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

Percentage Accurate: 27.5% → 32.0%

Time: 18.9s

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: 27.5% 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, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+222}:\\ \;\;\;\;t\_1 \cdot \cos \left(\frac{{\left(\sqrt[3]{\mathsf{fma}\left(2, a, 1\right) \cdot \left(t \cdot b\right)}\right)}^{3}}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (let* ((t_1 (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))))
   (*
    x_s
    (if (<= (* t_1 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))) 1e+222)
      (* t_1 (cos (/ (pow (cbrt (* (fma 2.0 a 1.0) (* t b))) 3.0) 16.0)))
      x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = x_m * 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+222) {
		tmp = t_1 * cos((pow(cbrt((fma(2.0, a, 1.0) * (t * b))), 3.0) / 16.0));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	t_1 = Float64(x_m * 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+222)
		tmp = Float64(t_1 * cos(Float64((cbrt(Float64(fma(2.0, a, 1.0) * Float64(t * b))) ^ 3.0) / 16.0)));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * 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+222], 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$95$m]), $MachinePrecision]]

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)))) < 1e222

   1. Initial program 52.9%

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

      1. add-cube-cbrt 53.2%

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

      2. pow3 53.2%

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

      3. associate-*l* 53.5%

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

      4. *-commutative 53.5%

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

      5. fma-undefine 53.5%

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

   4. Applied egg-rr 53.5%

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

   if 1e222 < (*.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 2.2%

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

      1. associate-*l* 2.2%

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

      2. *-commutative 2.2%

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

      3. *-commutative 2.2%

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

      4. associate-/l* 2.2%

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

      5. fma-define 2.2%

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

      6. associate-/l* 2.2%

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

      7. fma-define 2.2%

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

   3. Simplified 2.2%

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

   5. Taylor expanded in t around 0 13.4%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+222}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{{\left(\sqrt[3]{\mathsf{fma}\left(2, a, 1\right) \cdot \left(t \cdot b\right)}\right)}^{3}}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 32.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+222}:\\ \;\;\;\;t\_1 \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + {\left(\sqrt[3]{2 \cdot a}\right)}^{3}\right)\right)}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (let* ((t_1 (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))))
   (*
    x_s
    (if (<= (* t_1 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))) 1e+222)
      (* t_1 (cos (/ (* t (* b (+ 1.0 (pow (cbrt (* 2.0 a)) 3.0)))) 16.0)))
      x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = x_m * 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+222) {
		tmp = t_1 * cos(((t * (b * (1.0 + pow(cbrt((2.0 * a)), 3.0)))) / 16.0));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = x_m * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0));
	double tmp;
	if ((t_1 * Math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+222) {
		tmp = t_1 * Math.cos(((t * (b * (1.0 + Math.pow(Math.cbrt((2.0 * a)), 3.0)))) / 16.0));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	t_1 = Float64(x_m * 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+222)
		tmp = Float64(t_1 * cos(Float64(Float64(t * Float64(b * Float64(1.0 + (cbrt(Float64(2.0 * a)) ^ 3.0)))) / 16.0)));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * 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+222], N[(t$95$1 * N[Cos[N[(N[(t * N[(b * N[(1.0 + N[Power[N[Power[N[(2.0 * a), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]]

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)))) < 1e222

   1. Initial program 52.9%

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

      1. add-cube-cbrt 52.9%

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

      2. pow3 53.5%

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

      3. *-commutative 53.5%

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

   4. Applied egg-rr 53.5%

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

   if 1e222 < (*.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 2.2%

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

      1. associate-*l* 2.2%

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

      2. *-commutative 2.2%

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

      3. *-commutative 2.2%

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

      4. associate-/l* 2.2%

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

      5. fma-define 2.2%

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

      6. associate-/l* 2.2%

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

      7. fma-define 2.2%

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

   3. Simplified 2.2%

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

   5. Taylor expanded in t around 0 13.4%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+222}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + {\left(\sqrt[3]{2 \cdot a}\right)}^{3}\right)\right)}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 32.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;x\_m \cdot \left(\cos \left(\left(z \cdot \mathsf{fma}\left(y, 2, 1\right)\right) \cdot \frac{t}{16}\right) \cdot \cos \left(\frac{1}{\frac{16}{\mathsf{fma}\left(2, a, 1\right) \cdot \left(t \cdot b\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(e^{\log \left(\mathsf{log1p}\left(x\_m\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
        (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
       5e+302)
    (*
     x_m
     (*
      (cos (* (* z (fma y 2.0 1.0)) (/ t 16.0)))
      (cos (/ 1.0 (/ 16.0 (* (fma 2.0 a 1.0) (* t b)))))))
    (expm1 (exp (log (log1p x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 5e+302) {
		tmp = x_m * (cos(((z * fma(y, 2.0, 1.0)) * (t / 16.0))) * cos((1.0 / (16.0 / (fma(2.0, a, 1.0) * (t * b))))));
	} else {
		tmp = expm1(exp(log(log1p(x_m))));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x_m * 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+302)
		tmp = Float64(x_m * Float64(cos(Float64(Float64(z * fma(y, 2.0, 1.0)) * Float64(t / 16.0))) * cos(Float64(1.0 / Float64(16.0 / Float64(fma(2.0, a, 1.0) * Float64(t * b)))))));
	else
		tmp = expm1(exp(log(log1p(x_m))));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * 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+302], N[(x$95$m * N[(N[Cos[N[(N[(z * N[(y * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(t / 16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(1.0 / N[(16.0 / N[(N[(2.0 * a + 1.0), $MachinePrecision] * N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Exp[N[Exp[N[Log[N[Log[1 + x$95$m], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]]), $MachinePrecision]

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)))) < 5e302

   1. Initial program 52.4%

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

      1. associate-*l* 52.4%

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

      2. *-commutative 52.4%

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

      3. *-commutative 52.4%

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

      4. associate-/l* 52.4%

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

      5. fma-define 52.4%

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

      6. associate-/l* 52.4%

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

      7. fma-define 52.4%

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

   3. Simplified 52.4%

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

      1. associate-*r/ 52.4%

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

      2. fma-define 52.4%

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

      3. clear-num 52.4%

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

      4. associate-*l* 52.5%

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

      5. *-commutative 52.5%

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

      6. fma-undefine 52.5%

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

   6. Applied egg-rr 52.5%

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

   if 5e302 < (*.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 0.0%

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

   3. Simplified 2.3%

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

      1. expm1-log1p-u 1.6%

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

      2. associate-*r* 1.6%

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

      3. metadata-eval 1.6%

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

      4. div-inv 1.6%

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

      5. associate-*r* 1.8%

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

      6. *-commutative 1.8%

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

      7. div-inv 1.8%

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

      8. metadata-eval 1.8%

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

   6. Applied egg-rr 1.8%

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

   7. Taylor expanded in t around 0 5.6%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + x\right)}\right) \]
   8. Step-by-step derivation

      1. log1p-define 8.9%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x\right)}\right) \]

   9. Simplified 8.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x\right)}\right) \]
   10. Step-by-step derivation

       1. add-exp-log 6.6%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{e^{\log \left(\mathsf{log1p}\left(x\right)\right)}}\right) \]

   11. Applied egg-rr 6.6%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{e^{\log \left(\mathsf{log1p}\left(x\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;x \cdot \left(\cos \left(\left(z \cdot \mathsf{fma}\left(y, 2, 1\right)\right) \cdot \frac{t}{16}\right) \cdot \cos \left(\frac{1}{\frac{16}{\mathsf{fma}\left(2, a, 1\right) \cdot \left(t \cdot b\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(e^{\log \left(\mathsf{log1p}\left(x\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 32.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(e^{\log \left(\mathsf{log1p}\left(x\_m\right)\right)}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (let* ((t_1
         (*
          (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
          (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))))
   (* x_s (if (<= t_1 5e+302) t_1 (expm1 (exp (log (log1p x_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = (x_m * 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+302) {
		tmp = t_1;
	} else {
		tmp = expm1(exp(log(log1p(x_m))));
	}
	return x_s * tmp;
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = (x_m * 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+302) {
		tmp = t_1;
	} else {
		tmp = Math.expm1(Math.exp(Math.log(Math.log1p(x_m))));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	t_1 = (x_m * 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+302:
		tmp = t_1
	else:
		tmp = math.expm1(math.exp(math.log(math.log1p(x_m))))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	t_1 = Float64(Float64(x_m * 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+302)
		tmp = t_1;
	else
		tmp = expm1(exp(log(log1p(x_m))));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x$95$m * 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]}, N[(x$95$s * If[LessEqual[t$95$1, 5e+302], t$95$1, N[(Exp[N[Exp[N[Log[N[Log[1 + x$95$m], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]]), $MachinePrecision]]

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)))) < 5e302

   1. Initial program 52.4%

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

   if 5e302 < (*.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 0.0%

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

   3. Simplified 2.3%

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

      1. expm1-log1p-u 1.6%

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

      2. associate-*r* 1.6%

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

      3. metadata-eval 1.6%

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

      4. div-inv 1.6%

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

      5. associate-*r* 1.8%

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

      6. *-commutative 1.8%

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

      7. div-inv 1.8%

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

      8. metadata-eval 1.8%

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

   6. Applied egg-rr 1.8%

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

   7. Taylor expanded in t around 0 5.6%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + x\right)}\right) \]
   8. Step-by-step derivation

      1. log1p-define 8.9%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x\right)}\right) \]

   9. Simplified 8.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x\right)}\right) \]
   10. Step-by-step derivation

       1. add-exp-log 6.6%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{e^{\log \left(\mathsf{log1p}\left(x\right)\right)}}\right) \]

   11. Applied egg-rr 6.6%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{e^{\log \left(\mathsf{log1p}\left(x\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(e^{\log \left(\mathsf{log1p}\left(x\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(x\_m\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (let* ((t_1
         (*
          (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
          (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))))
   (* x_s (if (<= t_1 5e+302) t_1 (expm1 (log1p x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = (x_m * 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+302) {
		tmp = t_1;
	} else {
		tmp = expm1(log1p(x_m));
	}
	return x_s * tmp;
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = (x_m * 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+302) {
		tmp = t_1;
	} else {
		tmp = Math.expm1(Math.log1p(x_m));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	t_1 = (x_m * 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+302:
		tmp = t_1
	else:
		tmp = math.expm1(math.log1p(x_m))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	t_1 = Float64(Float64(x_m * 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+302)
		tmp = t_1;
	else
		tmp = expm1(log1p(x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x$95$m * 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]}, N[(x$95$s * If[LessEqual[t$95$1, 5e+302], t$95$1, N[(Exp[N[Log[1 + x$95$m], $MachinePrecision]] - 1), $MachinePrecision]]), $MachinePrecision]]

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)))) < 5e302

   1. Initial program 52.4%

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

   if 5e302 < (*.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 0.0%

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

   3. Simplified 2.3%

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

      1. expm1-log1p-u 1.6%

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

      2. associate-*r* 1.6%

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

      3. metadata-eval 1.6%

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

      4. div-inv 1.6%

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

      5. associate-*r* 1.8%

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

      6. *-commutative 1.8%

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

      7. div-inv 1.8%

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

      8. metadata-eval 1.8%

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

   6. Applied egg-rr 1.8%

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

   7. Taylor expanded in t around 0 5.6%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + x\right)}\right) \]
   8. Step-by-step derivation

      1. log1p-define 8.9%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x\right)}\right) \]

   9. Simplified 8.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 2.25 \cdot 10^{-51}:\\ \;\;\;\;x\_m \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(a \cdot -2 + -1\right)\right)\right)\right) \cdot \cos \left(0.0625 \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<= t 2.25e-51)
    (*
     x_m
     (*
      (cos (* 0.0625 (* b (* t (+ (* a -2.0) -1.0)))))
      (cos (* 0.0625 (* z t)))))
    x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 2.25e-51) {
		tmp = x_m * (cos((0.0625 * (b * (t * ((a * -2.0) + -1.0))))) * cos((0.0625 * (z * t))));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= 2.25d-51) then
        tmp = x_m * (cos((0.0625d0 * (b * (t * ((a * (-2.0d0)) + (-1.0d0)))))) * cos((0.0625d0 * (z * t))))
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 2.25e-51) {
		tmp = x_m * (Math.cos((0.0625 * (b * (t * ((a * -2.0) + -1.0))))) * Math.cos((0.0625 * (z * t))));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if t <= 2.25e-51:
		tmp = x_m * (math.cos((0.0625 * (b * (t * ((a * -2.0) + -1.0))))) * math.cos((0.0625 * (z * t))))
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (t <= 2.25e-51)
		tmp = Float64(x_m * 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_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 2.25e-51)
		tmp = x_m * (cos((0.0625 * (b * (t * ((a * -2.0) + -1.0))))) * cos((0.0625 * (z * t))));
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[t, 2.25e-51], N[(x$95$m * 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$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.24999999999999987e-51

   1. Initial program 36.3%

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

   3. Simplified 37.3%

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

   5. Taylor expanded in y around 0 36.7%

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

   if 2.24999999999999987e-51 < t

   1. Initial program 13.6%

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

      1. associate-*l* 13.6%

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

      2. *-commutative 13.6%

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

      3. *-commutative 13.6%

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

      4. associate-/l* 13.6%

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

      5. fma-define 13.6%

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

      6. associate-/l* 13.6%

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

      7. fma-define 13.6%

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

   3. Simplified 13.6%

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

   5. Taylor expanded in t around 0 19.9%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.25 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(a \cdot -2 + -1\right)\right)\right)\right) \cdot \cos \left(0.0625 \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 31.1% accurate, 225.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b) :precision binary64 (* x_s x_m))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * x_m;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_s * x_m
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * x_m;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	return x_s * x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	return Float64(x_s * x_m)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t, a, b)
	tmp = x_s * x_m;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * x$95$m), $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. 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%

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

   5. fma-define 30.5%

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

   6. associate-/l* 30.5%

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

   7. fma-define 30.5%

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

3. Simplified 30.5%

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

5. Taylor expanded in t around 0 32.8%

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

Developer Target 1: 30.5% 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 2024135 
(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))))