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

Percentage Accurate: 26.7% → 31.0%

Time: 18.7s

Alternatives: 6

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: 26.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 31.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)}\\ t_2 := \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot t_2 \leq 2 \cdot 10^{+49}:\\ \;\;\;\;t_2 \cdot \left(x \cdot \cos \left(\frac{t_1 \cdot \left(t_1 \cdot t_1\right)}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cbrt (* t (* z (fma 2.0 y 1.0)))))
        (t_2 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))))
   (if (<= (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) t_2) 2e+49)
     (* t_2 (* x (cos (/ (* t_1 (* t_1 t_1)) 16.0))))
     x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cbrt((t * (z * fma(2.0, y, 1.0))));
	double t_2 = cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_2) <= 2e+49) {
		tmp = t_2 * (x * cos(((t_1 * (t_1 * t_1)) / 16.0)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = cbrt(Float64(t * Float64(z * fma(2.0, y, 1.0))))
	t_2 = cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * t_2) <= 2e+49)
		tmp = Float64(t_2 * Float64(x * cos(Float64(Float64(t_1 * Float64(t_1 * t_1)) / 16.0))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[N[(t * N[(z * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], 2e+49], N[(t$95$2 * N[(x * N[Cos[N[(N[(t$95$1 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 1.99999999999999989e49

   1. Initial program 52.6%

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

      1. add-cbrt-cube 49.7%

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

      2. pow1/3 42.1%

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

      3. pow3 42.1%

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

      4. *-commutative 42.1%

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

      5. *-commutative 42.1%

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

      6. *-commutative 42.1%

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

      7. fma-def 42.1%

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

   3. Applied egg-rr 42.1%

      \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{{\left({\left(t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right)}^{3}\right)}^{0.3333333333333333}}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   4. Step-by-step derivation

      1. pow-pow 52.6%

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

      2. metadata-eval 52.6%

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

      3. pow1 52.6%

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

      4. add-cube-cbrt 53.4%

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

   5. Applied egg-rr 53.4%

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

   if 1.99999999999999989e49 < (*.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 11.0%

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

      1. associate-*l* 11.0%

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

      2. *-commutative 11.0%

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

      3. *-commutative 11.0%

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

      4. associate-*l/ 11.0%

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

      5. *-commutative 11.0%

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

      6. distribute-lft1-in 11.0%

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

      7. fma-def 11.0%

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

      8. *-commutative 11.0%

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

      9. associate-/l* 10.5%

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

      10. associate-/r/ 11.0%

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

      11. distribute-lft1-in 11.0%

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

      12. fma-def 11.0%

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

      13. *-commutative 11.0%

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

   3. Simplified 11.0%

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

   4. Taylor expanded in t around 0 14.2%

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

   5. Taylor expanded in t around 0 19.1%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+49}:\\ \;\;\;\;\cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \cdot \left(x \cdot \cos \left(\frac{\sqrt[3]{t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)} \cdot \left(\sqrt[3]{t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)} \cdot \sqrt[3]{t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)}\right)}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2: 30.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y \cdot 2, z, z\right)}}}\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 3.5 \cdot 10^{-69}:\\ \;\;\;\;x \cdot \left(\cos \left(t_1 \cdot {t_1}^{2}\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cbrt (/ t (/ 16.0 (fma (* y 2.0) z z))))))
   (if (<=
        (*
         (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
         (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
        3.5e-69)
     (*
      x
      (* (cos (* t_1 (pow t_1 2.0))) (cos (* (/ t 16.0) (fma (* 2.0 a) b b)))))
     x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cbrt((t / (16.0 / fma((y * 2.0), z, z))));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 3.5e-69) {
		tmp = x * (cos((t_1 * pow(t_1, 2.0))) * cos(((t / 16.0) * fma((2.0 * a), b, b))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = cbrt(Float64(t / Float64(16.0 / fma(Float64(y * 2.0), z, z))))
	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))) <= 3.5e-69)
		tmp = Float64(x * Float64(cos(Float64(t_1 * (t_1 ^ 2.0))) * cos(Float64(Float64(t / 16.0) * fma(Float64(2.0 * a), b, b)))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[N[(t / N[(16.0 / N[(N[(y * 2.0), $MachinePrecision] * z + z), $MachinePrecision]), $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[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.5e-69], N[(x * N[(N[Cos[N[(t$95$1 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t / 16.0), $MachinePrecision] * N[(N[(2.0 * a), $MachinePrecision] * b + b), $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))) < 3.5000000000000001e-69

   1. Initial program 54.9%

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

      1. associate-*l* 54.9%

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

      2. *-commutative 54.9%

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

      3. *-commutative 54.9%

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

      4. associate-*l/ 54.9%

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

      5. *-commutative 54.9%

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

      6. distribute-lft1-in 54.9%

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

      7. fma-def 54.9%

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

      8. *-commutative 54.9%

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

      9. associate-/l* 55.0%

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

      10. associate-/r/ 54.9%

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

      11. distribute-lft1-in 55.2%

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

      12. fma-def 54.9%

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

      13. *-commutative 54.9%

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

   3. Simplified 54.9%

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

      1. associate-*r/ 54.9%

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

      2. *-commutative 54.9%

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

   5. Applied egg-rr 54.9%

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

      1. associate-/l* 54.8%

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

   7. Simplified 54.8%

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

      1. add-cube-cbrt 55.4%

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

      2. pow2 55.4%

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

   9. Applied egg-rr 55.4%

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

   if 3.5000000000000001e-69 < (*.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 14.1%

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

      1. associate-*l* 14.1%

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

      2. *-commutative 14.1%

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

      3. *-commutative 14.1%

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

      4. associate-*l/ 14.1%

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

      5. *-commutative 14.1%

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

      6. distribute-lft1-in 14.1%

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

      7. fma-def 14.1%

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

      8. *-commutative 14.1%

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

      9. associate-/l* 13.5%

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

      10. associate-/r/ 14.1%

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

      11. distribute-lft1-in 14.0%

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

      12. fma-def 14.0%

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

      13. *-commutative 14.0%

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

   3. Simplified 14.0%

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

   4. Taylor expanded in t around 0 16.6%

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

   5. Taylor expanded in t around 0 21.4%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 3.5 \cdot 10^{-69}:\\ \;\;\;\;x \cdot \left(\cos \left(\sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y \cdot 2, z, z\right)}}} \cdot {\left(\sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y \cdot 2, z, z\right)}}}\right)}^{2}\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 31.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+255}:\\ \;\;\;\;x \cdot \left(\cos \left(t \cdot \frac{\mathsf{fma}\left(y \cdot 2, z, z\right)}{16}\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(b + 2 \cdot \left(a \cdot b\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
      5e+255)
   (*
    x
    (*
     (cos (* t (/ (fma (* y 2.0) z z) 16.0)))
     (cos (* 0.0625 (* t (+ b (* 2.0 (* 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 * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 5e+255) {
		tmp = x * (cos((t * (fma((y * 2.0), z, z) / 16.0))) * cos((0.0625 * (t * (b + (2.0 * (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(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 5e+255)
		tmp = Float64(x * Float64(cos(Float64(t * Float64(fma(Float64(y * 2.0), z, z) / 16.0))) * cos(Float64(0.0625 * Float64(t * Float64(b + Float64(2.0 * 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[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+255], N[(x * N[(N[Cos[N[(t * N[(N[(N[(y * 2.0), $MachinePrecision] * z + z), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.0625 * N[(t * N[(b + N[(2.0 * N[(a * b), $MachinePrecision]), $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))) < 5.0000000000000002e255

   1. Initial program 50.6%

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

      1. associate-*l* 50.6%

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

      2. *-commutative 50.6%

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

      3. *-commutative 50.6%

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

      4. associate-*l/ 50.6%

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

      5. *-commutative 50.6%

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

      6. distribute-lft1-in 50.6%

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

      7. fma-def 50.6%

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

      8. *-commutative 50.6%

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

      9. associate-/l* 50.2%

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

      10. associate-/r/ 50.6%

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

      11. distribute-lft1-in 50.7%

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

      12. fma-def 50.5%

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

      13. *-commutative 50.5%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in t around 0 50.7%

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

   if 5.0000000000000002e255 < (*.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 4.6%

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

      1. associate-*l* 4.6%

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

      2. *-commutative 4.6%

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

      3. *-commutative 4.6%

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

      4. associate-*l/ 4.6%

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

      5. *-commutative 4.6%

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

      6. distribute-lft1-in 4.6%

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

      7. fma-def 4.6%

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

      8. *-commutative 4.6%

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

      9. associate-/l* 4.5%

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

      10. associate-/r/ 4.6%

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

      11. distribute-lft1-in 4.6%

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

      12. fma-def 4.6%

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

      13. *-commutative 4.6%

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

   3. Simplified 4.6%

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

   4. Taylor expanded in t around 0 8.9%

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

   5. Taylor expanded in t around 0 14.7%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+255}:\\ \;\;\;\;x \cdot \left(\cos \left(t \cdot \frac{\mathsf{fma}\left(y \cdot 2, z, z\right)}{16}\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(b + 2 \cdot \left(a \cdot b\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 31.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

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 (* (+ 1.0 (* 2.0 a)) b)) 16.0)))))
   (if (<= t_1 5e+255) 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 * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (t_1 <= 5e+255) {
		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 * ((1.0d0 + (2.0d0 * a)) * b)) / 16.0d0))
    if (t_1 <= 5d+255) 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 * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (t_1 <= 5e+255) {
		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 * ((1.0 + (2.0 * a)) * b)) / 16.0))
	tmp = 0
	if t_1 <= 5e+255:
		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(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0)))
	tmp = 0.0
	if (t_1 <= 5e+255)
		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 * ((1.0 + (2.0 * a)) * b)) / 16.0));
	tmp = 0.0;
	if (t_1 <= 5e+255)
		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[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+255], 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))) < 5.0000000000000002e255

   1. Initial program 50.6%

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

   if 5.0000000000000002e255 < (*.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 4.6%

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

      1. associate-*l* 4.6%

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

      2. *-commutative 4.6%

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

      3. *-commutative 4.6%

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

      4. associate-*l/ 4.6%

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

      5. *-commutative 4.6%

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

      6. distribute-lft1-in 4.6%

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

      7. fma-def 4.6%

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

      8. *-commutative 4.6%

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

      9. associate-/l* 4.5%

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

      10. associate-/r/ 4.6%

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

      11. distribute-lft1-in 4.6%

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

      12. fma-def 4.6%

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

      13. *-commutative 4.6%

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

   3. Simplified 4.6%

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

   4. Taylor expanded in t around 0 8.9%

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

   5. Taylor expanded in t around 0 14.7%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 28.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.5%

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

2. Taylor expanded in y around 0 31.0%

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

3. Taylor expanded in t around 0 31.9%

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

4. Taylor expanded in a around 0 33.1%

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

5. Final simplification 33.1%

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

Alternative 6: 30.1% 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 30.5%

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

   1. associate-*l* 30.5%

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

   2. *-commutative 30.5%

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

   3. *-commutative 30.5%

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

   4. associate-*l/ 30.5%

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

   5. *-commutative 30.5%

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

   6. distribute-lft1-in 30.5%

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

   7. fma-def 30.5%

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

   8. *-commutative 30.5%

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

   9. associate-/l* 30.2%

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

   10. associate-/r/ 30.5%

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

   11. distribute-lft1-in 30.6%

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

   12. fma-def 30.4%

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

   13. *-commutative 30.4%

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

3. Simplified 30.4%

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

4. Taylor expanded in t around 0 30.4%

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

5. Taylor expanded in t around 0 33.1%

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

6. Final simplification 33.1%

   \[\leadsto x \]

Developer target: 29.5% 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 2023271 
(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))))