutative 38.3%

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

       4. associate-*r* 38.3%

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

       5. *-commutative 38.3%

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

       6. *-commutative 38.3%

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

       7. associate-*r* 38.3%

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

   12. Simplified 38.3%

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

       1. add-log-exp 38.3%

          \[\leadsto x \cdot \left(\cos \left(z \cdot \left(t \cdot 0.0625\right)\right) \cdot \color{blue}{\log \left(e^{\cos \left(t \cdot \left(b \cdot -0.0625\right)\right)}\right)}\right) \]

   14. Applied egg-rr 38.3%

       \[\leadsto x \cdot \left(\cos \left(z \cdot \left(t \cdot 0.0625\right)\right) \cdot \color{blue}{\log \left(e^{\cos \left(t \cdot \left(b \cdot -0.0625\right)\right)}\right)}\right) \]

   if 2.0499999999999999e34 < t

   1. Initial program 7.0%

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

   3. Simplified 7.2%

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

   5. Taylor expanded in a around 0 8.3%

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

      1. *-commutative 8.3%

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

      2. *-commutative 8.3%

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

   7. Simplified 8.3%

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

   8. Taylor expanded in t around 0 13.3%

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

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{+34}:\\ \;\;\;\;x \cdot \left(\cos \left(z \cdot \left(t \cdot 0.0625\right)\right) \cdot \log \left(e^{\cos \left(t \cdot \left(b \cdot -0.0625\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 29.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+34}:\\ \;\;\;\;x\_m \cdot \left(\cos \left(t \cdot \left(b \cdot -0.0625\right)\right) \cdot \cos \left(0.0625 \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<= t 1.25e+34)
    (* x_m (* (cos (* t (* b -0.0625))) (cos (* 0.0625 (* z t)))))
    x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 1.25e+34) {
		tmp = x_m * (cos((t * (b * -0.0625))) * cos((0.0625 * (z * t))));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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 <= 1.25d+34) then
        tmp = x_m * (cos((t * (b * (-0.0625d0)))) * cos((0.0625d0 * (z * t))))
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 1.25e+34) {
		tmp = x_m * (Math.cos((t * (b * -0.0625))) * Math.cos((0.0625 * (z * t))));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if t <= 1.25e+34:
		tmp = x_m * (math.cos((t * (b * -0.0625))) * math.cos((0.0625 * (z * t))))
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (t <= 1.25e+34)
		tmp = Float64(x_m * Float64(cos(Float64(t * Float64(b * -0.0625))) * cos(Float64(0.0625 * Float64(z * t)))));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 1.25e+34)
		tmp = x_m * (cos((t * (b * -0.0625))) * cos((0.0625 * (z * t))));
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[t, 1.25e+34], N[(x$95$m * N[(N[Cos[N[(t * N[(b * -0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.25e34

   1. Initial program 35.7%

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

   3. Simplified 37.2%

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

   5. Taylor expanded in a around 0 37.3%

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

      1. *-commutative 37.3%

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

      2. *-commutative 37.3%

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

   7. Simplified 37.3%

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

   8. Taylor expanded in y around 0 38.3%

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

      1. *-commutative 38.3%

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

      2. associate-*r* 38.3%

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

      3. *-commutative 38.3%

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

      4. *-commutative 38.3%

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

   10. Simplified 38.3%

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

   if 1.25e34 < t

   1. Initial program 7.0%

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

   3. Simplified 7.2%

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

   5. Taylor expanded in a around 0 8.3%

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

      1. *-commutative 8.3%

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

      2. *-commutative 8.3%

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

   7. Simplified 8.3%

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

   8. Taylor expanded in t around 0 13.3%

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

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+34}:\\ \;\;\;\;x \cdot \left(\cos \left(t \cdot \left(b \cdot -0.0625\right)\right) \cdot \cos \left(0.0625 \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 30.5% accurate, 225.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b) :precision binary64 (* x_s x_m))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * x_m;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_s * x_m
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * x_m;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	return x_s * x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	return Float64(x_s * x_m)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t, a, b)
	tmp = x_s * x_m;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 28.8%

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

3. Simplified 29.9%

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

5. Taylor expanded in a around 0 30.3%

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

   1. *-commutative 30.3%

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

   2. *-commutative 30.3%

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

7. Simplified 30.3%

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

8. Taylor expanded in t around 0 31.6%

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

9. Final simplification 31.6%

   \[\leadsto x \]
10. Add Preprocessing

Developer target: 30.1% 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 2024079 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (* 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))))