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

Percentage Accurate: 27.6% → 32.0%

Time: 19.4s

Alternatives: 7

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 27.6% 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: 32.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\left(t \cdot \left(b \cdot \mathsf{fma}\left(2, a, 1\right)\right)\right) \cdot 0.0625}\\ t_2 := x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\\ \mathbf{if}\;t\_2 \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+283}:\\ \;\;\;\;t\_2 \cdot \cos \left(t\_1 \cdot {t\_1}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cbrt (* (* t (* b (fma 2.0 a 1.0))) 0.0625)))
        (t_2 (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))))
   (if (<= (* t_2 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))) 2e+283)
     (* t_2 (cos (* t_1 (pow t_1 2.0))))
     x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cbrt(((t * (b * fma(2.0, a, 1.0))) * 0.0625));
	double t_2 = x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0));
	double tmp;
	if ((t_2 * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 2e+283) {
		tmp = t_2 * cos((t_1 * pow(t_1, 2.0)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = cbrt(Float64(Float64(t * Float64(b * fma(2.0, a, 1.0))) * 0.0625))
	t_2 = Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0)))
	tmp = 0.0
	if (Float64(t_2 * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 2e+283)
		tmp = Float64(t_2 * cos(Float64(t_1 * (t_1 ^ 2.0))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[N[(N[(t * N[(b * N[(2.0 * a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.0625), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[N[(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]), $MachinePrecision], 2e+283], N[(t$95$2 * N[Cos[N[(t$95$1 * N[Power[t$95$1, 2.0], $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 #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1.99999999999999991e283

   1. Initial program 52.7%

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

      1. expm1-log1p-u 49.9%

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

      2. associate-*l* 49.9%

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

      3. *-commutative 49.9%

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

      4. fma-undefine 49.9%

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

   4. Applied egg-rr 49.9%

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

      1. add-cube-cbrt 50.0%

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(b \cdot t\right)\right)\right)}{16}} \cdot \sqrt[3]{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(b \cdot t\right)\right)\right)}{16}}\right) \cdot \sqrt[3]{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(b \cdot t\right)\right)\right)}{16}}\right)} \]

      2. pow2 50.0%

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\color{blue}{{\left(\sqrt[3]{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(b \cdot t\right)\right)\right)}{16}}\right)}^{2}} \cdot \sqrt[3]{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(b \cdot t\right)\right)\right)}{16}}\right) \]

      3. div-inv 50.0%

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left({\left(\sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(b \cdot t\right)\right)\right) \cdot \frac{1}{16}}}\right)}^{2} \cdot \sqrt[3]{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(b \cdot t\right)\right)\right)}{16}}\right) \]

      4. expm1-log1p-u 50.5%

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left({\left(\sqrt[3]{\color{blue}{\left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(b \cdot t\right)\right)} \cdot \frac{1}{16}}\right)}^{2} \cdot \sqrt[3]{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(b \cdot t\right)\right)\right)}{16}}\right) \]

      5. associate-*r* 50.6%

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left({\left(\sqrt[3]{\color{blue}{\left(\left(\mathsf{fma}\left(2, a, 1\right) \cdot b\right) \cdot t\right)} \cdot \frac{1}{16}}\right)}^{2} \cdot \sqrt[3]{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(b \cdot t\right)\right)\right)}{16}}\right) \]

      6. metadata-eval 50.6%

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left({\left(\sqrt[3]{\left(\left(\mathsf{fma}\left(2, a, 1\right) \cdot b\right) \cdot t\right) \cdot \color{blue}{0.0625}}\right)}^{2} \cdot \sqrt[3]{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(b \cdot t\right)\right)\right)}{16}}\right) \]

      7. div-inv 50.6%

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left({\left(\sqrt[3]{\left(\left(\mathsf{fma}\left(2, a, 1\right) \cdot b\right) \cdot t\right) \cdot 0.0625}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(b \cdot t\right)\right)\right) \cdot \frac{1}{16}}}\right) \]

      8. expm1-log1p-u 53.3%

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

      9. associate-*r* 53.1%

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

      10. metadata-eval 53.1%

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

   6. Applied egg-rr 53.1%

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

   if 1.99999999999999991e283 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 2.8%

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

   3. Simplified 3.8%

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

   5. Taylor expanded in t around 0 12.1%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\sqrt[3]{\left(t \cdot \left(b \cdot \mathsf{fma}\left(2, a, 1\right)\right)\right) \cdot 0.0625} \cdot {\left(\sqrt[3]{\left(t \cdot \left(b \cdot \mathsf{fma}\left(2, a, 1\right)\right)\right) \cdot 0.0625}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 31.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{0.0625 \cdot \left(t \cdot b\right)}\\ t_2 := x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\\ \mathbf{if}\;t\_2 \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+269}:\\ \;\;\;\;t\_2 \cdot \cos \left(t\_1 \cdot {t\_1}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cbrt (* 0.0625 (* t b))))
        (t_2 (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))))
   (if (<= (* t_2 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))) 1e+269)
     (* t_2 (cos (* t_1 (pow t_1 2.0))))
     x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cbrt((0.0625 * (t * b)));
	double t_2 = x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0));
	double tmp;
	if ((t_2 * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+269) {
		tmp = t_2 * cos((t_1 * pow(t_1, 2.0)));
	} else {
		tmp = x;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.cbrt((0.0625 * (t * b)));
	double t_2 = x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0));
	double tmp;
	if ((t_2 * Math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+269) {
		tmp = t_2 * Math.cos((t_1 * Math.pow(t_1, 2.0)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = cbrt(Float64(0.0625 * Float64(t * b)))
	t_2 = Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0)))
	tmp = 0.0
	if (Float64(t_2 * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 1e+269)
		tmp = Float64(t_2 * cos(Float64(t_1 * (t_1 ^ 2.0))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[N[(0.0625 * N[(t * b), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[N[(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]), $MachinePrecision], 1e+269], N[(t$95$2 * N[Cos[N[(t$95$1 * N[Power[t$95$1, 2.0], $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 #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1e269

   1. Initial program 52.8%

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

      1. expm1-log1p-u 49.9%

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

      2. associate-*l* 49.9%

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

      3. *-commutative 49.9%

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

      4. fma-undefine 49.9%

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

   4. Applied egg-rr 49.9%

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

   5. Taylor expanded in a around 0 50.8%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\mathsf{expm1}\left(\color{blue}{\log \left(1 + b \cdot t\right)}\right)}{16}\right) \]
   6. Step-by-step derivation

      1. log1p-define 50.8%

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

   7. Simplified 50.8%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(b \cdot t\right)}\right)}{16}\right) \]
   8. Step-by-step derivation

      1. expm1-log1p-u 51.7%

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

      2. add-cube-cbrt 53.1%

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

      3. pow2 53.1%

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

      4. div-inv 53.1%

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

      5. metadata-eval 53.1%

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

      6. div-inv 53.1%

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

      7. metadata-eval 53.1%

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

   9. Applied egg-rr 53.1%

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

   if 1e269 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 4.5%

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

   3. Simplified 5.4%

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

   5. Taylor expanded in t around 0 13.6%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+269}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\sqrt[3]{0.0625 \cdot \left(t \cdot b\right)} \cdot {\left(\sqrt[3]{0.0625 \cdot \left(t \cdot b\right)}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 31.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\\ \mathbf{if}\;t\_1 \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+151}:\\ \;\;\;\;t\_1 \cdot \cos \left(\frac{{\left(\sqrt[3]{\mathsf{fma}\left(2, a, 1\right) \cdot \left(t \cdot b\right)}\right)}^{3}}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))))
   (if (<= (* t_1 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))) 1e+151)
     (* t_1 (cos (/ (pow (cbrt (* (fma 2.0 a 1.0) (* t b))) 3.0) 16.0)))
     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));
	double tmp;
	if ((t_1 * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+151) {
		tmp = t_1 * cos((pow(cbrt((fma(2.0, a, 1.0) * (t * b))), 3.0) / 16.0));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0)))
	tmp = 0.0
	if (Float64(t_1 * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 1e+151)
		tmp = Float64(t_1 * cos(Float64((cbrt(Float64(fma(2.0, a, 1.0) * Float64(t * b))) ^ 3.0) / 16.0)));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[LessEqual[N[(t$95$1 * N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+151], N[(t$95$1 * N[Cos[N[(N[Power[N[Power[N[(N[(2.0 * a + 1.0), $MachinePrecision] * N[(t * b), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1.00000000000000002e151

   1. Initial program 52.7%

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

      1. add-cube-cbrt 52.5%

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

      2. pow3 52.8%

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

      3. associate-*l* 52.8%

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

      4. *-commutative 52.8%

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

      5. fma-undefine 52.8%

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

   4. Applied egg-rr 52.8%

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

   if 1.00000000000000002e151 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 10.1%

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

   3. Simplified 10.8%

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

   5. Taylor expanded in t around 0 18.0%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+151}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{{\left(\sqrt[3]{\mathsf{fma}\left(2, a, 1\right) \cdot \left(t \cdot b\right)}\right)}^{3}}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 31.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+151}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \frac{t}{16}\right)\right) \cdot \left(x \cdot \cos \left(t \cdot \left(\mathsf{fma}\left(2, a, 1\right) \cdot \frac{b}{16}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

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+151)
   (*
    (cos (* (fma y 2.0 1.0) (* z (/ t 16.0))))
    (* x (cos (* t (* (fma 2.0 a 1.0) (/ b 16.0))))))
   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+151) {
		tmp = cos((fma(y, 2.0, 1.0) * (z * (t / 16.0)))) * (x * cos((t * (fma(2.0, a, 1.0) * (b / 16.0)))));
	} 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+151)
		tmp = Float64(cos(Float64(fma(y, 2.0, 1.0) * Float64(z * Float64(t / 16.0)))) * Float64(x * cos(Float64(t * Float64(fma(2.0, a, 1.0) * Float64(b / 16.0))))));
	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+151], N[(N[Cos[N[(N[(y * 2.0 + 1.0), $MachinePrecision] * N[(z * N[(t / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * N[Cos[N[(t * N[(N[(2.0 * a + 1.0), $MachinePrecision] * N[(b / 16.0), $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 #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1.00000000000000002e151

   1. Initial program 52.7%

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

   3. Simplified 52.8%

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

   if 1.00000000000000002e151 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 10.1%

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

   3. Simplified 10.8%

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

   5. Taylor expanded in t around 0 18.0%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+151}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \frac{t}{16}\right)\right) \cdot \left(x \cdot \cos \left(t \cdot \left(\mathsf{fma}\left(2, a, 1\right) \cdot \frac{b}{16}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 31.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{if}\;t\_1 \leq 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

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 1e+151) 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 <= 1e+151) {
		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 <= 1d+151) 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 <= 1e+151) {
		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 <= 1e+151:
		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 <= 1e+151)
		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 <= 1e+151)
		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, 1e+151], t$95$1, x]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1.00000000000000002e151

   1. Initial program 52.7%

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

   if 1.00000000000000002e151 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 10.1%

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

   3. Simplified 10.8%

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

   5. Taylor expanded in t around 0 18.0%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+151}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.4 \cdot 10^{-59}:\\ \;\;\;\;x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(a \cdot -2 + -1\right)\right)\right)\right) \cdot \cos \left(0.0625 \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 5.4e-59)
   (*
    x
    (*
     (cos (* 0.0625 (* b (* t (+ (* a -2.0) -1.0)))))
     (cos (* 0.0625 (* z t)))))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 5.4e-59) {
		tmp = x * (cos((0.0625 * (b * (t * ((a * -2.0) + -1.0))))) * cos((0.0625 * (z * t))));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= 5.4d-59) then
        tmp = x * (cos((0.0625d0 * (b * (t * ((a * (-2.0d0)) + (-1.0d0)))))) * cos((0.0625d0 * (z * t))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 5.4e-59) {
		tmp = x * (Math.cos((0.0625 * (b * (t * ((a * -2.0) + -1.0))))) * Math.cos((0.0625 * (z * t))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 5.4e-59:
		tmp = x * (math.cos((0.0625 * (b * (t * ((a * -2.0) + -1.0))))) * math.cos((0.0625 * (z * t))))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 5.4e-59)
		tmp = Float64(x * Float64(cos(Float64(0.0625 * Float64(b * Float64(t * Float64(Float64(a * -2.0) + -1.0))))) * cos(Float64(0.0625 * Float64(z * t)))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 5.4e-59)
		tmp = x * (cos((0.0625 * (b * (t * ((a * -2.0) + -1.0))))) * cos((0.0625 * (z * t))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 5.4e-59], N[(x * N[(N[Cos[N[(0.0625 * N[(b * N[(t * N[(N[(a * -2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < 5.3999999999999998e-59

   1. Initial program 38.6%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in y around 0 39.0%

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

   if 5.3999999999999998e-59 < t

   1. Initial program 10.4%

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

   3. Simplified 10.2%

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

   5. Taylor expanded in t around 0 15.3%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.4 \cdot 10^{-59}:\\ \;\;\;\;x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(a \cdot -2 + -1\right)\right)\right)\right) \cdot \cos \left(0.0625 \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 30.6% 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 30.9%

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

3. Simplified 31.1%

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

5. Taylor expanded in t around 0 32.9%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 30.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024137 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (* x (cos (* (/ b 16) (/ t (+ (- 1 (* a 2)) (pow (* a 2) 2)))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))