Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.4% → 99.5%

Time: 9.6s

Alternatives: 12

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: 96.4% 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2.9 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{\frac{z_m}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{x}{z_m}\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (* z_s (if (<= z_m 2.9e-71) (/ x (/ z_m t_0)) (* t_0 (/ x z_m))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double t_0 = sin(y) / y;
	double tmp;
	if (z_m <= 2.9e-71) {
		tmp = x / (z_m / t_0);
	} else {
		tmp = t_0 * (x / z_m);
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (z_m <= 2.9d-71) then
        tmp = x / (z_m / t_0)
    else
        tmp = t_0 * (x / z_m)
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (z_m <= 2.9e-71) {
		tmp = x / (z_m / t_0);
	} else {
		tmp = t_0 * (x / z_m);
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	t_0 = math.sin(y) / y
	tmp = 0
	if z_m <= 2.9e-71:
		tmp = x / (z_m / t_0)
	else:
		tmp = t_0 * (x / z_m)
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (z_m <= 2.9e-71)
		tmp = Float64(x / Float64(z_m / t_0));
	else
		tmp = Float64(t_0 * Float64(x / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (z_m <= 2.9e-71)
		tmp = x / (z_m / t_0);
	else
		tmp = t_0 * (x / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(z$95$s * If[LessEqual[z$95$m, 2.9e-71], N[(x / N[(z$95$m / t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.8999999999999999e-71

   1. Initial program 92.9%

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

      1. associate-/l* 96.9%

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

   3. Simplified 96.9%

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

   if 2.8999999999999999e-71 < 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%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{z_m \cdot y}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 7e-13) (/ x z_m) (* x (/ (sin y) (* z_m y))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 7e-13) {
		tmp = x / z_m;
	} else {
		tmp = x * (sin(y) / (z_m * y));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 7d-13) then
        tmp = x / z_m
    else
        tmp = x * (sin(y) / (z_m * y))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 7e-13) {
		tmp = x / z_m;
	} else {
		tmp = x * (Math.sin(y) / (z_m * y));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 7e-13:
		tmp = x / z_m
	else:
		tmp = x * (math.sin(y) / (z_m * y))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 7e-13)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(x * Float64(sin(y) / Float64(z_m * y)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 7e-13)
		tmp = x / z_m;
	else
		tmp = x * (sin(y) / (z_m * y));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 7e-13], N[(x / z$95$m), $MachinePrecision], N[(x * N[(N[Sin[y], $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 7.0000000000000005e-13

   1. Initial program 96.4%

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

      1. associate-*r/ 97.6%

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

      2. associate-/l/ 87.4%

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

      3. *-commutative 87.4%

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

   3. Simplified 87.4%

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

   5. Taylor expanded in y around 0 66.9%

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

   if 7.0000000000000005e-13 < y

   1. Initial program 90.9%

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

      1. associate-*r/ 89.2%

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

      2. associate-/l/ 88.1%

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

      3. *-commutative 88.1%

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

   3. Simplified 88.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 3.8e-33) (/ x z_m) (* (sin y) (/ (/ x y) z_m)))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 3.8e-33) {
		tmp = x / z_m;
	} else {
		tmp = sin(y) * ((x / y) / z_m);
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 3.8d-33) then
        tmp = x / z_m
    else
        tmp = sin(y) * ((x / y) / z_m)
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 3.8e-33) {
		tmp = x / z_m;
	} else {
		tmp = Math.sin(y) * ((x / y) / z_m);
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 3.8e-33:
		tmp = x / z_m
	else:
		tmp = math.sin(y) * ((x / y) / z_m)
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 3.8e-33)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(sin(y) * Float64(Float64(x / y) / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 3.8e-33)
		tmp = x / z_m;
	else
		tmp = sin(y) * ((x / y) / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 3.8e-33], N[(x / z$95$m), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(N[(x / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.79999999999999994e-33

   1. Initial program 96.3%

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

      1. associate-*r/ 97.6%

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

      2. associate-/l/ 87.3%

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

      3. *-commutative 87.3%

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

   3. Simplified 87.3%

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

   5. Taylor expanded in y around 0 66.7%

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

   if 3.79999999999999994e-33 < y

   1. Initial program 91.0%

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

      1. *-lft-identity 91.0%

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

      2. metadata-eval 91.0%

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

      3. times-frac 91.0%

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

      4. neg-mul-1 91.0%

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

      5. distribute-lft-neg-out 91.0%

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

      6. associate-*r/ 90.9%

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

      7. associate-*l/ 90.9%

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

      8. *-commutative 90.9%

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

      9. times-frac 91.1%

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

      10. remove-double-neg 91.1%

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

      11. distribute-frac-neg 91.1%

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

      12. sin-neg 91.1%

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

      13. sin-neg 91.1%

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

      14. neg-mul-1 91.1%

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

      15. associate-/l* 90.9%

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

      16. associate-/r/ 91.1%

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

      17. distribute-lft-neg-in 91.1%

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

      18. metadata-eval 91.1%

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

      19. metadata-eval 91.1%

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

      20. neg-mul-1 91.1%

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

      21. sin-neg 91.1%

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

      22. *-commutative 91.1%

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

   3. Simplified 91.1%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 1.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m) :precision binary64 (* z_s (* (/ (sin y) y) (/ x z_m))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * ((sin(y) / y) * (x / z_m));
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = z_s * ((sin(y) / y) * (x / z_m))
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	return z_s * ((Math.sin(y) / y) * (x / z_m));
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	return z_s * ((math.sin(y) / y) * (x / z_m))

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(Float64(sin(y) / y) * Float64(x / z_m)))
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m)
	tmp = z_s * ((sin(y) / y) * (x / z_m));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.8%

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

   1. *-commutative 94.8%

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

   2. associate-*r/ 96.5%

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

3. Simplified 96.5%

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

5. Final simplification 96.5%

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

Alternative 5: 62.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{1}{x \cdot \frac{-1}{z_m \cdot \left(-y\right)}}}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 1.95e-7) (/ x z_m) (/ y (/ 1.0 (* x (/ -1.0 (* z_m (- y)))))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1.95e-7) {
		tmp = x / z_m;
	} else {
		tmp = y / (1.0 / (x * (-1.0 / (z_m * -y))));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 1.95d-7) then
        tmp = x / z_m
    else
        tmp = y / (1.0d0 / (x * ((-1.0d0) / (z_m * -y))))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1.95e-7) {
		tmp = x / z_m;
	} else {
		tmp = y / (1.0 / (x * (-1.0 / (z_m * -y))));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 1.95e-7:
		tmp = x / z_m
	else:
		tmp = y / (1.0 / (x * (-1.0 / (z_m * -y))))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 1.95e-7)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(y / Float64(1.0 / Float64(x * Float64(-1.0 / Float64(z_m * Float64(-y))))));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 1.95e-7)
		tmp = x / z_m;
	else
		tmp = y / (1.0 / (x * (-1.0 / (z_m * -y))));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 1.95e-7], N[(x / z$95$m), $MachinePrecision], N[(y / N[(1.0 / N[(x * N[(-1.0 / N[(z$95$m * (-y)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.95000000000000012e-7

   1. Initial program 96.4%

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

      1. associate-*r/ 97.6%

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

      2. associate-/l/ 87.4%

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

      3. *-commutative 87.4%

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

   3. Simplified 87.4%

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

   5. Taylor expanded in y around 0 66.9%

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

   if 1.95000000000000012e-7 < y

   1. Initial program 90.9%

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

      1. associate-*r/ 89.2%

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

      2. associate-/l/ 88.1%

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

      3. *-commutative 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in y around 0 16.1%

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

      1. *-un-lft-identity 16.1%

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

      2. div-inv 16.1%

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

      3. *-commutative 16.1%

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

      4. *-inverses 16.1%

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

      5. times-frac 18.2%

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

      6. *-commutative 18.2%

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

      7. associate-/r* 15.9%

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

      8. *-commutative 15.9%

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

      9. associate-*r/ 21.8%

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

      10. associate-/l* 27.1%

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

      11. div-inv 27.1%

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

      12. clear-num 27.1%

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

   7. Applied egg-rr 27.1%

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

      1. associate-*r/ 27.2%

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

      2. *-commutative 27.2%

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

      3. clear-num 27.2%

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

   9. Applied egg-rr 27.2%

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

       1. clear-num 27.2%

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

       2. frac-2neg 27.2%

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

       3. associate-/r/ 27.2%

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

       4. distribute-rgt-neg-in 27.2%

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

   11. Applied egg-rr 27.2%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{1}{x \cdot \frac{-1}{z \cdot \left(-y\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.2% accurate, 8.9× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 1e+99) (/ 1.0 (/ z_m x)) (* x (/ y (* z_m y))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1e+99) {
		tmp = 1.0 / (z_m / x);
	} else {
		tmp = x * (y / (z_m * y));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 1d+99) then
        tmp = 1.0d0 / (z_m / x)
    else
        tmp = x * (y / (z_m * y))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1e+99) {
		tmp = 1.0 / (z_m / x);
	} else {
		tmp = x * (y / (z_m * y));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 1e+99:
		tmp = 1.0 / (z_m / x)
	else:
		tmp = x * (y / (z_m * y))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 1e+99)
		tmp = Float64(1.0 / Float64(z_m / x));
	else
		tmp = Float64(x * Float64(y / Float64(z_m * y)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 1e+99)
		tmp = 1.0 / (z_m / x);
	else
		tmp = x * (y / (z_m * y));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 1e+99], N[(1.0 / N[(z$95$m / x), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.9999999999999997e98

   1. Initial program 96.0%

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

      1. associate-*r/ 97.4%

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

      2. associate-/l/ 88.5%

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

      3. *-commutative 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in y around 0 59.9%

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

      1. div-inv 60.1%

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

      2. clear-num 60.4%

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

   7. Applied egg-rr 60.4%

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

   if 9.9999999999999997e98 < y

   1. Initial program 89.0%

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

      1. associate-*r/ 84.6%

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

      2. associate-/l/ 82.9%

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

      3. *-commutative 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in y around 0 23.7%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+99}:\\ \;\;\;\;\frac{1}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z_m \cdot y}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 8e+15) (/ x z_m) (* y (/ x (* z_m y))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 8e+15) {
		tmp = x / z_m;
	} else {
		tmp = y * (x / (z_m * y));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 8d+15) then
        tmp = x / z_m
    else
        tmp = y * (x / (z_m * y))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 8e+15) {
		tmp = x / z_m;
	} else {
		tmp = y * (x / (z_m * y));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 8e+15:
		tmp = x / z_m
	else:
		tmp = y * (x / (z_m * y))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 8e+15)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(y * Float64(x / Float64(z_m * y)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 8e+15)
		tmp = x / z_m;
	else
		tmp = y * (x / (z_m * y));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 8e+15], N[(x / z$95$m), $MachinePrecision], N[(y * N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 8e15

   1. Initial program 96.4%

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

      1. associate-*r/ 97.6%

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

      2. associate-/l/ 87.5%

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

      3. *-commutative 87.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in y around 0 66.2%

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

   if 8e15 < y

   1. Initial program 90.6%

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

      1. *-lft-identity 90.6%

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

      2. metadata-eval 90.6%

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

      3. times-frac 90.6%

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

      4. neg-mul-1 90.6%

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

      5. distribute-lft-neg-out 90.6%

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

      6. associate-*r/ 90.5%

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

      7. associate-*l/ 90.5%

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

      8. *-commutative 90.5%

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

      9. times-frac 90.7%

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

      10. remove-double-neg 90.7%

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

      11. distribute-frac-neg 90.7%

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

      12. sin-neg 90.7%

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

      13. sin-neg 90.7%

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

      14. neg-mul-1 90.7%

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

      15. associate-/l* 90.6%

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

      16. associate-/r/ 90.7%

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

      17. distribute-lft-neg-in 90.7%

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

      18. metadata-eval 90.7%

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

      19. metadata-eval 90.7%

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

      20. neg-mul-1 90.7%

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

      21. sin-neg 90.7%

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

      22. *-commutative 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in y around 0 26.8%

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

   6. Taylor expanded in x around 0 26.9%

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

      1. *-commutative 26.9%

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

   8. Simplified 26.9%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 0.12:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{z_m}{x}}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 0.12) (/ x z_m) (/ y (* y (/ z_m x))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 0.12) {
		tmp = x / z_m;
	} else {
		tmp = y / (y * (z_m / x));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 0.12d0) then
        tmp = x / z_m
    else
        tmp = y / (y * (z_m / x))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 0.12) {
		tmp = x / z_m;
	} else {
		tmp = y / (y * (z_m / x));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 0.12:
		tmp = x / z_m
	else:
		tmp = y / (y * (z_m / x))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 0.12)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(y / Float64(y * Float64(z_m / x)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 0.12)
		tmp = x / z_m;
	else
		tmp = y / (y * (z_m / x));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 0.12], N[(x / z$95$m), $MachinePrecision], N[(y / N[(y * N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 0.12

   1. Initial program 96.4%

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

      1. associate-*r/ 97.6%

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

      2. associate-/l/ 87.4%

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

      3. *-commutative 87.4%

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

   3. Simplified 87.4%

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

   5. Taylor expanded in y around 0 66.9%

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

   if 0.12 < y

   1. Initial program 90.9%

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

      1. associate-*r/ 89.2%

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

      2. associate-/l/ 88.1%

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

      3. *-commutative 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in y around 0 16.1%

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

      1. *-un-lft-identity 16.1%

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

      2. div-inv 16.1%

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

      3. *-commutative 16.1%

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

      4. *-inverses 16.1%

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

      5. times-frac 18.2%

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

      6. *-commutative 18.2%

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

      7. associate-/r* 15.9%

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

      8. *-commutative 15.9%

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

      9. associate-*r/ 21.8%

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

      10. associate-/l* 27.1%

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

      11. div-inv 27.1%

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

      12. clear-num 27.1%

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

   7. Applied egg-rr 27.1%

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

      1. clear-num 27.1%

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

      2. associate-/r/ 27.1%

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

      3. associate-/l/ 26.3%

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

      4. clear-num 26.3%

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

      5. associate-/l/ 26.4%

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

      6. *-commutative 26.4%

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

   9. Applied egg-rr 26.4%

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

       1. clear-num 27.2%

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

       2. associate-*l/ 27.2%

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

       3. *-un-lft-identity 27.2%

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

       4. *-un-lft-identity 27.2%

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

       5. times-frac 26.8%

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

       6. /-rgt-identity 26.8%

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

   11. Applied egg-rr 26.8%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.12:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.18 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z_m \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 1.18e-15) (/ x z_m) (/ y (* z_m (/ y x))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1.18e-15) {
		tmp = x / z_m;
	} else {
		tmp = y / (z_m * (y / x));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 1.18d-15) then
        tmp = x / z_m
    else
        tmp = y / (z_m * (y / x))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1.18e-15) {
		tmp = x / z_m;
	} else {
		tmp = y / (z_m * (y / x));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 1.18e-15:
		tmp = x / z_m
	else:
		tmp = y / (z_m * (y / x))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 1.18e-15)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(y / Float64(z_m * Float64(y / x)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 1.18e-15)
		tmp = x / z_m;
	else
		tmp = y / (z_m * (y / x));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 1.18e-15], N[(x / z$95$m), $MachinePrecision], N[(y / N[(z$95$m * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.18000000000000004e-15

   1. Initial program 96.4%

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

      1. associate-*r/ 97.6%

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

      2. associate-/l/ 87.4%

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

      3. *-commutative 87.4%

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

   3. Simplified 87.4%

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

   5. Taylor expanded in y around 0 66.9%

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

   if 1.18000000000000004e-15 < y

   1. Initial program 90.9%

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

      1. associate-*r/ 89.2%

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

      2. associate-/l/ 88.1%

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

      3. *-commutative 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in y around 0 16.1%

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

      1. *-un-lft-identity 16.1%

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

      2. div-inv 16.1%

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

      3. *-commutative 16.1%

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

      4. *-inverses 16.1%

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

      5. times-frac 18.2%

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

      6. *-commutative 18.2%

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

      7. associate-/r* 15.9%

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

      8. *-commutative 15.9%

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

      9. associate-*r/ 21.8%

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

      10. associate-/l* 27.1%

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

      11. div-inv 27.1%

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

      12. clear-num 27.1%

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

   7. Applied egg-rr 27.1%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.18 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 0.0014:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z_m \cdot y}{x}}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 0.0014) (/ x z_m) (/ y (/ (* z_m y) x)))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 0.0014) {
		tmp = x / z_m;
	} else {
		tmp = y / ((z_m * y) / x);
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 0.0014d0) then
        tmp = x / z_m
    else
        tmp = y / ((z_m * y) / x)
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 0.0014) {
		tmp = x / z_m;
	} else {
		tmp = y / ((z_m * y) / x);
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 0.0014:
		tmp = x / z_m
	else:
		tmp = y / ((z_m * y) / x)
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 0.0014)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(y / Float64(Float64(z_m * y) / x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 0.0014)
		tmp = x / z_m;
	else
		tmp = y / ((z_m * y) / x);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 0.0014], N[(x / z$95$m), $MachinePrecision], N[(y / N[(N[(z$95$m * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 0.00139999999999999999

   1. Initial program 96.4%

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

      1. associate-*r/ 97.6%

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

      2. associate-/l/ 87.4%

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

      3. *-commutative 87.4%

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

   3. Simplified 87.4%

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

   5. Taylor expanded in y around 0 66.9%

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

   if 0.00139999999999999999 < y

   1. Initial program 90.9%

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

      1. associate-*r/ 89.2%

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

      2. associate-/l/ 88.1%

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

      3. *-commutative 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in y around 0 16.1%

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

      1. *-un-lft-identity 16.1%

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

      2. div-inv 16.1%

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

      3. *-commutative 16.1%

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

      4. *-inverses 16.1%

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

      5. times-frac 18.2%

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

      6. *-commutative 18.2%

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

      7. associate-/r* 15.9%

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

      8. *-commutative 15.9%

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

      9. associate-*r/ 21.8%

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

      10. associate-/l* 27.1%

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

      11. div-inv 27.1%

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

      12. clear-num 27.1%

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

   7. Applied egg-rr 27.1%

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

   8. Taylor expanded in z around 0 27.2%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0014:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.3% accurate, 21.4× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m) :precision binary64 (* z_s (/ 1.0 (/ z_m x))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * (1.0 / (z_m / x));
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = z_s * (1.0d0 / (z_m / x))
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	return z_s * (1.0 / (z_m / x));
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	return z_s * (1.0 / (z_m / x))

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(1.0 / Float64(z_m / x)))
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m)
	tmp = z_s * (1.0 / (z_m / x));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * N[(1.0 / N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.8%

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

   1. associate-*r/ 95.2%

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

   2. associate-/l/ 87.6%

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

   3. *-commutative 87.6%

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

3. Simplified 87.6%

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

5. Taylor expanded in y around 0 52.5%

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

   1. div-inv 52.6%

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

   2. clear-num 52.9%

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

7. Applied egg-rr 52.9%

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

8. Final simplification 52.9%

   \[\leadsto \frac{1}{\frac{z}{x}} \]
9. Add Preprocessing

Alternative 12: 58.4% accurate, 35.7× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m) :precision binary64 (* z_s (/ x z_m)))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * (x / z_m);
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = z_s * (x / z_m)
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	return z_s * (x / z_m);
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	return z_s * (x / z_m)

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(x / z_m))
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m)
	tmp = z_s * (x / z_m);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.8%

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

   1. associate-*r/ 95.2%

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

   2. associate-/l/ 87.6%

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

   3. *-commutative 87.6%

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

3. Simplified 87.6%

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

5. Taylor expanded in y around 0 52.6%

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

6. Final simplification 52.6%

   \[\leadsto \frac{x}{z} \]
7. Add Preprocessing

Developer target: 99.5% 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 2024020 
(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))