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

Percentage Accurate: 28.1% → 32.5%

Time: 30.5s

Alternatives: 5

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 32.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\frac{16}{t}}\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+283}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{z}{{t_1}^{2}} \cdot \frac{\mathsf{fma}\left(2, y, 1\right)}{t_1}\right) \cdot \cos \left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{\frac{16}{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cbrt (/ 16.0 t))))
   (if (<=
        (*
         (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
         (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
        1e+283)
     (*
      x
      (*
       (cos (* (/ z (pow t_1 2.0)) (/ (fma 2.0 y 1.0) 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((16.0 / t));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+283) {
		tmp = x * (cos(((z / pow(t_1, 2.0)) * (fma(2.0, y, 1.0) / 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(16.0 / t))
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 1e+283)
		tmp = Float64(x * Float64(cos(Float64(Float64(z / (t_1 ^ 2.0)) * Float64(fma(2.0, y, 1.0) / 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[(16.0 / t), $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[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+283], N[(x * N[(N[Cos[N[(N[(z / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * y + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(b * N[(a * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(16.0 / t), $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))) < 9.99999999999999955e282

   1. Initial program 45.7%

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

      1. *-commutative 45.7%

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

      2. associate-*l* 45.7%

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

      3. cos-neg 45.7%

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

      4. distribute-frac-neg 45.7%

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

      5. distribute-lft-neg-in 45.7%

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

      6. distribute-rgt-neg-out 45.7%

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

      7. associate-*l* 45.7%

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

      8. *-commutative 45.7%

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

      9. associate-*l* 45.7%

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

   3. Simplified 45.8%

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

      1. *-commutative 45.8%

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

      2. add-cube-cbrt 46.8%

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

      3. times-frac 46.9%

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

      4. pow2 46.9%

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

      5. fma-def 46.9%

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

      6. *-commutative 46.9%

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

      7. fma-def 46.9%

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

   5. Applied egg-rr 46.9%

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

   if 9.99999999999999955e282 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

   2. Taylor expanded in y around 0 1.6%

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

   3. Taylor expanded in t around 0 12.7%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+283}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{z}{{\left(\sqrt[3]{\frac{16}{t}}\right)}^{2}} \cdot \frac{\mathsf{fma}\left(2, y, 1\right)}{\sqrt[3]{\frac{16}{t}}}\right) \cdot \cos \left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{\frac{16}{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2: 32.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+283}:\\ \;\;\;\;x \cdot \left(\left|\cos \left(t \cdot \left(0.0625 \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right)\right)\right| \cdot \cos \left(\frac{b \cdot \mathsf{fma}\left(2, a, 1\right)}{\frac{16}{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
      1e+283)
   (*
    x
    (*
     (fabs (cos (* t (* 0.0625 (* z (fma 2.0 y 1.0))))))
     (cos (/ (* b (fma 2.0 a 1.0)) (/ 16.0 t)))))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+283) {
		tmp = x * (fabs(cos((t * (0.0625 * (z * fma(2.0, y, 1.0)))))) * cos(((b * fma(2.0, a, 1.0)) / (16.0 / t))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 1e+283)
		tmp = Float64(x * Float64(abs(cos(Float64(t * Float64(0.0625 * Float64(z * fma(2.0, y, 1.0)))))) * cos(Float64(Float64(b * fma(2.0, a, 1.0)) / Float64(16.0 / t)))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+283], N[(x * N[(N[Abs[N[Cos[N[(t * N[(0.0625 * N[(z * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(b * N[(2.0 * a + 1.0), $MachinePrecision]), $MachinePrecision] / N[(16.0 / t), $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))) < 9.99999999999999955e282

   1. Initial program 45.7%

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

   3. Simplified 45.2%

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

      1. add-sqr-sqrt 43.2%

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

      2. sqrt-unprod 45.8%

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

      3. pow2 45.8%

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

      4. *-commutative 45.8%

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

      5. div-inv 45.8%

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

      6. fma-def 45.8%

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

      7. *-commutative 45.8%

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

      8. fma-def 45.8%

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

      9. metadata-eval 45.8%

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

   5. Applied egg-rr 45.8%

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

      1. associate-*r* 46.2%

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

      2. associate-*r* 46.2%

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

      3. *-commutative 46.2%

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

      4. unpow2 46.2%

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

      5. rem-sqrt-square 46.2%

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

      6. associate-*l* 46.2%

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

      7. associate-*l* 46.2%

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

      8. associate-*l* 46.3%

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

      9. *-commutative 46.3%

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

      10. *-commutative 46.3%

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

      11. associate-*r* 46.3%

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

   7. Simplified 46.3%

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

      1. *-commutative 46.3%

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

      2. fma-udef 46.3%

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

      3. *-commutative 46.3%

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

      4. fma-def 46.3%

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

      5. clear-num 46.3%

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

      6. div-inv 46.8%

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

      7. fma-def 46.8%

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

      8. *-commutative 46.8%

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

      9. fma-udef 46.8%

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

      10. *-commutative 46.8%

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

   9. Applied egg-rr 46.8%

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

   if 9.99999999999999955e282 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

   2. Taylor expanded in y around 0 1.6%

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

   3. Taylor expanded in t around 0 12.7%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+283}:\\ \;\;\;\;x \cdot \left(\left|\cos \left(t \cdot \left(0.0625 \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right)\right)\right| \cdot \cos \left(\frac{b \cdot \mathsf{fma}\left(2, a, 1\right)}{\frac{16}{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 32.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+283}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{\frac{16}{t}}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
      1e+283)
   (* x (* (cos (/ (* b (fma a 2.0 1.0)) (/ 16.0 t))) (log1p (expm1 1.0))))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+283) {
		tmp = x * (cos(((b * fma(a, 2.0, 1.0)) / (16.0 / t))) * log1p(expm1(1.0)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 1e+283)
		tmp = Float64(x * Float64(cos(Float64(Float64(b * fma(a, 2.0, 1.0)) / Float64(16.0 / t))) * log1p(expm1(1.0))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+283], N[(x * N[(N[Cos[N[(N[(b * N[(a * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(16.0 / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Log[1 + N[(Exp[1.0] - 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))) < 9.99999999999999955e282

   1. Initial program 45.7%

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

      1. *-commutative 45.7%

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

      2. associate-*l* 45.7%

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

      3. cos-neg 45.7%

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

      4. distribute-frac-neg 45.7%

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

      5. distribute-lft-neg-in 45.7%

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

      6. distribute-rgt-neg-out 45.7%

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

      7. associate-*l* 45.7%

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

      8. *-commutative 45.7%

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

      9. associate-*l* 45.7%

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

   3. Simplified 45.8%

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

      1. fma-def 45.8%

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

      2. associate-/l* 45.9%

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

      3. log1p-expm1-u 45.9%

         \[\leadsto x \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right)\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right) \]

      4. div-inv 45.9%

         \[\leadsto x \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \color{blue}{\left(\left(\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)}\right)\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right) \]

      5. fma-def 45.9%

         \[\leadsto x \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(\left(\color{blue}{\mathsf{fma}\left(y, 2, 1\right)} \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right) \]

      6. associate-*l* 45.8%

         \[\leadsto x \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\color{blue}{\left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot t\right)\right)} \cdot \frac{1}{16}\right)\right)\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right) \]

      7. fma-def 45.8%

         \[\leadsto x \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(\color{blue}{\left(y \cdot 2 + 1\right)} \cdot \left(z \cdot t\right)\right) \cdot \frac{1}{16}\right)\right)\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right) \]

      8. *-commutative 45.8%

         \[\leadsto x \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(\left(\color{blue}{2 \cdot y} + 1\right) \cdot \left(z \cdot t\right)\right) \cdot \frac{1}{16}\right)\right)\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right) \]

      9. fma-def 45.8%

         \[\leadsto x \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(\color{blue}{\mathsf{fma}\left(2, y, 1\right)} \cdot \left(z \cdot t\right)\right) \cdot \frac{1}{16}\right)\right)\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right) \]

      10. metadata-eval 45.8%

          \[\leadsto x \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(z \cdot t\right)\right) \cdot \color{blue}{0.0625}\right)\right)\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right) \]

   5. Applied egg-rr 45.8%

      \[\leadsto x \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(z \cdot t\right)\right) \cdot 0.0625\right)\right)\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right) \]

   6. Taylor expanded in z around 0 46.4%

      \[\leadsto x \cdot \left(\mathsf{log1p}\left(\color{blue}{e^{1} - 1}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right) \]
   7. Step-by-step derivation

      1. expm1-def 46.4%

         \[\leadsto x \cdot \left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(1\right)}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right) \]

   8. Simplified 46.4%

      \[\leadsto x \cdot \left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(1\right)}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right) \]

   if 9.99999999999999955e282 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

   2. Taylor expanded in y around 0 1.6%

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

   3. Taylor expanded in t around 0 12.7%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+283}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{\frac{16}{t}}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 32.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+283}:\\ \;\;\;\;x \cdot \cos \left(0.0625 \cdot \left(t \cdot b + 2 \cdot \left(a \cdot \left(t \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
      1e+283)
   (* x (cos (* 0.0625 (+ (* t b) (* 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 * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+283) {
		tmp = x * cos((0.0625 * ((t * b) + (2.0 * (a * (t * b))))));
	} 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 (((x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos(((t * ((1.0d0 + (2.0d0 * a)) * b)) / 16.0d0))) <= 1d+283) then
        tmp = x * cos((0.0625d0 * ((t * b) + (2.0d0 * (a * (t * b))))))
    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 (((x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+283) {
		tmp = x * Math.cos((0.0625 * ((t * b) + (2.0 * (a * (t * b))))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+283:
		tmp = x * math.cos((0.0625 * ((t * b) + (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(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 1e+283)
		tmp = Float64(x * cos(Float64(0.0625 * Float64(Float64(t * b) + Float64(2.0 * Float64(a * Float64(t * b)))))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+283)
		tmp = x * cos((0.0625 * ((t * b) + (2.0 * (a * (t * b))))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+283], N[(x * N[Cos[N[(0.0625 * N[(N[(t * b), $MachinePrecision] + 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))) < 9.99999999999999955e282

   1. Initial program 45.7%

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

   2. Taylor expanded in y around 0 45.2%

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

   3. Taylor expanded in z around 0 45.5%

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

      1. associate-*r* 45.5%

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

      2. metadata-eval 45.5%

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

      3. cancel-sign-sub-inv 45.5%

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

      4. associate-*r* 45.5%

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

      5. associate-*r* 45.5%

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

      6. cancel-sign-sub-inv 45.5%

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

      7. metadata-eval 45.5%

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

      8. associate-*r* 45.5%

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

      9. associate-*r* 46.3%

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

      10. *-commutative 46.3%

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

      11. +-commutative 46.3%

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

      12. fma-udef 46.3%

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

   5. Simplified 46.3%

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

   6. Taylor expanded in a around 0 46.3%

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

   if 9.99999999999999955e282 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

   2. Taylor expanded in y around 0 1.6%

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

   3. Taylor expanded in t around 0 12.7%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+283}:\\ \;\;\;\;x \cdot \cos \left(0.0625 \cdot \left(t \cdot b + 2 \cdot \left(a \cdot \left(t \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 31.4% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

2. Taylor expanded in y around 0 29.4%

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

3. Taylor expanded in t around 0 32.2%

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

4. Final simplification 32.2%

   \[\leadsto x \]

Developer target: 31.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023320 
(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))))