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

Percentage Accurate: 27.9% → 32.3%

Time: 28.0s

Alternatives: 4

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: 27.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 32.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\mathsf{fma}\left(2, y, 1\right) \cdot \left(z \cdot 0.0625\right)}\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 4.2 \cdot 10^{+259}:\\ \;\;\;\;\left(x \cdot \cos \left({t_1}^{2} \cdot \left(t \cdot t_1\right)\right)\right) \cdot \cos \left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{\frac{16}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cbrt (* (fma 2.0 y 1.0) (* z 0.0625)))))
   (if (<=
        (*
         (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
         (cos (/ (* t (* b (+ 1.0 (* 2.0 a)))) 16.0)))
        4.2e+259)
     (*
      (* x (cos (* (pow t_1 2.0) (* t t_1))))
      (cos (/ (* b (fma a 2.0 1.0)) (/ 16.0 t))))
     x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cbrt((fma(2.0, y, 1.0) * (z * 0.0625)));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * (b * (1.0 + (2.0 * a)))) / 16.0))) <= 4.2e+259) {
		tmp = (x * cos((pow(t_1, 2.0) * (t * t_1)))) * cos(((b * fma(a, 2.0, 1.0)) / (16.0 / t)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = cbrt(Float64(fma(2.0, y, 1.0) * Float64(z * 0.0625)))
	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(b * Float64(1.0 + Float64(2.0 * a)))) / 16.0))) <= 4.2e+259)
		tmp = Float64(Float64(x * cos(Float64((t_1 ^ 2.0) * Float64(t * t_1)))) * cos(Float64(Float64(b * fma(a, 2.0, 1.0)) / Float64(16.0 / t))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[N[(N[(2.0 * y + 1.0), $MachinePrecision] * N[(z * 0.0625), $MachinePrecision]), $MachinePrecision], 1/3], $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] * N[Cos[N[(N[(t * N[(b * N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4.2e+259], N[(N[(x * N[Cos[N[(N[Power[t$95$1, 2.0], $MachinePrecision] * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(b * N[(a * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(16.0 / t), $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))) < 4.20000000000000011e259

   1. Initial program 41.9%

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

      1. *-commutative 41.9%

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

      2. associate-*l* 41.9%

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

      3. cos-neg 41.9%

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

      4. distribute-frac-neg 41.9%

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

      5. distribute-lft-neg-in 41.9%

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

      6. distribute-rgt-neg-out 41.9%

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

      7. associate-*l* 41.9%

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

      8. *-commutative 41.9%

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

   3. Simplified 42.1%

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

      1. associate-/r/ 42.1%

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

      2. add-cube-cbrt 42.8%

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

      3. associate-*l* 43.1%

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

      4. pow2 43.1%

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

      5. div-inv 43.1%

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

      6. fma-def 43.1%

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

      7. associate-*l* 43.1%

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

      8. *-commutative 43.1%

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

      9. fma-def 43.1%

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

      10. metadata-eval 43.1%

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

   5. Applied egg-rr 43.1%

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

   if 4.20000000000000011e259 < (*.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 2.6%

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

      1. associate-*l* 2.6%

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

      2. *-commutative 2.6%

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

      3. *-commutative 2.6%

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

      4. associate-*l/ 2.6%

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

      5. fma-def 2.6%

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

      6. associate-*l/ 2.6%

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

      7. *-commutative 2.6%

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

      8. fma-def 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in b around 0 8.4%

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

   5. Taylor expanded in z around 0 13.6%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 4.2 \cdot 10^{+259}:\\ \;\;\;\;\left(x \cdot \cos \left({\left(\sqrt[3]{\mathsf{fma}\left(2, y, 1\right) \cdot \left(z \cdot 0.0625\right)}\right)}^{2} \cdot \left(t \cdot \sqrt[3]{\mathsf{fma}\left(2, y, 1\right) \cdot \left(z \cdot 0.0625\right)}\right)\right)\right) \cdot \cos \left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{\frac{16}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2: 32.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t_1}{16}\right) \leq 2 \cdot 10^{+289}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{1}{\frac{\frac{\frac{16}{t}}{z}}{\mathsf{fma}\left(2, y, 1\right)}}\right) \cdot \cos \left(0.0625 \cdot t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* b (+ 1.0 (* 2.0 a))))))
   (if (<=
        (*
         (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
         (cos (/ t_1 16.0)))
        2e+289)
     (*
      x
      (*
       (cos (/ 1.0 (/ (/ (/ 16.0 t) z) (fma 2.0 y 1.0))))
       (cos (* 0.0625 t_1))))
     x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (b * (1.0 + (2.0 * a)));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((t_1 / 16.0))) <= 2e+289) {
		tmp = x * (cos((1.0 / (((16.0 / t) / z) / fma(2.0, y, 1.0)))) * cos((0.0625 * t_1)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(b * Float64(1.0 + Float64(2.0 * a))))
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(t_1 / 16.0))) <= 2e+289)
		tmp = Float64(x * Float64(cos(Float64(1.0 / Float64(Float64(Float64(16.0 / t) / z) / fma(2.0, y, 1.0)))) * cos(Float64(0.0625 * t_1))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(b * N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $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] * N[Cos[N[(t$95$1 / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+289], 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[(0.0625 * t$95$1), $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))) < 2.0000000000000001e289

   1. Initial program 41.2%

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

      1. associate-*l* 41.2%

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

      2. *-commutative 41.2%

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

      3. *-commutative 41.2%

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

      4. associate-*l/ 41.2%

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

      5. fma-def 41.2%

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

      6. associate-*l/ 41.2%

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

      7. *-commutative 41.2%

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

      8. fma-def 41.2%

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

   3. Simplified 41.2%

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

      1. associate-/r/ 41.5%

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

      2. fma-def 41.5%

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

      3. associate-/l* 41.1%

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

      4. clear-num 41.8%

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

      5. *-commutative 41.8%

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

      6. fma-def 41.8%

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

   5. Applied egg-rr 41.8%

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

   6. Taylor expanded in b around 0 41.8%

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

   if 2.0000000000000001e289 < (*.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 1.0%

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

      1. associate-*l* 1.0%

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

      2. *-commutative 1.0%

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

      3. *-commutative 1.0%

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

      4. associate-*l/ 1.0%

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

      5. fma-def 1.0%

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

      6. associate-*l/ 1.0%

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

      7. *-commutative 1.0%

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

      8. fma-def 1.0%

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

   3. Simplified 1.0%

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

   4. Taylor expanded in b around 0 7.3%

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

   5. Taylor expanded in z around 0 12.9%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 2 \cdot 10^{+289}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{1}{\frac{\frac{\frac{16}{t}}{z}}{\mathsf{fma}\left(2, y, 1\right)}}\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 31.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot t_1 \leq 2 \cdot 10^{+289}:\\ \;\;\;\;t_1 \cdot \left(x \cdot \cos \left(\frac{z \cdot t}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* t (* b (+ 1.0 (* 2.0 a)))) 16.0))))
   (if (<= (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) t_1) 2e+289)
     (* 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 * (b * (1.0 + (2.0 * a)))) / 16.0));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 2e+289) {
		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 * (b * (1.0d0 + (2.0d0 * a)))) / 16.0d0))
    if (((x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * t_1) <= 2d+289) 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 * (b * (1.0 + (2.0 * a)))) / 16.0));
	double tmp;
	if (((x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 2e+289) {
		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 * (b * (1.0 + (2.0 * a)))) / 16.0))
	tmp = 0
	if ((x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 2e+289:
		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(b * Float64(1.0 + Float64(2.0 * a)))) / 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) <= 2e+289)
		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 * (b * (1.0 + (2.0 * a)))) / 16.0));
	tmp = 0.0;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 2e+289)
		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[(b * N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $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], 2e+289], 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))) < 2.0000000000000001e289

   1. Initial program 41.2%

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

   2. Taylor expanded in y around 0 41.4%

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

   if 2.0000000000000001e289 < (*.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 1.0%

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

      1. associate-*l* 1.0%

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

      2. *-commutative 1.0%

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

      3. *-commutative 1.0%

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

      4. associate-*l/ 1.0%

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

      5. fma-def 1.0%

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

      6. associate-*l/ 1.0%

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

      7. *-commutative 1.0%

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

      8. fma-def 1.0%

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

   3. Simplified 1.0%

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

   4. Taylor expanded in b around 0 7.3%

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

   5. Taylor expanded in z around 0 12.9%

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

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \cdot \left(x \cdot \cos \left(\frac{z \cdot t}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 31.2% 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 25.0%

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

   1. associate-*l* 25.0%

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

   2. *-commutative 25.0%

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

   3. *-commutative 25.0%

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

   4. associate-*l/ 25.0%

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

   5. fma-def 25.0%

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

   6. associate-*l/ 25.0%

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

   7. *-commutative 25.0%

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

   8. fma-def 25.0%

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

3. Simplified 25.0%

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

4. Taylor expanded in b around 0 26.8%

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

5. Taylor expanded in z around 0 28.0%

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

6. Final simplification 28.0%

   \[\leadsto x \]

Developer target: 30.8% 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 2023279 
(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))))