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

Percentage Accurate: 28.1% → 32.6%

Time: 27.5s

Alternatives: 11

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: 28.1% 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: 32.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{b \cdot \mathsf{fma}\left(2, a, 1\right)}\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 4 \cdot 10^{+302}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{z \cdot \mathsf{fma}\left(y, 2, 1\right)}{\frac{16}{t}}\right) \cdot \cos \left(\frac{{t_1}^{2}}{16} \cdot \frac{t_1}{\frac{1}{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cbrt (* b (fma 2.0 a 1.0)))))
   (if (<=
        (*
         (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
         (cos (/ (* t (* b (+ 1.0 (* 2.0 a)))) 16.0)))
        4e+302)
     (*
      x
      (*
       (cos (/ (* z (fma y 2.0 1.0)) (/ 16.0 t)))
       (cos (* (/ (pow t_1 2.0) 16.0) (/ t_1 (/ 1.0 t))))))
     x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cbrt((b * fma(2.0, a, 1.0)));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * (b * (1.0 + (2.0 * a)))) / 16.0))) <= 4e+302) {
		tmp = x * (cos(((z * fma(y, 2.0, 1.0)) / (16.0 / t))) * cos(((pow(t_1, 2.0) / 16.0) * (t_1 / (1.0 / t)))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = cbrt(Float64(b * fma(2.0, a, 1.0)))
	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))) <= 4e+302)
		tmp = Float64(x * Float64(cos(Float64(Float64(z * fma(y, 2.0, 1.0)) / Float64(16.0 / t))) * cos(Float64(Float64((t_1 ^ 2.0) / 16.0) * Float64(t_1 / Float64(1.0 / t))))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[N[(b * N[(2.0 * a + 1.0), $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], 4e+302], N[(x * N[(N[Cos[N[(N[(z * N[(y * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(16.0 / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] / 16.0), $MachinePrecision] * N[(t$95$1 / N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 4.0000000000000003e302

   1. Initial program 49.8%

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

      1. *-commutative 49.8%

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

      2. associate-*l* 49.8%

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

      3. cos-neg 49.8%

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

      4. distribute-frac-neg 49.8%

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

      5. distribute-lft-neg-in 49.8%

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

      6. distribute-rgt-neg-out 49.8%

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

      7. associate-*l* 49.8%

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

      8. *-commutative 49.8%

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

      9. associate-*l* 49.8%

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

   3. Simplified 50.5%

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

      1. fma-def 50.5%

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

      2. add-cube-cbrt 51.0%

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

      3. div-inv 51.0%

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

      4. times-frac 51.2%

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

      5. pow2 51.2%

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

      6. *-commutative 51.2%

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

      7. *-commutative 51.2%

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

      8. fma-udef 51.2%

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

      9. *-commutative 51.2%

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

      10. *-commutative 51.2%

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

      11. fma-udef 51.2%

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

   5. Applied egg-rr 51.2%

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

   if 4.0000000000000003e302 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

   3. Simplified 2.3%

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

   4. Taylor expanded in t around 0 5.8%

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

   5. Taylor expanded in t around 0 11.5%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 4 \cdot 10^{+302}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{z \cdot \mathsf{fma}\left(y, 2, 1\right)}{\frac{16}{t}}\right) \cdot \cos \left(\frac{{\left(\sqrt[3]{b \cdot \mathsf{fma}\left(2, a, 1\right)}\right)}^{2}}{16} \cdot \frac{\sqrt[3]{b \cdot \mathsf{fma}\left(2, a, 1\right)}}{\frac{1}{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2: 32.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 4 \cdot 10^{+302}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \left({\left(\sqrt[3]{\left(b \cdot \mathsf{fma}\left(2, a, 1\right)\right) \cdot \left(t \cdot 0.0625\right)}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* b (+ 1.0 (* 2.0 a)))) 16.0)))
      4e+302)
   (*
    x
    (*
     (cos (* (/ (fma y 2.0 1.0) 16.0) (* z t)))
     (cos (pow (cbrt (* (* b (fma 2.0 a 1.0)) (* t 0.0625))) 3.0))))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * (b * (1.0 + (2.0 * a)))) / 16.0))) <= 4e+302) {
		tmp = x * (cos(((fma(y, 2.0, 1.0) / 16.0) * (z * t))) * cos(pow(cbrt(((b * fma(2.0, a, 1.0)) * (t * 0.0625))), 3.0)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(b * Float64(1.0 + Float64(2.0 * a)))) / 16.0))) <= 4e+302)
		tmp = Float64(x * Float64(cos(Float64(Float64(fma(y, 2.0, 1.0) / 16.0) * Float64(z * t))) * cos((cbrt(Float64(Float64(b * fma(2.0, a, 1.0)) * Float64(t * 0.0625))) ^ 3.0))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(b * N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e+302], N[(x * N[(N[Cos[N[(N[(N[(y * 2.0 + 1.0), $MachinePrecision] / 16.0), $MachinePrecision] * N[(z * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[Power[N[Power[N[(N[(b * N[(2.0 * a + 1.0), $MachinePrecision]), $MachinePrecision] * N[(t * 0.0625), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $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))) < 4.0000000000000003e302

   1. Initial program 49.8%

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

   3. Simplified 49.8%

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

      1. associate-*l/ 49.8%

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

      2. fma-udef 49.8%

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

      3. *-commutative 49.8%

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

      4. *-commutative 49.8%

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

      5. add-cube-cbrt 50.3%

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

      6. pow3 50.8%

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

      7. *-commutative 50.8%

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

      8. *-commutative 50.8%

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

      9. fma-udef 50.8%

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

      10. associate-*l/ 50.8%

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

      11. *-commutative 50.8%

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

      12. *-commutative 50.8%

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

      13. div-inv 50.8%

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

      14. metadata-eval 50.8%

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

   5. Applied egg-rr 50.8%

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

   if 4.0000000000000003e302 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

   3. Simplified 2.3%

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

   4. Taylor expanded in t around 0 5.8%

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

   5. Taylor expanded in t around 0 11.5%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 4 \cdot 10^{+302}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \left({\left(\sqrt[3]{\left(b \cdot \mathsf{fma}\left(2, a, 1\right)\right) \cdot \left(t \cdot 0.0625\right)}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 32.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 4 \cdot 10^{+302}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \left(a \cdot \left(\left(t \cdot b\right) \cdot \sqrt[3]{0.001953125}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* b (+ 1.0 (* 2.0 a)))) 16.0)))
      4e+302)
   (*
    x
    (*
     (cos (/ (fma y 2.0 1.0) (/ (/ 16.0 t) z)))
     (cos (* a (* (* t b) (cbrt 0.001953125))))))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * (b * (1.0 + (2.0 * a)))) / 16.0))) <= 4e+302) {
		tmp = x * (cos((fma(y, 2.0, 1.0) / ((16.0 / t) / z))) * cos((a * ((t * b) * cbrt(0.001953125)))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(b * Float64(1.0 + Float64(2.0 * a)))) / 16.0))) <= 4e+302)
		tmp = Float64(x * Float64(cos(Float64(fma(y, 2.0, 1.0) / Float64(Float64(16.0 / t) / z))) * cos(Float64(a * Float64(Float64(t * b) * cbrt(0.001953125))))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(b * N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e+302], N[(x * N[(N[Cos[N[(N[(y * 2.0 + 1.0), $MachinePrecision] / N[(N[(16.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(a * N[(N[(t * b), $MachinePrecision] * N[Power[0.001953125, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 4.0000000000000003e302

   1. Initial program 49.8%

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

   3. Simplified 49.8%

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

      1. add-cbrt-cube 48.4%

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

      2. pow1/3 40.5%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \color{blue}{\left({\left(\left(\left(\frac{t \cdot \mathsf{fma}\left(2, a, 1\right)}{16} \cdot b\right) \cdot \left(\frac{t \cdot \mathsf{fma}\left(2, a, 1\right)}{16} \cdot b\right)\right) \cdot \left(\frac{t \cdot \mathsf{fma}\left(2, a, 1\right)}{16} \cdot b\right)\right)}^{0.3333333333333333}\right)}\right) \]

      3. pow3 40.5%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \left({\color{blue}{\left({\left(\frac{t \cdot \mathsf{fma}\left(2, a, 1\right)}{16} \cdot b\right)}^{3}\right)}}^{0.3333333333333333}\right)\right) \]

      4. div-inv 40.5%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \left({\left({\left(\color{blue}{\left(\left(t \cdot \mathsf{fma}\left(2, a, 1\right)\right) \cdot \frac{1}{16}\right)} \cdot b\right)}^{3}\right)}^{0.3333333333333333}\right)\right) \]

      5. metadata-eval 40.5%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \left({\left({\left(\left(\left(t \cdot \mathsf{fma}\left(2, a, 1\right)\right) \cdot \color{blue}{0.0625}\right) \cdot b\right)}^{3}\right)}^{0.3333333333333333}\right)\right) \]

   5. Applied egg-rr 40.5%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \color{blue}{\left({\left({\left(\left(\left(t \cdot \mathsf{fma}\left(2, a, 1\right)\right) \cdot 0.0625\right) \cdot b\right)}^{3}\right)}^{0.3333333333333333}\right)}\right) \]

   6. Taylor expanded in a around inf 50.9%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \color{blue}{\left(a \cdot \left(b \cdot \left(t \cdot \sqrt[3]{0.001953125}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. associate-*r* 50.7%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \left(a \cdot \color{blue}{\left(\left(b \cdot t\right) \cdot \sqrt[3]{0.001953125}\right)}\right)\right) \]

      2. *-commutative 50.7%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \left(a \cdot \left(\color{blue}{\left(t \cdot b\right)} \cdot \sqrt[3]{0.001953125}\right)\right)\right) \]

   8. Simplified 50.7%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \color{blue}{\left(a \cdot \left(\left(t \cdot b\right) \cdot \sqrt[3]{0.001953125}\right)\right)}\right) \]

   if 4.0000000000000003e302 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

   3. Simplified 2.3%

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

   4. Taylor expanded in t around 0 5.8%

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

   5. Taylor expanded in t around 0 11.5%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 4 \cdot 10^{+302}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \left(a \cdot \left(\left(t \cdot b\right) \cdot \sqrt[3]{0.001953125}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 32.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 4 \cdot 10^{+302}:\\ \;\;\;\;x \cdot \left(\cos \left(\left(t \cdot 0.0625\right) \cdot \frac{\mathsf{fma}\left(2, y, 1\right)}{\frac{1}{z}}\right) \cdot \cos \left(0.0625 \cdot \left(2 \cdot \left(a \cdot \left(t \cdot b\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* b (+ 1.0 (* 2.0 a)))) 16.0)))
      4e+302)
   (*
    x
    (*
     (cos (* (* t 0.0625) (/ (fma 2.0 y 1.0) (/ 1.0 z))))
     (cos (* 0.0625 (* 2.0 (* a (* t b)))))))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * (b * (1.0 + (2.0 * a)))) / 16.0))) <= 4e+302) {
		tmp = x * (cos(((t * 0.0625) * (fma(2.0, y, 1.0) / (1.0 / z)))) * cos((0.0625 * (2.0 * (a * (t * b))))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(b * Float64(1.0 + Float64(2.0 * a)))) / 16.0))) <= 4e+302)
		tmp = Float64(x * Float64(cos(Float64(Float64(t * 0.0625) * Float64(fma(2.0, y, 1.0) / Float64(1.0 / z)))) * cos(Float64(0.0625 * Float64(2.0 * Float64(a * Float64(t * b)))))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(b * N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e+302], N[(x * N[(N[Cos[N[(N[(t * 0.0625), $MachinePrecision] * N[(N[(2.0 * y + 1.0), $MachinePrecision] / N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.0625 * N[(2.0 * N[(a * N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 4.0000000000000003e302

   1. Initial program 49.8%

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

   3. Simplified 49.8%

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

   4. Taylor expanded in t around 0 49.8%

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

      1. *-un-lft-identity 49.8%

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

      2. div-inv 49.8%

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

      3. times-frac 49.4%

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

      4. clear-num 49.8%

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

      5. div-inv 49.8%

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

      6. metadata-eval 49.8%

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

      7. fma-def 49.8%

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

      8. *-commutative 49.8%

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

      9. fma-def 49.8%

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

   6. Applied egg-rr 49.8%

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

   7. Taylor expanded in a around inf 49.9%

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

      1. *-commutative 49.9%

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

   9. Simplified 49.9%

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

   if 4.0000000000000003e302 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

   3. Simplified 2.3%

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

   4. Taylor expanded in t around 0 5.8%

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

   5. Taylor expanded in t around 0 11.5%

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

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 4 \cdot 10^{+302}:\\ \;\;\;\;x \cdot \left(\cos \left(\left(t \cdot 0.0625\right) \cdot \frac{\mathsf{fma}\left(2, y, 1\right)}{\frac{1}{z}}\right) \cdot \cos \left(0.0625 \cdot \left(2 \cdot \left(a \cdot \left(t \cdot b\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 32.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 4 \cdot 10^{+302}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{z \cdot \mathsf{fma}\left(y, 2, 1\right)}{\frac{16}{t}}\right) \cdot \cos \left(\left(a \cdot \left(t \cdot b\right)\right) \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* b (+ 1.0 (* 2.0 a)))) 16.0)))
      4e+302)
   (*
    x
    (*
     (cos (/ (* z (fma y 2.0 1.0)) (/ 16.0 t)))
     (cos (* (* a (* t b)) 0.125))))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * (b * (1.0 + (2.0 * a)))) / 16.0))) <= 4e+302) {
		tmp = x * (cos(((z * fma(y, 2.0, 1.0)) / (16.0 / t))) * cos(((a * (t * b)) * 0.125)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(b * Float64(1.0 + Float64(2.0 * a)))) / 16.0))) <= 4e+302)
		tmp = Float64(x * Float64(cos(Float64(Float64(z * fma(y, 2.0, 1.0)) / Float64(16.0 / t))) * cos(Float64(Float64(a * Float64(t * b)) * 0.125))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(b * N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e+302], N[(x * N[(N[Cos[N[(N[(z * N[(y * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(16.0 / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(a * N[(t * b), $MachinePrecision]), $MachinePrecision] * 0.125), $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))) < 4.0000000000000003e302

   1. Initial program 49.8%

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

      1. *-commutative 49.8%

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

      2. associate-*l* 49.8%

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

      3. cos-neg 49.8%

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

      4. distribute-frac-neg 49.8%

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

      5. distribute-lft-neg-in 49.8%

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

      6. distribute-rgt-neg-out 49.8%

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

      7. associate-*l* 49.8%

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

      8. *-commutative 49.8%

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

      9. associate-*l* 49.8%

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

   3. Simplified 50.5%

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

      1. fma-def 50.5%

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

      2. *-commutative 50.5%

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

      3. div-inv 50.5%

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

      4. times-frac 50.0%

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

      5. *-commutative 50.0%

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

      6. fma-udef 50.0%

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

   5. Applied egg-rr 50.0%

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

   6. Taylor expanded in a around inf 49.8%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \color{blue}{\left(0.125 \cdot \left(a \cdot \left(b \cdot t\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 49.8%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \left(0.125 \cdot \left(a \cdot \color{blue}{\left(t \cdot b\right)}\right)\right)\right) \]

   8. Simplified 49.8%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \color{blue}{\left(0.125 \cdot \left(a \cdot \left(t \cdot b\right)\right)\right)}\right) \]

   if 4.0000000000000003e302 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

   3. Simplified 2.3%

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

   4. Taylor expanded in t around 0 5.8%

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

   5. Taylor expanded in t around 0 11.5%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 4 \cdot 10^{+302}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{z \cdot \mathsf{fma}\left(y, 2, 1\right)}{\frac{16}{t}}\right) \cdot \cos \left(\left(a \cdot \left(t \cdot b\right)\right) \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 32.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 4 \cdot 10^{+302}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \left(0.125 \cdot \left(t \cdot \left(a \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* b (+ 1.0 (* 2.0 a)))) 16.0)))
      4e+302)
   (*
    x
    (*
     (cos (/ (fma y 2.0 1.0) (/ (/ 16.0 t) z)))
     (cos (* 0.125 (* t (* a b))))))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * (b * (1.0 + (2.0 * a)))) / 16.0))) <= 4e+302) {
		tmp = x * (cos((fma(y, 2.0, 1.0) / ((16.0 / t) / z))) * cos((0.125 * (t * (a * b)))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(b * Float64(1.0 + Float64(2.0 * a)))) / 16.0))) <= 4e+302)
		tmp = Float64(x * Float64(cos(Float64(fma(y, 2.0, 1.0) / Float64(Float64(16.0 / t) / z))) * cos(Float64(0.125 * Float64(t * Float64(a * b))))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(b * N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e+302], N[(x * N[(N[Cos[N[(N[(y * 2.0 + 1.0), $MachinePrecision] / N[(N[(16.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.125 * N[(t * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 4.0000000000000003e302

   1. Initial program 49.8%

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

   3. Simplified 49.8%

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

   4. Taylor expanded in a around inf 49.9%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \color{blue}{\left(0.125 \cdot \left(a \cdot \left(b \cdot t\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 49.9%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \left(0.125 \cdot \color{blue}{\left(\left(b \cdot t\right) \cdot a\right)}\right)\right) \]

      2. *-commutative 49.9%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \left(0.125 \cdot \left(\color{blue}{\left(t \cdot b\right)} \cdot a\right)\right)\right) \]

      3. associate-*l* 49.9%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \left(0.125 \cdot \color{blue}{\left(t \cdot \left(b \cdot a\right)\right)}\right)\right) \]

   6. Simplified 49.9%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \color{blue}{\left(0.125 \cdot \left(t \cdot \left(b \cdot a\right)\right)\right)}\right) \]

   if 4.0000000000000003e302 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

   3. Simplified 2.3%

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

   4. Taylor expanded in t around 0 5.8%

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

   5. Taylor expanded in t around 0 11.5%

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

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 4 \cdot 10^{+302}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \cos \left(0.125 \cdot \left(t \cdot \left(a \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 32.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right)\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{+302}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (*
          (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
          (cos (/ (* t (* b (+ 1.0 (* 2.0 a)))) 16.0)))))
   (if (<= t_1 4e+302) t_1 x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * (b * (1.0 + (2.0 * a)))) / 16.0));
	double tmp;
	if (t_1 <= 4e+302) {
		tmp = t_1;
	} 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 = (x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos(((t * (b * (1.0d0 + (2.0d0 * a)))) / 16.0d0))
    if (t_1 <= 4d+302) then
        tmp = t_1
    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 = (x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos(((t * (b * (1.0 + (2.0 * a)))) / 16.0));
	double tmp;
	if (t_1 <= 4e+302) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos(((t * (b * (1.0 + (2.0 * a)))) / 16.0))
	tmp = 0
	if t_1 <= 4e+302:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = 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)))
	tmp = 0.0
	if (t_1 <= 4e+302)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * (b * (1.0 + (2.0 * a)))) / 16.0));
	tmp = 0.0;
	if (t_1 <= 4e+302)
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 4e+302], t$95$1, 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))) < 4.0000000000000003e302

   1. Initial program 49.8%

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

   if 4.0000000000000003e302 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

   3. Simplified 2.3%

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

   4. Taylor expanded in t around 0 5.8%

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

   5. Taylor expanded in t around 0 11.5%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 4 \cdot 10^{+302}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 29.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{+38}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 2.1e+38)
   (*
    (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
    (cos (/ (* b (* t (+ 1.0 (* 2.0 a)))) 16.0)))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 2.1e+38) {
		tmp = (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((b * (t * (1.0 + (2.0 * a)))) / 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) :: tmp
    if (t <= 2.1d+38) then
        tmp = (x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos(((b * (t * (1.0d0 + (2.0d0 * a)))) / 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 tmp;
	if (t <= 2.1e+38) {
		tmp = (x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos(((b * (t * (1.0 + (2.0 * a)))) / 16.0));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 2.1e+38:
		tmp = (x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos(((b * (t * (1.0 + (2.0 * a)))) / 16.0))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 2.1e+38)
		tmp = Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(b * Float64(t * Float64(1.0 + Float64(2.0 * a)))) / 16.0)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 2.1e+38)
		tmp = (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((b * (t * (1.0 + (2.0 * a)))) / 16.0));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 2.1e+38], 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[(b * N[(t * N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < 2.1e38

   1. Initial program 38.2%

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

   2. Taylor expanded in b around 0 39.1%

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

   if 2.1e38 < t

   1. Initial program 8.0%

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

   3. Simplified 7.3%

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

   4. Taylor expanded in t around 0 12.1%

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

   5. Taylor expanded in t around 0 13.2%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{+38}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 30.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{+59}:\\ \;\;\;\;x \cdot \cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{16}{z \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 7.8e+59) (* x (cos (/ (fma y 2.0 1.0) (/ 16.0 (* z t))))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 7.8e+59) {
		tmp = x * cos((fma(y, 2.0, 1.0) / (16.0 / (z * t))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 7.8e+59)
		tmp = Float64(x * cos(Float64(fma(y, 2.0, 1.0) / Float64(16.0 / Float64(z * t)))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 7.8e+59], N[(x * N[Cos[N[(N[(y * 2.0 + 1.0), $MachinePrecision] / N[(16.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < 7.80000000000000043e59

   1. Initial program 37.6%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in t around 0 39.2%

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

   5. Taylor expanded in t around 0 39.0%

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

   if 7.80000000000000043e59 < t

   1. Initial program 8.4%

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

   3. Simplified 7.4%

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

   4. Taylor expanded in t around 0 11.8%

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

   5. Taylor expanded in t around 0 13.5%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{+59}:\\ \;\;\;\;x \cdot \cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{16}{z \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 30.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \cos \left(z \cdot \left(t \cdot 0.0625\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* x (cos (* z (* t 0.0625)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * cos((z * (t * 0.0625)));
}

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((z * (t * 0.0625d0)))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.cos((z * (t * 0.0625)));
}

Python

def code(x, y, z, t, a, b):
	return x * math.cos((z * (t * 0.0625)))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * cos(Float64(z * Float64(t * 0.0625))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * cos((z * (t * 0.0625)));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Cos[N[(z * N[(t * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.7%

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

3. Simplified 32.5%

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

4. Taylor expanded in t around 0 33.7%

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

5. Taylor expanded in y around 0 33.6%

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

   1. associate-*r* 33.6%

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

7. Simplified 33.6%

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

8. Final simplification 33.6%

   \[\leadsto x \cdot \cos \left(z \cdot \left(t \cdot 0.0625\right)\right) \]

Alternative 11: 31.3% 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 31.7%

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

3. Simplified 32.5%

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

4. Taylor expanded in t around 0 33.7%

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

5. Taylor expanded in t around 0 33.4%

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

6. Final simplification 33.4%

   \[\leadsto x \]

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 2023318 
(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))))