Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 98.4%

Time: 6.1s

Alternatives: 7

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: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 8.5 \cdot 10^{-49}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x\_m}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ \end{array} \end{array} \]

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 8.5e-49)
    (* (sin y) (/ (/ x_m z) y))
    (/ (* x_m (/ (sin y) y)) 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 <= 8.5e-49) {
		tmp = sin(y) * ((x_m / z) / y);
	} else {
		tmp = (x_m * (sin(y) / y)) / 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 <= 8.5d-49) then
        tmp = sin(y) * ((x_m / z) / y)
    else
        tmp = (x_m * (sin(y) / y)) / 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 <= 8.5e-49) {
		tmp = Math.sin(y) * ((x_m / z) / y);
	} else {
		tmp = (x_m * (Math.sin(y) / y)) / 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 <= 8.5e-49:
		tmp = math.sin(y) * ((x_m / z) / y)
	else:
		tmp = (x_m * (math.sin(y) / y)) / 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 <= 8.5e-49)
		tmp = Float64(sin(y) * Float64(Float64(x_m / z) / y));
	else
		tmp = Float64(Float64(x_m * Float64(sin(y) / y)) / 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 <= 8.5e-49)
		tmp = sin(y) * ((x_m / z) / y);
	else
		tmp = (x_m * (sin(y) / y)) / 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, 8.5e-49], N[(N[Sin[y], $MachinePrecision] * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 8.50000000000000069e-49

   1. Initial program 90.6%

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

      1. associate-*l/ 96.8%

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

      2. associate-*r/ 85.7%

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

      3. *-commutative 85.7%

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

      4. associate-/l* 92.2%

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

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\sin y \cdot \frac{\frac{x}{z}}{y}} \]
   4. Add Preprocessing

   if 8.50000000000000069e-49 < x

   1. Initial program 99.8%

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

Alternative 2: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-38}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+219}:\\ \;\;\;\;x\_m \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x\_m}{z}}{y}\\ \end{array} \end{array} \]

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 (<= y 3e-38)
    (/ x_m z)
    (if (<= y 4.3e+219)
      (* x_m (/ (sin y) (* y z)))
      (* (sin y) (/ (/ x_m z) 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 (y <= 3e-38) {
		tmp = x_m / z;
	} else if (y <= 4.3e+219) {
		tmp = x_m * (sin(y) / (y * z));
	} else {
		tmp = sin(y) * ((x_m / z) / 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 (y <= 3d-38) then
        tmp = x_m / z
    else if (y <= 4.3d+219) then
        tmp = x_m * (sin(y) / (y * z))
    else
        tmp = sin(y) * ((x_m / z) / 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 (y <= 3e-38) {
		tmp = x_m / z;
	} else if (y <= 4.3e+219) {
		tmp = x_m * (Math.sin(y) / (y * z));
	} else {
		tmp = Math.sin(y) * ((x_m / z) / 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 y <= 3e-38:
		tmp = x_m / z
	elif y <= 4.3e+219:
		tmp = x_m * (math.sin(y) / (y * z))
	else:
		tmp = math.sin(y) * ((x_m / z) / 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 (y <= 3e-38)
		tmp = Float64(x_m / z);
	elseif (y <= 4.3e+219)
		tmp = Float64(x_m * Float64(sin(y) / Float64(y * z)));
	else
		tmp = Float64(sin(y) * Float64(Float64(x_m / z) / 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 (y <= 3e-38)
		tmp = x_m / z;
	elseif (y <= 4.3e+219)
		tmp = x_m * (sin(y) / (y * z));
	else
		tmp = sin(y) * ((x_m / z) / 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[y, 3e-38], N[(x$95$m / z), $MachinePrecision], If[LessEqual[y, 4.3e+219], N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 2.99999999999999989e-38

   1. Initial program 94.5%

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

      1. associate-*l/ 98.3%

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

      2. associate-*r/ 84.3%

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

      3. *-commutative 84.3%

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

      4. associate-/l* 90.5%

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

   3. Simplified 90.5%

      \[\leadsto \color{blue}{\sin y \cdot \frac{\frac{x}{z}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 73.9%

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

   if 2.99999999999999989e-38 < y < 4.2999999999999997e219

   1. Initial program 95.9%

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

      1. associate-*l/ 90.9%

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

      2. associate-*r/ 90.9%

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

      3. *-commutative 90.9%

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

      4. associate-/l* 91.0%

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

   3. Simplified 91.0%

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

      1. associate-/l/ 94.3%

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

      2. associate-*r/ 94.3%

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

      3. *-commutative 94.3%

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

      4. *-commutative 94.3%

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

      5. associate-/l/ 95.9%

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

      6. associate-*r/ 95.9%

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

      7. associate-/l* 94.1%

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

      8. *-commutative 94.1%

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

      9. associate-/l/ 94.2%

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

   6. Applied egg-rr 94.2%

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

   if 4.2999999999999997e219 < y

   1. Initial program 74.7%

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

      1. associate-*l/ 86.0%

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

      2. associate-*r/ 86.2%

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

      3. *-commutative 86.2%

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

      4. associate-/l* 86.2%

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

   3. Simplified 86.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+219}:\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x\_m}{z}}{y}\\ \end{array} \end{array} \]

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 (<= y 3.8e-48) (/ x_m z) (* (sin y) (/ (/ x_m z) 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 (y <= 3.8e-48) {
		tmp = x_m / z;
	} else {
		tmp = sin(y) * ((x_m / z) / 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 (y <= 3.8d-48) then
        tmp = x_m / z
    else
        tmp = sin(y) * ((x_m / z) / 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 (y <= 3.8e-48) {
		tmp = x_m / z;
	} else {
		tmp = Math.sin(y) * ((x_m / z) / 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 y <= 3.8e-48:
		tmp = x_m / z
	else:
		tmp = math.sin(y) * ((x_m / z) / 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 (y <= 3.8e-48)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(sin(y) * Float64(Float64(x_m / z) / 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 (y <= 3.8e-48)
		tmp = x_m / z;
	else
		tmp = sin(y) * ((x_m / z) / 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[y, 3.8e-48], N[(x$95$m / z), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.80000000000000002e-48

   1. Initial program 94.4%

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

      1. associate-*l/ 98.3%

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

      2. associate-*r/ 84.1%

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

      3. *-commutative 84.1%

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

      4. associate-/l* 90.4%

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

   3. Simplified 90.4%

      \[\leadsto \color{blue}{\sin y \cdot \frac{\frac{x}{z}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 73.6%

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

   if 3.80000000000000002e-48 < y

   1. Initial program 90.2%

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

      1. associate-*l/ 89.8%

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

      2. associate-*r/ 89.8%

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

      3. *-commutative 89.8%

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

      4. associate-/l* 89.9%

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

   3. Simplified 89.9%

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

Alternative 4: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x\_m}{y}}{z}\\ \end{array} \end{array} \]

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 (<= y 2.8e-8) (/ x_m z) (* (sin y) (/ (/ x_m y) 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 (y <= 2.8e-8) {
		tmp = x_m / z;
	} else {
		tmp = sin(y) * ((x_m / y) / 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 (y <= 2.8d-8) then
        tmp = x_m / z
    else
        tmp = sin(y) * ((x_m / y) / 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 (y <= 2.8e-8) {
		tmp = x_m / z;
	} else {
		tmp = Math.sin(y) * ((x_m / y) / 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 y <= 2.8e-8:
		tmp = x_m / z
	else:
		tmp = math.sin(y) * ((x_m / y) / 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 (y <= 2.8e-8)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(sin(y) * Float64(Float64(x_m / y) / 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 (y <= 2.8e-8)
		tmp = x_m / z;
	else
		tmp = sin(y) * ((x_m / y) / 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[y, 2.8e-8], N[(x$95$m / z), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(N[(x$95$m / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.7999999999999999e-8

   1. Initial program 94.6%

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

      1. associate-*l/ 98.4%

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

      2. associate-*r/ 84.6%

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

      3. *-commutative 84.6%

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

      4. associate-/l* 90.7%

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

   3. Simplified 90.7%

      \[\leadsto \color{blue}{\sin y \cdot \frac{\frac{x}{z}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 74.3%

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

   if 2.7999999999999999e-8 < y

   1. Initial program 89.5%

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

      1. associate-*l/ 89.1%

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

      2. associate-*r/ 89.1%

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

      3. *-commutative 89.1%

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

      4. associate-/l* 89.2%

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

   3. Simplified 89.2%

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

      1. associate-/l/ 94.0%

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

      2. associate-/r* 89.6%

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

   6. Applied egg-rr 89.6%

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

Alternative 5: 62.2% accurate, 8.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 260000000:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{y \cdot z}\\ \end{array} \end{array} \]

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 (<= y 260000000.0) (/ x_m z) (* y (/ x_m (* y 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 (y <= 260000000.0) {
		tmp = x_m / z;
	} else {
		tmp = y * (x_m / (y * 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 (y <= 260000000.0d0) then
        tmp = x_m / z
    else
        tmp = y * (x_m / (y * 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 (y <= 260000000.0) {
		tmp = x_m / z;
	} else {
		tmp = y * (x_m / (y * 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 y <= 260000000.0:
		tmp = x_m / z
	else:
		tmp = y * (x_m / (y * 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 (y <= 260000000.0)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(y * Float64(x_m / Float64(y * 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 (y <= 260000000.0)
		tmp = x_m / z;
	else
		tmp = y * (x_m / (y * 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[y, 260000000.0], N[(x$95$m / z), $MachinePrecision], N[(y * N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.6e8

   1. Initial program 94.7%

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

      1. associate-*l/ 98.4%

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

      2. associate-*r/ 85.0%

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

      3. *-commutative 85.0%

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

      4. associate-/l* 90.9%

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

   3. Simplified 90.9%

      \[\leadsto \color{blue}{\sin y \cdot \frac{\frac{x}{z}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 72.7%

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

   if 2.6e8 < y

   1. Initial program 88.6%

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

      1. associate-*l/ 88.2%

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

      2. associate-*r/ 88.2%

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

      3. *-commutative 88.2%

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

      4. associate-/l* 88.3%

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

   3. Simplified 88.3%

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

      1. associate-*r/ 88.2%

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

      2. div-inv 88.3%

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

      3. associate-*r* 88.2%

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

      4. *-commutative 88.2%

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

      5. associate-*l/ 88.5%

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

      6. associate-*r/ 88.6%

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

      7. clear-num 88.6%

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

      8. un-div-inv 88.6%

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

   6. Applied egg-rr 88.6%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}} \cdot \frac{1}{z}} \]
   7. Step-by-step derivation

      1. *-commutative 88.6%

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

      2. frac-times 93.5%

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

      3. associate-/l* 93.5%

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

      4. associate-*l/ 93.4%

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

      5. clear-num 93.4%

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

      6. associate-/r* 94.0%

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

      7. associate-*l/ 88.3%

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

   8. Applied egg-rr 88.3%

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

   9. Taylor expanded in y around 0 8.4%

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

       1. associate-/l* 13.0%

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

       2. frac-times 16.7%

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

       3. *-commutative 16.7%

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

       4. associate-/l* 28.1%

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

       5. *-commutative 28.1%

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

   11. Applied egg-rr 28.1%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 260000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-38}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y}{y \cdot z}\\ \end{array} \end{array} \]

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 (<= y 3e-38) (/ x_m z) (* x_m (/ y (* y 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 (y <= 3e-38) {
		tmp = x_m / z;
	} else {
		tmp = x_m * (y / (y * 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 (y <= 3d-38) then
        tmp = x_m / z
    else
        tmp = x_m * (y / (y * 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 (y <= 3e-38) {
		tmp = x_m / z;
	} else {
		tmp = x_m * (y / (y * 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 y <= 3e-38:
		tmp = x_m / z
	else:
		tmp = x_m * (y / (y * 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 (y <= 3e-38)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(x_m * Float64(y / Float64(y * 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 (y <= 3e-38)
		tmp = x_m / z;
	else
		tmp = x_m * (y / (y * 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[y, 3e-38], N[(x$95$m / z), $MachinePrecision], N[(x$95$m * N[(y / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.99999999999999989e-38

   1. Initial program 94.5%

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

      1. associate-*l/ 98.3%

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

      2. associate-*r/ 84.3%

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

      3. *-commutative 84.3%

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

      4. associate-/l* 90.5%

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

   3. Simplified 90.5%

      \[\leadsto \color{blue}{\sin y \cdot \frac{\frac{x}{z}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 73.9%

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

   if 2.99999999999999989e-38 < y

   1. Initial program 89.9%

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

      1. associate-*l/ 89.5%

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

      2. associate-*r/ 89.6%

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

      3. *-commutative 89.6%

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

      4. associate-/l* 89.7%

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

   3. Simplified 89.7%

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

      1. associate-*r/ 89.6%

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

      2. div-inv 89.6%

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

      3. associate-*r* 89.5%

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

      4. *-commutative 89.5%

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

      5. associate-*l/ 89.9%

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

      6. associate-*r/ 89.9%

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

      7. clear-num 89.8%

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

      8. un-div-inv 89.9%

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

   6. Applied egg-rr 89.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\sin y}} \cdot \frac{1}{z}} \]
   7. Step-by-step derivation

      1. *-commutative 89.9%

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

      2. frac-times 94.2%

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

      3. associate-/l* 94.2%

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

      4. associate-*l/ 94.1%

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

      5. clear-num 94.1%

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

      6. associate-/r* 94.7%

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

      7. associate-*l/ 89.6%

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

   8. Applied egg-rr 89.6%

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

   9. Taylor expanded in y around 0 13.2%

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

       1. *-commutative 13.2%

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

       2. associate-/l* 13.3%

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

       3. associate-/l/ 20.6%

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

       4. associate-/l* 21.9%

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

       5. *-commutative 21.9%

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

   11. Applied egg-rr 21.9%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.4% accurate, 35.7× 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 93.1%

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

   1. associate-*l/ 95.8%

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

   2. associate-*r/ 85.9%

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

   3. *-commutative 85.9%

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

   4. associate-/l* 90.3%

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

3. Simplified 90.3%

   \[\leadsto \color{blue}{\sin y \cdot \frac{\frac{x}{z}}{y}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 56.2%

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

Developer target: 99.6% accurate, 0.9× 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 2024096 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :alt
  (if (< z -4.2173720203427147e-29) (/ (* x (/ 1.0 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1.0 (/ y (sin y)))) z)))

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