Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 99.7%

Time: 7.2s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot \frac{\sin y}{y}}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))

C

double code(double x, double y, double z) {
	return (x * (sin(y) / y)) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (sin(y) / y)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * (Math.sin(y) / y)) / z;
}

Python

def code(x, y, z):
	return (x * (math.sin(y) / y)) / z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(sin(y) / y)) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (sin(y) / y)) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \frac{\sin y}{y}}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))

C

double code(double x, double y, double z) {
	return (x * (sin(y) / y)) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (sin(y) / y)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * (Math.sin(y) / y)) / z;
}

Python

def code(x, y, z):
	return (x * (math.sin(y) / y)) / z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(sin(y) / y)) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (sin(y) / y)) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x\_m}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y} \cdot x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2e-15) (/ x_m (* z (/ y (sin y)))) (/ (* (/ (sin y) y) x_m) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2e-15) {
		tmp = x_m / (z * (y / sin(y)));
	} else {
		tmp = ((sin(y) / y) * x_m) / z;
	}
	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)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 2d-15) then
        tmp = x_m / (z * (y / sin(y)))
    else
        tmp = ((sin(y) / y) * x_m) / z
    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 tmp;
	if (x_m <= 2e-15) {
		tmp = x_m / (z * (y / Math.sin(y)));
	} else {
		tmp = ((Math.sin(y) / y) * x_m) / z;
	}
	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):
	tmp = 0
	if x_m <= 2e-15:
		tmp = x_m / (z * (y / math.sin(y)))
	else:
		tmp = ((math.sin(y) / y) * x_m) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 2e-15)
		tmp = Float64(x_m / Float64(z * Float64(y / sin(y))));
	else
		tmp = Float64(Float64(Float64(sin(y) / y) * x_m) / z);
	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)
	tmp = 0.0;
	if (x_m <= 2e-15)
		tmp = x_m / (z * (y / sin(y)));
	else
		tmp = ((sin(y) / y) * x_m) / z;
	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_] := N[(x$95$s * If[LessEqual[x$95$m, 2e-15], N[(x$95$m / N[(z * N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000002e-15

   1. Initial program 96.2%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sin y}{y}} \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

      6. div-inv N/A

         \[\leadsto \frac{x}{\color{blue}{z \cdot \frac{1}{\frac{\sin y}{y}}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{x}{z \cdot \frac{1}{\color{blue}{\frac{\sin y}{y}}}} \]

      8. clear-num N/A

         \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{y}{\sin y}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

      11. lower-/.f64 96.3

          \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y}} \cdot z} \]

   4. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y} \cdot z}} \]

   if 2.0000000000000002e-15 < x

   1. Initial program 99.6%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.5:\\ \;\;\;\;\frac{x\_m}{y} \cdot \frac{\sin y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 0.5)
    (* (/ x_m y) (/ (sin y) z))
    (/
     x_m
     (*
      (fma (fma 0.019444444444444445 (* y y) 0.16666666666666666) (* y y) 1.0)
      z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 0.5) {
		tmp = (x_m / y) * (sin(y) / z);
	} else {
		tmp = x_m / (fma(fma(0.019444444444444445, (y * y), 0.16666666666666666), (y * y), 1.0) * z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.5)
		tmp = Float64(Float64(x_m / y) * Float64(sin(y) / z));
	else
		tmp = Float64(x_m / Float64(fma(fma(0.019444444444444445, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * z));
	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_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.5], N[(N[(x$95$m / y), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(N[(0.019444444444444445 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.5

   1. Initial program 94.6%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y} \cdot x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y}} \cdot x}{z} \]

      5. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y \cdot \frac{1}{y}\right)} \cdot x}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sin y \cdot \left(\frac{1}{y} \cdot x\right)}}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z}} \cdot \left(\frac{1}{y} \cdot x\right) \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{1 \cdot x}{y}} \]

      11. *-lft-identity N/A

          \[\leadsto \frac{\sin y}{z} \cdot \frac{\color{blue}{x}}{y} \]

      12. lower-/.f64 94.7

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{x}{y}} \]

   4. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \frac{x}{y}} \]

   if 0.5 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sin y}{y}} \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

      6. div-inv N/A

         \[\leadsto \frac{x}{\color{blue}{z \cdot \frac{1}{\frac{\sin y}{y}}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{x}{z \cdot \frac{1}{\color{blue}{\frac{\sin y}{y}}}} \]

      8. clear-num N/A

         \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{y}{\sin y}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

      11. lower-/.f64 100.0

          \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y}} \cdot z} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y} \cdot z}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)} \cdot z} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) + 1\right)} \cdot z} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(\color{blue}{\left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) \cdot {y}^{2}} + 1\right) \cdot z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}, {y}^{2}, 1\right)} \cdot z} \]

      4. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\frac{7}{360} \cdot {y}^{2} + \frac{1}{6}}, {y}^{2}, 1\right) \cdot z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7}{360}, {y}^{2}, \frac{1}{6}\right)}, {y}^{2}, 1\right) \cdot z} \]

      6. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{y \cdot y}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{y \cdot y}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot z} \]

      8. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, y \cdot y, \frac{1}{6}\right), \color{blue}{y \cdot y}, 1\right) \cdot z} \]

      9. lower-*.f64 100.0

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \cdot z} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)} \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.5:\\ \;\;\;\;\frac{x}{y} \cdot \frac{\sin y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.5:\\ \;\;\;\;\frac{\sin y \cdot x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 0.5)
    (/ (* (sin y) x_m) (* z y))
    (/
     x_m
     (*
      (fma (fma 0.019444444444444445 (* y y) 0.16666666666666666) (* y y) 1.0)
      z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 0.5) {
		tmp = (sin(y) * x_m) / (z * y);
	} else {
		tmp = x_m / (fma(fma(0.019444444444444445, (y * y), 0.16666666666666666), (y * y), 1.0) * z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.5)
		tmp = Float64(Float64(sin(y) * x_m) / Float64(z * y));
	else
		tmp = Float64(x_m / Float64(fma(fma(0.019444444444444445, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * z));
	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_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.5], N[(N[(N[Sin[y], $MachinePrecision] * x$95$m), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(N[(0.019444444444444445 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.5

   1. Initial program 94.6%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{\sin y}{y}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot \sin y}{y}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \sin y}{z \cdot y}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \sin y}{z \cdot y}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin y \cdot x}}{z \cdot y} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin y \cdot x}}{z \cdot y} \]

      9. lower-*.f64 89.8

         \[\leadsto \frac{\sin y \cdot x}{\color{blue}{z \cdot y}} \]

   4. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\frac{\sin y \cdot x}{z \cdot y}} \]

   if 0.5 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sin y}{y}} \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

      6. div-inv N/A

         \[\leadsto \frac{x}{\color{blue}{z \cdot \frac{1}{\frac{\sin y}{y}}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{x}{z \cdot \frac{1}{\color{blue}{\frac{\sin y}{y}}}} \]

      8. clear-num N/A

         \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{y}{\sin y}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

      11. lower-/.f64 100.0

          \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y}} \cdot z} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y} \cdot z}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)} \cdot z} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) + 1\right)} \cdot z} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(\color{blue}{\left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) \cdot {y}^{2}} + 1\right) \cdot z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}, {y}^{2}, 1\right)} \cdot z} \]

      4. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\frac{7}{360} \cdot {y}^{2} + \frac{1}{6}}, {y}^{2}, 1\right) \cdot z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7}{360}, {y}^{2}, \frac{1}{6}\right)}, {y}^{2}, 1\right) \cdot z} \]

      6. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{y \cdot y}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{y \cdot y}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot z} \]

      8. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, y \cdot y, \frac{1}{6}\right), \color{blue}{y \cdot y}, 1\right) \cdot z} \]

      9. lower-*.f64 100.0

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \cdot z} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)} \cdot z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 95.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.5:\\ \;\;\;\;\frac{x\_m}{z \cdot y} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 0.5)
    (* (/ x_m (* z y)) (sin y))
    (/
     x_m
     (*
      (fma (fma 0.019444444444444445 (* y y) 0.16666666666666666) (* y y) 1.0)
      z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 0.5) {
		tmp = (x_m / (z * y)) * sin(y);
	} else {
		tmp = x_m / (fma(fma(0.019444444444444445, (y * y), 0.16666666666666666), (y * y), 1.0) * z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.5)
		tmp = Float64(Float64(x_m / Float64(z * y)) * sin(y));
	else
		tmp = Float64(x_m / Float64(fma(fma(0.019444444444444445, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * z));
	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_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.5], N[(N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(N[(0.019444444444444445 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.5

   1. Initial program 94.6%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y} \cdot x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y}} \cdot x}{z} \]

      5. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y \cdot \frac{1}{y}\right)} \cdot x}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sin y \cdot \left(\frac{1}{y} \cdot x\right)}}{z} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\sin y \cdot \frac{\frac{1}{y} \cdot x}{z}} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{y} \cdot x}{z} \cdot \sin y} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{y} \cdot x}{z} \cdot \sin y} \]

      10. div-inv N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{y} \cdot x\right) \cdot \frac{1}{z}\right)} \cdot \sin y \]

      11. associate-*l/ N/A

          \[\leadsto \left(\color{blue}{\frac{1 \cdot x}{y}} \cdot \frac{1}{z}\right) \cdot \sin y \]

      12. *-lft-identity N/A

          \[\leadsto \left(\frac{\color{blue}{x}}{y} \cdot \frac{1}{z}\right) \cdot \sin y \]

      13. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot \frac{1}{z}}{y}} \cdot \sin y \]

      14. div-inv N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{z}}}{y} \cdot \sin y \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{z}}{y}} \cdot \sin y \]

      16. lower-/.f64 91.9

          \[\leadsto \frac{\color{blue}{\frac{x}{z}}}{y} \cdot \sin y \]

   4. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{y} \cdot \sin y} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{z}}{y}} \cdot \sin y \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{z}}}{y} \cdot \sin y \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot z}} \cdot \sin y \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot z}} \cdot \sin y \]

      5. lower-*.f64 89.8

         \[\leadsto \frac{x}{\color{blue}{y \cdot z}} \cdot \sin y \]

   6. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot z}} \cdot \sin y \]

   if 0.5 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sin y}{y}} \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

      6. div-inv N/A

         \[\leadsto \frac{x}{\color{blue}{z \cdot \frac{1}{\frac{\sin y}{y}}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{x}{z \cdot \frac{1}{\color{blue}{\frac{\sin y}{y}}}} \]

      8. clear-num N/A

         \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{y}{\sin y}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

      11. lower-/.f64 100.0

          \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y}} \cdot z} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y} \cdot z}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)} \cdot z} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) + 1\right)} \cdot z} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(\color{blue}{\left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) \cdot {y}^{2}} + 1\right) \cdot z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}, {y}^{2}, 1\right)} \cdot z} \]

      4. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\frac{7}{360} \cdot {y}^{2} + \frac{1}{6}}, {y}^{2}, 1\right) \cdot z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7}{360}, {y}^{2}, \frac{1}{6}\right)}, {y}^{2}, 1\right) \cdot z} \]

      6. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{y \cdot y}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{y \cdot y}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot z} \]

      8. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, y \cdot y, \frac{1}{6}\right), \color{blue}{y \cdot y}, 1\right) \cdot z} \]

      9. lower-*.f64 100.0

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \cdot z} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)} \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.5:\\ \;\;\;\;\frac{x}{z \cdot y} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-99}:\\ \;\;\;\;\frac{x\_m}{\left(0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 1e-99)
    (/ x_m (* (* 0.16666666666666666 (* y y)) z))
    (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 1e-99) {
		tmp = x_m / ((0.16666666666666666 * (y * y)) * z);
	} else {
		tmp = x_m / z;
	}
	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)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((sin(y) / y) <= 1d-99) then
        tmp = x_m / ((0.16666666666666666d0 * (y * y)) * z)
    else
        tmp = x_m / z
    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 tmp;
	if ((Math.sin(y) / y) <= 1e-99) {
		tmp = x_m / ((0.16666666666666666 * (y * y)) * z);
	} else {
		tmp = x_m / z;
	}
	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):
	tmp = 0
	if (math.sin(y) / y) <= 1e-99:
		tmp = x_m / ((0.16666666666666666 * (y * y)) * z)
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 1e-99)
		tmp = Float64(x_m / Float64(Float64(0.16666666666666666 * Float64(y * y)) * z));
	else
		tmp = Float64(x_m / z);
	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)
	tmp = 0.0;
	if ((sin(y) / y) <= 1e-99)
		tmp = x_m / ((0.16666666666666666 * (y * y)) * z);
	else
		tmp = x_m / z;
	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_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 1e-99], N[(x$95$m / N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 1e-99

   1. Initial program 94.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sin y}{y}} \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

      6. div-inv N/A

         \[\leadsto \frac{x}{\color{blue}{z \cdot \frac{1}{\frac{\sin y}{y}}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{x}{z \cdot \frac{1}{\color{blue}{\frac{\sin y}{y}}}} \]

      8. clear-num N/A

         \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{y}{\sin y}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

      11. lower-/.f64 89.6

          \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y}} \cdot z} \]

   4. Applied rewrites 89.6%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y} \cdot z}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \cdot z} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \cdot z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, 1\right)} \cdot z} \]

      3. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \cdot z} \]

      4. lower-*.f64 32.9

         \[\leadsto \frac{x}{\mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \cdot z} \]

   7. Applied rewrites 32.9%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \cdot z} \]

   8. Taylor expanded in y around inf

      \[\leadsto \frac{x}{\left(\frac{1}{6} \cdot \color{blue}{{y}^{2}}\right) \cdot z} \]
   9. Step-by-step derivation

   10. Applied rewrites 32.9%

       \[\leadsto \frac{x}{\left(0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot z} \]

   if 1e-99 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.3%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 90.9

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

   5. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 5.7 \cdot 10^{+48}:\\ \;\;\;\;\frac{x\_m}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{y} \cdot \sin y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z 5.7e+48) (/ x_m (* z (/ y (sin y)))) (* (/ (/ x_m z) y) (sin y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= 5.7e+48) {
		tmp = x_m / (z * (y / sin(y)));
	} else {
		tmp = ((x_m / z) / y) * sin(y);
	}
	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)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 5.7d+48) then
        tmp = x_m / (z * (y / sin(y)))
    else
        tmp = ((x_m / z) / y) * sin(y)
    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 tmp;
	if (z <= 5.7e+48) {
		tmp = x_m / (z * (y / Math.sin(y)));
	} else {
		tmp = ((x_m / z) / y) * Math.sin(y);
	}
	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):
	tmp = 0
	if z <= 5.7e+48:
		tmp = x_m / (z * (y / math.sin(y)))
	else:
		tmp = ((x_m / z) / y) * math.sin(y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= 5.7e+48)
		tmp = Float64(x_m / Float64(z * Float64(y / sin(y))));
	else
		tmp = Float64(Float64(Float64(x_m / z) / y) * sin(y));
	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)
	tmp = 0.0;
	if (z <= 5.7e+48)
		tmp = x_m / (z * (y / sin(y)));
	else
		tmp = ((x_m / z) / y) * sin(y);
	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_] := N[(x$95$s * If[LessEqual[z, 5.7e+48], N[(x$95$m / N[(z * N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.69999999999999968e48

   1. Initial program 96.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sin y}{y}} \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

      6. div-inv N/A

         \[\leadsto \frac{x}{\color{blue}{z \cdot \frac{1}{\frac{\sin y}{y}}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{x}{z \cdot \frac{1}{\color{blue}{\frac{\sin y}{y}}}} \]

      8. clear-num N/A

         \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{y}{\sin y}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

      11. lower-/.f64 96.5

          \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y}} \cdot z} \]

   4. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y} \cdot z}} \]

   if 5.69999999999999968e48 < z

   1. Initial program 99.9%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y} \cdot x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y}} \cdot x}{z} \]

      5. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y \cdot \frac{1}{y}\right)} \cdot x}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sin y \cdot \left(\frac{1}{y} \cdot x\right)}}{z} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\sin y \cdot \frac{\frac{1}{y} \cdot x}{z}} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{y} \cdot x}{z} \cdot \sin y} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{y} \cdot x}{z} \cdot \sin y} \]

      10. div-inv N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{y} \cdot x\right) \cdot \frac{1}{z}\right)} \cdot \sin y \]

      11. associate-*l/ N/A

          \[\leadsto \left(\color{blue}{\frac{1 \cdot x}{y}} \cdot \frac{1}{z}\right) \cdot \sin y \]

      12. *-lft-identity N/A

          \[\leadsto \left(\frac{\color{blue}{x}}{y} \cdot \frac{1}{z}\right) \cdot \sin y \]

      13. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot \frac{1}{z}}{y}} \cdot \sin y \]

      14. div-inv N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{z}}}{y} \cdot \sin y \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{z}}{y}} \cdot \sin y \]

      16. lower-/.f64 97.9

          \[\leadsto \frac{\color{blue}{\frac{x}{z}}}{y} \cdot \sin y \]

   4. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{y} \cdot \sin y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.7 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{y} \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{1}{\frac{z}{x\_m} \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (/ 1.0 (* (/ z x_m) (fma (* y y) 0.16666666666666666 1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (1.0 / ((z / x_m) * fma((y * y), 0.16666666666666666, 1.0)));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(1.0 / Float64(Float64(z / x_m) * fma(Float64(y * y), 0.16666666666666666, 1.0))))
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_] := N[(x$95$s * N[(1.0 / N[(N[(z / x$95$m), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.2%

   \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sin y}{y}} \]

   4. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

   6. div-inv N/A

      \[\leadsto \frac{x}{\color{blue}{z \cdot \frac{1}{\frac{\sin y}{y}}}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{x}{z \cdot \frac{1}{\color{blue}{\frac{\sin y}{y}}}} \]

   8. clear-num N/A

      \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{y}{\sin y}}} \]

   9. *-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

   11. lower-/.f64 94.6

       \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y}} \cdot z} \]

4. Applied rewrites 94.6%

   \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y} \cdot z}} \]

5. Taylor expanded in y around 0

   \[\leadsto \frac{x}{\color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \cdot z} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \cdot z} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, 1\right)} \cdot z} \]

   3. unpow2 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \cdot z} \]

   4. lower-*.f64 65.0

      \[\leadsto \frac{x}{\mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \cdot z} \]

7. Applied rewrites 65.0%

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \cdot z} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot z}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot z}{x}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot z}{x}}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot z}}{x}} \]

   5. associate-/l* N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot \frac{z}{x}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot \frac{z}{x}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{1}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(y \cdot y, \frac{1}{6}, 1\right)\right)\right) \cdot \color{blue}{\frac{z}{x}}} \]

9. Applied rewrites 65.1%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \frac{z}{x}}} \]

10. Final simplification 65.1%

    \[\leadsto \frac{1}{\frac{z}{x} \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)} \]
11. Add Preprocessing

Alternative 8: 66.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot z, 0.16666666666666666, z\right)} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (/ x_m (fma (* (* y y) z) 0.16666666666666666 z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (x_m / fma(((y * y) * z), 0.16666666666666666, z));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(x_m / fma(Float64(Float64(y * y) * z), 0.16666666666666666, z)))
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_] := N[(x$95$s * N[(x$95$m / N[(N[(N[(y * y), $MachinePrecision] * z), $MachinePrecision] * 0.16666666666666666 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.2%

   \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sin y}{y}} \]

   4. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

   6. div-inv N/A

      \[\leadsto \frac{x}{\color{blue}{z \cdot \frac{1}{\frac{\sin y}{y}}}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{x}{z \cdot \frac{1}{\color{blue}{\frac{\sin y}{y}}}} \]

   8. clear-num N/A

      \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{y}{\sin y}}} \]

   9. *-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

   11. lower-/.f64 94.6

       \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y}} \cdot z} \]

4. Applied rewrites 94.6%

   \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y} \cdot z}} \]

5. Taylor expanded in y around 0

   \[\leadsto \frac{x}{\color{blue}{z + \frac{1}{6} \cdot \left({y}^{2} \cdot z\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot z\right) + z}} \]

   2. *-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{\left({y}^{2} \cdot z\right) \cdot \frac{1}{6}} + z} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({y}^{2} \cdot z, \frac{1}{6}, z\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{y}^{2} \cdot z}, \frac{1}{6}, z\right)} \]

   5. unpow2 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\left(y \cdot y\right)} \cdot z, \frac{1}{6}, z\right)} \]

   6. lower-*.f64 65.0

      \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\left(y \cdot y\right)} \cdot z, 0.16666666666666666, z\right)} \]

7. Applied rewrites 65.0%

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot z, 0.16666666666666666, z\right)}} \]
8. Add Preprocessing

Alternative 9: 58.0% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (x_m / z)
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) {
	return x_s * (x_m / z);
}

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (x_m / z);
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_] := N[(x$95$s * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.2%

   \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{x}{z}} \]
4. Step-by-step derivation

   1. lower-/.f64 58.4

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

5. Applied rewrites 58.4%

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

Developer Target 1: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\sin y}\\ t_1 := \frac{x \cdot \frac{1}{t\_0}}{z}\\ \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (sin y))) (t_1 (/ (* x (/ 1.0 t_0)) z)))
   (if (< z -4.2173720203427147e-29)
     t_1
     (if (< z 4.446702369113811e+64) (/ x (* z t_0)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = y / sin(y);
	double t_1 = (x * (1.0 / t_0)) / z;
	double tmp;
	if (z < -4.2173720203427147e-29) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x / (z * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y / sin(y)
    t_1 = (x * (1.0d0 / t_0)) / z
    if (z < (-4.2173720203427147d-29)) then
        tmp = t_1
    else if (z < 4.446702369113811d+64) then
        tmp = x / (z * t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / Math.sin(y);
	double t_1 = (x * (1.0 / t_0)) / z;
	double tmp;
	if (z < -4.2173720203427147e-29) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x / (z * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / math.sin(y)
	t_1 = (x * (1.0 / t_0)) / z
	tmp = 0
	if z < -4.2173720203427147e-29:
		tmp = t_1
	elif z < 4.446702369113811e+64:
		tmp = x / (z * t_0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / sin(y))
	t_1 = Float64(Float64(x * Float64(1.0 / t_0)) / z)
	tmp = 0.0
	if (z < -4.2173720203427147e-29)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = Float64(x / Float64(z * t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / sin(y);
	t_1 = (x * (1.0 / t_0)) / z;
	tmp = 0.0;
	if (z < -4.2173720203427147e-29)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = x / (z * t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Less[z, -4.2173720203427147e-29], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x / N[(z * t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024295 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -42173720203427147/1000000000000000000000000000000000000000000000) (/ (* x (/ 1 (/ y (sin y)))) z) (if (< z 44467023691138110000000000000000000000000000000000000000000000000) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1 (/ y (sin y)))) z))))

  (/ (* x (/ (sin y) y)) z))