Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.0% → 99.5%

Time: 9.2s

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: 96.0% 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

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 10^{+63}:\\ \;\;\;\;\frac{x}{\frac{z_m}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z_m}\\ \end{array} \end{array} \end{array} \]

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 1e+63) (/ x (/ z_m t_0)) (/ (* x t_0) 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 <= 1e+63) {
		tmp = x / (z_m / t_0);
	} else {
		tmp = (x * t_0) / 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 <= 1d+63) then
        tmp = x / (z_m / t_0)
    else
        tmp = (x * t_0) / 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 <= 1e+63) {
		tmp = x / (z_m / t_0);
	} else {
		tmp = (x * t_0) / 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 <= 1e+63:
		tmp = x / (z_m / t_0)
	else:
		tmp = (x * t_0) / 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 <= 1e+63)
		tmp = Float64(x / Float64(z_m / t_0));
	else
		tmp = Float64(Float64(x * t_0) / 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 <= 1e+63)
		tmp = x / (z_m / t_0);
	else
		tmp = (x * t_0) / 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, 1e+63], N[(x / N[(z$95$m / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.00000000000000006e63

   1. Initial program 96.9%

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

      1. associate-/l* 97.6%

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

   3. Simplified 97.6%

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

   if 1.00000000000000006e63 < z

   1. Initial program 99.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+63}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ z_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 5 \cdot 10^{-223}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{x}{z_m}\\ \end{array} \end{array} \end{array} \]

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 (<= t_0 5e-223) (* (sin y) (/ (/ x y) z_m)) (* 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 (t_0 <= 5e-223) {
		tmp = sin(y) * ((x / y) / z_m);
	} 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 (t_0 <= 5d-223) then
        tmp = sin(y) * ((x / y) / z_m)
    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 (t_0 <= 5e-223) {
		tmp = Math.sin(y) * ((x / y) / z_m);
	} 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 t_0 <= 5e-223:
		tmp = math.sin(y) * ((x / y) / z_m)
	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 (t_0 <= 5e-223)
		tmp = Float64(sin(y) * Float64(Float64(x / y) / z_m));
	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 (t_0 <= 5e-223)
		tmp = sin(y) * ((x / y) / z_m);
	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[t$95$0, 5e-223], N[(N[Sin[y], $MachinePrecision] * N[(N[(x / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 5.00000000000000024e-223

   1. Initial program 96.1%

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

      1. *-lft-identity 96.1%

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

      2. metadata-eval 96.1%

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

      3. times-frac 96.1%

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

      4. neg-mul-1 96.1%

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

      5. distribute-lft-neg-out 96.1%

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

      6. associate-*r/ 96.1%

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

      7. associate-*l/ 96.2%

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

      8. *-commutative 96.2%

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

      9. times-frac 95.1%

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

      10. remove-double-neg 95.1%

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

      11. distribute-frac-neg 95.1%

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

      12. sin-neg 95.1%

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

      13. sin-neg 95.1%

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

      14. neg-mul-1 95.1%

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

      15. associate-/l* 95.0%

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

      16. associate-/r/ 95.1%

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

      17. distribute-lft-neg-in 95.1%

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

      18. metadata-eval 95.1%

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

      19. metadata-eval 95.1%

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

      20. neg-mul-1 95.1%

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

      21. sin-neg 95.1%

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

      22. *-commutative 95.1%

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

   3. Simplified 95.1%

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

   if 5.00000000000000024e-223 < (/.f64 (sin.f64 y) y)

   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/ 98.8%

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

   3. Simplified 98.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 5 \cdot 10^{-223}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+205}:\\ \;\;\;\;x \cdot \frac{\sin y}{z_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z_m}\\ \end{array} \end{array} \]

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 1.5e-8)
    (/ x z_m)
    (if (<= y 6e+205)
      (* x (/ (sin y) (* z_m y)))
      (* (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 <= 1.5e-8) {
		tmp = x / z_m;
	} else if (y <= 6e+205) {
		tmp = x * (sin(y) / (z_m * y));
	} 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 <= 1.5d-8) then
        tmp = x / z_m
    else if (y <= 6d+205) then
        tmp = x * (sin(y) / (z_m * y))
    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 <= 1.5e-8) {
		tmp = x / z_m;
	} else if (y <= 6e+205) {
		tmp = x * (Math.sin(y) / (z_m * y));
	} 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 <= 1.5e-8:
		tmp = x / z_m
	elif y <= 6e+205:
		tmp = x * (math.sin(y) / (z_m * y))
	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 <= 1.5e-8)
		tmp = Float64(x / z_m);
	elseif (y <= 6e+205)
		tmp = Float64(x * Float64(sin(y) / Float64(z_m * y)));
	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 <= 1.5e-8)
		tmp = x / z_m;
	elseif (y <= 6e+205)
		tmp = x * (sin(y) / (z_m * y));
	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, 1.5e-8], N[(x / z$95$m), $MachinePrecision], If[LessEqual[y, 6e+205], N[(x * N[(N[Sin[y], $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(N[(x / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 1.49999999999999987e-8

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0 77.2%

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

   if 1.49999999999999987e-8 < y < 5.9999999999999999e205

   1. Initial program 97.1%

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

      1. associate-*r/ 99.5%

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

      2. associate-/l/ 99.8%

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

      3. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   if 5.9999999999999999e205 < y

   1. Initial program 92.4%

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

      1. *-lft-identity 92.4%

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

      2. metadata-eval 92.4%

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

      3. times-frac 92.4%

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

      4. neg-mul-1 92.4%

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

      5. distribute-lft-neg-out 92.4%

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

      6. associate-*r/ 92.2%

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

      7. associate-*l/ 92.6%

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

      8. *-commutative 92.6%

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

      9. times-frac 92.5%

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

      10. remove-double-neg 92.5%

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

      11. distribute-frac-neg 92.5%

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

      12. sin-neg 92.5%

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

      13. sin-neg 92.5%

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

      14. neg-mul-1 92.5%

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

      15. associate-/l* 92.5%

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

      16. associate-/r/ 92.5%

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

      17. distribute-lft-neg-in 92.5%

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

      18. metadata-eval 92.5%

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

      19. metadata-eval 92.5%

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

      20. neg-mul-1 92.5%

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

      21. sin-neg 92.5%

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

      22. *-commutative 92.5%

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

   3. Simplified 92.5%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+205}:\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{z_m \cdot y}\\ \end{array} \end{array} \]

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 2.2e-8) (/ 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 <= 2.2e-8) {
		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 <= 2.2d-8) 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 <= 2.2e-8) {
		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 <= 2.2e-8:
		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 <= 2.2e-8)
		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 <= 2.2e-8)
		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, 2.2e-8], 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 < 2.1999999999999998e-8

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0 77.2%

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

   if 2.1999999999999998e-8 < y

   1. Initial program 95.4%

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

      1. associate-*r/ 93.7%

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

      2. associate-/l/ 93.8%

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

      3. *-commutative 93.8%

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

   3. Simplified 93.8%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{\frac{z_m}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{x}{z_m}\\ \end{array} \end{array} \end{array} \]

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 2e+52) (/ 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 <= 2e+52) {
		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 <= 2d+52) 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 <= 2e+52) {
		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 <= 2e+52:
		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 <= 2e+52)
		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 <= 2e+52)
		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, 2e+52], 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 < 2e52

   1. Initial program 96.9%

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

      1. associate-/l* 97.5%

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

   3. Simplified 97.5%

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

   if 2e52 < z

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(z_m \cdot \left(y \cdot 0.16666666666666666 + \frac{1}{y}\right)\right)}\\ \end{array} \end{array} \]

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 3.8e-24)
    (/ x z_m)
    (/ x (* y (* z_m (+ (* y 0.16666666666666666) (/ 1.0 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 <= 3.8e-24) {
		tmp = x / z_m;
	} else {
		tmp = x / (y * (z_m * ((y * 0.16666666666666666) + (1.0 / 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 <= 3.8d-24) then
        tmp = x / z_m
    else
        tmp = x / (y * (z_m * ((y * 0.16666666666666666d0) + (1.0d0 / 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 <= 3.8e-24) {
		tmp = x / z_m;
	} else {
		tmp = x / (y * (z_m * ((y * 0.16666666666666666) + (1.0 / 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 <= 3.8e-24:
		tmp = x / z_m
	else:
		tmp = x / (y * (z_m * ((y * 0.16666666666666666) + (1.0 / 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 <= 3.8e-24)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(x / Float64(y * Float64(z_m * Float64(Float64(y * 0.16666666666666666) + Float64(1.0 / 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 <= 3.8e-24)
		tmp = x / z_m;
	else
		tmp = x / (y * (z_m * ((y * 0.16666666666666666) + (1.0 / 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, 3.8e-24], N[(x / z$95$m), $MachinePrecision], N[(x / N[(y * N[(z$95$m * N[(N[(y * 0.16666666666666666), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.80000000000000026e-24

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0 76.7%

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

   if 3.80000000000000026e-24 < y

   1. Initial program 95.7%

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

      1. associate-*r/ 94.1%

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

      2. associate-/l/ 94.2%

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

      3. *-commutative 94.2%

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

   3. Simplified 94.2%

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

      1. associate-/r* 94.1%

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

      2. clear-num 93.9%

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

      3. div-inv 94.0%

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

      4. associate-/r/ 93.9%

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

      5. associate-/r* 91.7%

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

   6. Applied egg-rr 91.7%

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

   7. Taylor expanded in y around 0 35.2%

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

   8. Taylor expanded in z around 0 33.8%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(z \cdot \left(y \cdot 0.16666666666666666 + \frac{1}{y}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\left(z_m \cdot y\right) \cdot 0.16666666666666666 + \frac{z_m}{y}}}{y}\\ \end{array} \end{array} \]

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 4.1e-14)
    (/ x z_m)
    (/ (/ x (+ (* (* z_m y) 0.16666666666666666) (/ z_m y))) 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 <= 4.1e-14) {
		tmp = x / z_m;
	} else {
		tmp = (x / (((z_m * y) * 0.16666666666666666) + (z_m / y))) / 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 <= 4.1d-14) then
        tmp = x / z_m
    else
        tmp = (x / (((z_m * y) * 0.16666666666666666d0) + (z_m / y))) / 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 <= 4.1e-14) {
		tmp = x / z_m;
	} else {
		tmp = (x / (((z_m * y) * 0.16666666666666666) + (z_m / y))) / 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 <= 4.1e-14:
		tmp = x / z_m
	else:
		tmp = (x / (((z_m * y) * 0.16666666666666666) + (z_m / y))) / 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 <= 4.1e-14)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(Float64(x / Float64(Float64(Float64(z_m * y) * 0.16666666666666666) + Float64(z_m / y))) / 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 <= 4.1e-14)
		tmp = x / z_m;
	else
		tmp = (x / (((z_m * y) * 0.16666666666666666) + (z_m / y))) / 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, 4.1e-14], N[(x / z$95$m), $MachinePrecision], N[(N[(x / N[(N[(N[(z$95$m * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + N[(z$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.1000000000000002e-14

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0 77.2%

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

   if 4.1000000000000002e-14 < y

   1. Initial program 95.4%

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

      1. associate-*r/ 93.7%

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

      2. associate-/l/ 93.8%

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

      3. *-commutative 93.8%

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

   3. Simplified 93.8%

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

      1. associate-/r* 93.7%

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

      2. clear-num 93.6%

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

      3. div-inv 93.6%

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

      4. associate-/r/ 93.6%

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

      5. associate-/r* 91.2%

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

   6. Applied egg-rr 91.2%

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

   7. Taylor expanded in y around 0 31.2%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\left(z \cdot y\right) \cdot 0.16666666666666666 + \frac{z}{y}}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot \frac{x}{z_m \cdot y}}{y}\\ \end{array} \end{array} \]

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 2.5) (/ x z_m) (/ (* 6.0 (/ x (* z_m y))) 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 <= 2.5) {
		tmp = x / z_m;
	} else {
		tmp = (6.0 * (x / (z_m * y))) / 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 <= 2.5d0) then
        tmp = x / z_m
    else
        tmp = (6.0d0 * (x / (z_m * y))) / 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 <= 2.5) {
		tmp = x / z_m;
	} else {
		tmp = (6.0 * (x / (z_m * y))) / 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 <= 2.5:
		tmp = x / z_m
	else:
		tmp = (6.0 * (x / (z_m * y))) / 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 <= 2.5)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(Float64(6.0 * Float64(x / Float64(z_m * y))) / 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 <= 2.5)
		tmp = x / z_m;
	else
		tmp = (6.0 * (x / (z_m * y))) / 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, 2.5], N[(x / z$95$m), $MachinePrecision], N[(N[(6.0 * N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.5

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0 77.0%

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

   if 2.5 < y

   1. Initial program 95.3%

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

      1. associate-*r/ 93.5%

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

      2. associate-/l/ 93.6%

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

      3. *-commutative 93.6%

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

   3. Simplified 93.6%

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

      1. associate-/r* 93.5%

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

      2. clear-num 93.3%

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

      3. div-inv 93.4%

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

      4. associate-/r/ 93.4%

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

      5. associate-/r* 90.9%

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

   6. Applied egg-rr 90.9%

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

   7. Taylor expanded in y around 0 29.3%

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

   8. Taylor expanded in y around inf 29.3%

      \[\leadsto \frac{\color{blue}{6 \cdot \frac{x}{y \cdot z}}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot \frac{x}{z \cdot y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.05 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z_m \cdot y}\\ \end{array} \end{array} \]

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 3.05e-9) (/ 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 <= 3.05e-9) {
		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 <= 3.05d-9) 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 <= 3.05e-9) {
		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 <= 3.05e-9:
		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 <= 3.05e-9)
		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 <= 3.05e-9)
		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, 3.05e-9], 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 < 3.05e-9

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0 77.2%

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

   if 3.05e-9 < y

   1. Initial program 95.4%

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

      1. associate-*r/ 93.7%

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

      2. associate-/l/ 93.8%

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

      3. *-commutative 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in y around 0 20.6%

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

      1. un-div-inv 20.6%

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

      2. clear-num 20.6%

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

   7. Applied egg-rr 20.6%

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

      1. *-un-lft-identity 20.6%

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

      2. *-inverses 20.6%

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

      3. times-frac 23.3%

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

      4. associate-/r* 23.7%

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

      5. clear-num 23.7%

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

      6. associate-/r/ 29.5%

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

   9. Applied egg-rr 29.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.05 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.1% accurate, 8.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 0.00037:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{z_m}{x}}\\ \end{array} \end{array} \]

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 0.00037) (/ 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 <= 0.00037) {
		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 <= 0.00037d0) 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 <= 0.00037) {
		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 <= 0.00037:
		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 <= 0.00037)
		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 <= 0.00037)
		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, 0.00037], 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 < 3.6999999999999999e-4

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0 77.2%

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

   if 3.6999999999999999e-4 < y

   1. Initial program 95.4%

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

      1. associate-*r/ 93.6%

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

      2. associate-/l/ 93.7%

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

      3. *-commutative 93.7%

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

   3. Simplified 93.7%

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

   5. Taylor expanded in y around 0 19.7%

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

      1. un-div-inv 19.7%

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

      2. clear-num 19.7%

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

   7. Applied egg-rr 19.7%

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

      1. *-un-lft-identity 19.7%

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

      2. *-inverses 19.7%

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

      3. times-frac 22.5%

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

      4. associate-/r* 22.8%

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

      5. clear-num 22.8%

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

      6. associate-/r/ 28.7%

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

   9. Applied egg-rr 28.7%

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

       1. *-commutative 28.7%

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

       2. clear-num 29.1%

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

       3. un-div-inv 29.1%

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

       4. *-un-lft-identity 29.1%

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

       5. times-frac 28.9%

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

       6. /-rgt-identity 28.9%

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

   11. Applied egg-rr 28.9%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00037:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.2% accurate, 8.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z_m \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

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 2.1e-8) (/ 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 <= 2.1e-8) {
		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 <= 2.1d-8) 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 <= 2.1e-8) {
		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 <= 2.1e-8:
		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 <= 2.1e-8)
		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 <= 2.1e-8)
		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, 2.1e-8], 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 < 2.09999999999999994e-8

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0 77.2%

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

   if 2.09999999999999994e-8 < y

   1. Initial program 95.4%

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

      1. associate-*r/ 93.7%

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

      2. associate-/l/ 93.8%

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

      3. *-commutative 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in y around 0 20.6%

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

      1. un-div-inv 20.6%

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

      2. clear-num 20.6%

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

   7. Applied egg-rr 20.6%

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

      1. *-un-lft-identity 20.6%

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

      2. *-inverses 20.6%

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

      3. times-frac 23.3%

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

      4. associate-/r* 23.7%

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

      5. clear-num 23.7%

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

      6. associate-/r/ 29.5%

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

   9. Applied egg-rr 29.5%

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

       1. associate-*l/ 23.3%

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

       2. *-commutative 23.3%

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

       3. associate-/l/ 20.4%

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

       4. associate-*l/ 20.4%

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

       5. associate-/l* 23.3%

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

       6. associate-/l/ 29.8%

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

   11. Applied egg-rr 29.8%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.8% 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 97.4%

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

3. Taylor expanded in y around 0 63.0%

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

4. Final simplification 63.0%

   \[\leadsto \frac{x}{z} \]
5. 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 2024024 
(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))