Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 99.7%

Time: 7.0s

Alternatives: 10

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: 96.1% 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

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{x_m}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m \cdot t_0}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (* x_s (if (<= x_m 1.8e-47) (/ x_m (/ z t_0)) (/ (* x_m t_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 t_0 = sin(y) / y;
	double tmp;
	if (x_m <= 1.8e-47) {
		tmp = x_m / (z / t_0);
	} else {
		tmp = (x_m * t_0) / 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) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (x_m <= 1.8d-47) then
        tmp = x_m / (z / t_0)
    else
        tmp = (x_m * t_0) / 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 t_0 = Math.sin(y) / y;
	double tmp;
	if (x_m <= 1.8e-47) {
		tmp = x_m / (z / t_0);
	} else {
		tmp = (x_m * t_0) / 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):
	t_0 = math.sin(y) / y
	tmp = 0
	if x_m <= 1.8e-47:
		tmp = x_m / (z / t_0)
	else:
		tmp = (x_m * t_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)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (x_m <= 1.8e-47)
		tmp = Float64(x_m / Float64(z / t_0));
	else
		tmp = Float64(Float64(x_m * t_0) / 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)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (x_m <= 1.8e-47)
		tmp = x_m / (z / t_0);
	else
		tmp = (x_m * t_0) / 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_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 1.8e-47], N[(x$95$m / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.79999999999999995e-47

   1. Initial program 95.8%

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

      1. associate-/l* 94.7%

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

   3. Simplified 94.7%

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

   if 1.79999999999999995e-47 < x

   1. Initial program 99.8%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Alternative 2: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-26}:\\ \;\;\;\;\frac{x_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \frac{\sin y}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= y 9e-26) (/ x_m z) (* x_m (/ (sin y) (* 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 <= 9e-26) {
		tmp = x_m / z;
	} else {
		tmp = x_m * (sin(y) / (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 <= 9d-26) then
        tmp = x_m / z
    else
        tmp = x_m * (sin(y) / (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 <= 9e-26) {
		tmp = x_m / z;
	} else {
		tmp = x_m * (Math.sin(y) / (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 <= 9e-26:
		tmp = x_m / z
	else:
		tmp = x_m * (math.sin(y) / (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 <= 9e-26)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(x_m * Float64(sin(y) / Float64(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 <= 9e-26)
		tmp = x_m / z;
	else
		tmp = x_m * (sin(y) / (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, 9e-26], N[(x$95$m / z), $MachinePrecision], N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 8.9999999999999998e-26

   1. Initial program 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-*r/ 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in y around 0 72.6%

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

   if 8.9999999999999998e-26 < y

   1. Initial program 94.8%

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

      1. associate-*r/ 87.9%

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

      2. associate-/l/ 88.0%

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

      3. *-commutative 88.0%

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

   3. Simplified 88.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \end{array} \]

Alternative 3: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x_m}{y}}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= y 5.5e-11) (/ 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 <= 5.5e-11) {
		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 <= 5.5d-11) 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 <= 5.5e-11) {
		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 <= 5.5e-11:
		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 <= 5.5e-11)
		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 <= 5.5e-11)
		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, 5.5e-11], 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 < 5.49999999999999975e-11

   1. Initial program 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-*r/ 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in y around 0 73.4%

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

   if 5.49999999999999975e-11 < y

   1. Initial program 94.4%

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

      1. *-lft-identity 94.4%

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

      2. metadata-eval 94.4%

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

      3. times-frac 94.4%

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

      4. neg-mul-1 94.4%

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

      5. distribute-lft-neg-out 94.4%

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

      6. associate-*r/ 94.4%

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

      7. associate-*l/ 94.5%

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

      8. *-commutative 94.5%

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

      9. times-frac 94.3%

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

      10. remove-double-neg 94.3%

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

      11. distribute-frac-neg 94.3%

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

      12. sin-neg 94.3%

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

      13. sin-neg 94.3%

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

      14. neg-mul-1 94.3%

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

      15. associate-/l* 94.2%

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

      16. associate-/r/ 94.3%

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

      17. distribute-lft-neg-in 94.3%

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

      18. metadata-eval 94.3%

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

      19. metadata-eval 94.3%

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

      20. neg-mul-1 94.3%

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

      21. sin-neg 94.3%

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

      22. *-commutative 94.3%

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

   3. Simplified 94.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]

Alternative 4: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.2:\\ \;\;\;\;\frac{x_m}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{x_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (* x_s (if (<= z 0.2) (/ x_m (/ z t_0)) (* t_0 (/ 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 t_0 = sin(y) / y;
	double tmp;
	if (z <= 0.2) {
		tmp = x_m / (z / t_0);
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (z <= 0.2d0) then
        tmp = x_m / (z / t_0)
    else
        tmp = t_0 * (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 t_0 = Math.sin(y) / y;
	double tmp;
	if (z <= 0.2) {
		tmp = x_m / (z / t_0);
	} else {
		tmp = t_0 * (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):
	t_0 = math.sin(y) / y
	tmp = 0
	if z <= 0.2:
		tmp = x_m / (z / t_0)
	else:
		tmp = t_0 * (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)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (z <= 0.2)
		tmp = Float64(x_m / Float64(z / t_0));
	else
		tmp = Float64(t_0 * 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)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (z <= 0.2)
		tmp = x_m / (z / t_0);
	else
		tmp = t_0 * (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_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, 0.2], N[(x$95$m / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.20000000000000001

   1. Initial program 96.1%

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

      1. associate-/l* 94.2%

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

   3. Simplified 94.2%

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

   if 0.20000000000000001 < z

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.2:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 5: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \left(\frac{\sin y}{y} \cdot \frac{x_m}{z}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (* (/ (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) {
	return x_s * ((sin(y) / y) * (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 * ((sin(y) / y) * (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 * ((Math.sin(y) / y) * (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 * ((math.sin(y) / y) * (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(Float64(sin(y) / y) * 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 * ((sin(y) / y) * (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[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*r/ 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 6: 63.0% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{x_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x_m}{z}}{y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= y 5.5e-21) (/ x_m z) (* 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 <= 5.5e-21) {
		tmp = x_m / z;
	} else {
		tmp = 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 <= 5.5d-21) then
        tmp = x_m / z
    else
        tmp = 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 <= 5.5e-21) {
		tmp = x_m / z;
	} else {
		tmp = 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 <= 5.5e-21:
		tmp = x_m / z
	else:
		tmp = 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 <= 5.5e-21)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(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 <= 5.5e-21)
		tmp = x_m / z;
	else
		tmp = 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, 5.5e-21], N[(x$95$m / z), $MachinePrecision], N[(y * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.49999999999999977e-21

   1. Initial program 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-*r/ 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in y around 0 72.9%

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

   if 5.49999999999999977e-21 < y

   1. Initial program 94.7%

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

      1. associate-/l* 87.6%

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

   3. Simplified 87.6%

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

      1. associate-/r/ 98.3%

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

      2. frac-times 87.7%

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

      3. *-commutative 87.7%

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

      4. times-frac 94.8%

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

   5. Applied egg-rr 94.8%

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

   6. Taylor expanded in y around 0 22.0%

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

      1. frac-times 27.3%

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

      2. associate-/l* 37.7%

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

      3. associate-*l/ 37.3%

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

      4. *-commutative 37.3%

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

   8. Applied egg-rr 37.3%

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

      1. clear-num 37.3%

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

      2. associate-/r/ 37.3%

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

      3. *-commutative 37.3%

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

      4. *-un-lft-identity 37.3%

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

      5. *-inverses 37.3%

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

      6. associate-/r/ 37.4%

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

      7. associate-/r/ 34.4%

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

      8. associate-/r* 34.4%

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

      9. frac-times 37.2%

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

      10. *-commutative 37.2%

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

      11. clear-num 37.2%

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

      12. times-frac 37.3%

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

      13. *-inverses 37.3%

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

      14. *-un-lft-identity 37.3%

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

   10. Applied egg-rr 37.3%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{y}\\ \end{array} \]

Alternative 7: 63.3% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{z}{x_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= y 5.5e-11) (/ x_m z) (/ y (* y (/ z x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 5.5e-11) {
		tmp = x_m / z;
	} else {
		tmp = y / (y * (z / x_m));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    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 <= 5.5d-11) then
        tmp = x_m / z
    else
        tmp = y / (y * (z / x_m))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 5.5e-11) {
		tmp = x_m / z;
	} else {
		tmp = y / (y * (z / x_m));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= 5.5e-11:
		tmp = x_m / z
	else:
		tmp = y / (y * (z / x_m))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 5.5e-11)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(y / Float64(y * Float64(z / x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= 5.5e-11)
		tmp = x_m / z;
	else
		tmp = y / (y * (z / x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 5.5e-11], N[(x$95$m / z), $MachinePrecision], N[(y / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.49999999999999975e-11

   1. Initial program 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-*r/ 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in y around 0 73.4%

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

   if 5.49999999999999975e-11 < y

   1. Initial program 94.4%

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

      1. associate-/l* 87.1%

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

   3. Simplified 87.1%

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

      1. associate-/r/ 98.2%

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

      2. frac-times 87.3%

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

      3. *-commutative 87.3%

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

      4. times-frac 94.6%

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

   5. Applied egg-rr 94.6%

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

   6. Taylor expanded in y around 0 18.8%

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

      1. frac-times 24.4%

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

      2. associate-/l* 35.2%

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

      3. associate-*l/ 34.7%

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

      4. *-commutative 34.7%

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

   8. Applied egg-rr 34.7%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 8: 63.4% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= y 5.5e-11) (/ x_m z) (/ y (* z (/ y x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 5.5e-11) {
		tmp = x_m / z;
	} else {
		tmp = y / (z * (y / x_m));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    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 <= 5.5d-11) then
        tmp = x_m / z
    else
        tmp = y / (z * (y / x_m))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 5.5e-11) {
		tmp = x_m / z;
	} else {
		tmp = y / (z * (y / x_m));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= 5.5e-11:
		tmp = x_m / z
	else:
		tmp = y / (z * (y / x_m))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 5.5e-11)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(y / Float64(z * Float64(y / x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= 5.5e-11)
		tmp = x_m / z;
	else
		tmp = y / (z * (y / x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 5.5e-11], N[(x$95$m / z), $MachinePrecision], N[(y / N[(z * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.49999999999999975e-11

   1. Initial program 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-*r/ 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in y around 0 73.4%

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

   if 5.49999999999999975e-11 < y

   1. Initial program 94.4%

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

      1. associate-/l* 87.1%

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

   3. Simplified 87.1%

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

      1. associate-/r/ 98.2%

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

      2. frac-times 87.3%

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

      3. *-commutative 87.3%

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

      4. times-frac 94.6%

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

   5. Applied egg-rr 94.6%

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

   6. Taylor expanded in y around 0 18.8%

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

      1. frac-times 24.4%

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

      2. associate-/l* 35.2%

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

      3. associate-*l/ 34.7%

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

      4. *-commutative 34.7%

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

   8. Applied egg-rr 34.7%

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

   9. Taylor expanded in y around 0 35.2%

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

       1. associate-*l/ 34.8%

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

       2. *-commutative 34.8%

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

   11. Simplified 34.8%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 9: 63.2% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 10^{+72}:\\ \;\;\;\;\frac{x_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot y}{x_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= y 1e+72) (/ x_m z) (/ y (/ (* z y) x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1e+72) {
		tmp = x_m / z;
	} else {
		tmp = y / ((z * y) / x_m);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    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 <= 1d+72) then
        tmp = x_m / z
    else
        tmp = y / ((z * y) / x_m)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1e+72) {
		tmp = x_m / z;
	} else {
		tmp = y / ((z * y) / x_m);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= 1e+72:
		tmp = x_m / z
	else:
		tmp = y / ((z * y) / x_m)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 1e+72)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(y / Float64(Float64(z * y) / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= 1e+72)
		tmp = x_m / z;
	else
		tmp = y / ((z * y) / x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 1e+72], N[(x$95$m / z), $MachinePrecision], N[(y / N[(N[(z * y), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999944e71

   1. Initial program 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*r/ 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in y around 0 67.1%

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

   if 9.99999999999999944e71 < y

   1. Initial program 93.8%

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

      1. associate-/l* 86.5%

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

   3. Simplified 86.5%

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

      1. associate-/r/ 97.6%

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

      2. frac-times 86.6%

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

      3. *-commutative 86.6%

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

      4. times-frac 93.9%

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

   5. Applied egg-rr 93.9%

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

   6. Taylor expanded in y around 0 17.3%

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

      1. frac-times 26.0%

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

      2. associate-/l* 42.3%

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

      3. associate-*l/ 41.5%

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

      4. *-commutative 41.5%

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

   8. Applied egg-rr 41.5%

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

   9. Taylor expanded in y around 0 42.3%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+72}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot y}{x}}\\ \end{array} \]

Alternative 10: 59.7% 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 1 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 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*r/ 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in y around 0 57.1%

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

5. Final simplification 57.1%

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

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 2023332 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (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))