Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 99.8%

Time: 6.7s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{\sin y}{z_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z_m}}{\frac{y}{\sin y}}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2e-20) (/ x (/ y (/ (sin y) z_m))) (/ (/ x z_m) (/ y (sin 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 (z_m <= 2e-20) {
		tmp = x / (y / (sin(y) / z_m));
	} else {
		tmp = (x / z_m) / (y / sin(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 (z_m <= 2d-20) then
        tmp = x / (y / (sin(y) / z_m))
    else
        tmp = (x / z_m) / (y / sin(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 (z_m <= 2e-20) {
		tmp = x / (y / (Math.sin(y) / z_m));
	} else {
		tmp = (x / z_m) / (y / Math.sin(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 z_m <= 2e-20:
		tmp = x / (y / (math.sin(y) / z_m))
	else:
		tmp = (x / z_m) / (y / math.sin(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 (z_m <= 2e-20)
		tmp = Float64(x / Float64(y / Float64(sin(y) / z_m)));
	else
		tmp = Float64(Float64(x / z_m) / Float64(y / sin(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 (z_m <= 2e-20)
		tmp = x / (y / (sin(y) / z_m));
	else
		tmp = (x / z_m) / (y / sin(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[z$95$m, 2e-20], N[(x / N[(y / N[(N[Sin[y], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.99999999999999989e-20

   1. Initial program 95.7%

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

      1. associate-*r/ 97.1%

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

      2. associate-/l/ 84.0%

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

      3. *-commutative 84.0%

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

   3. Simplified 84.0%

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

      1. clear-num 83.9%

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

      2. un-div-inv 84.0%

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

      3. associate-/l* 93.9%

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

   5. Applied egg-rr 93.9%

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

   if 1.99999999999999989e-20 < z

   1. Initial program 99.8%

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

      1. associate-*r/ 92.4%

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

      2. associate-/l/ 90.6%

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

      3. *-commutative 90.6%

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

   3. Simplified 90.6%

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

      1. associate-*r/ 87.9%

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

      2. *-commutative 87.9%

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

      3. frac-times 99.8%

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

      4. clear-num 99.8%

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

      5. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{\sin y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \end{array} \]

Alternative 2: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{z_m \cdot y}\\ \end{array} \end{array} \]

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 5e-21) (/ 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 <= 5e-21) {
		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 <= 5d-21) 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 <= 5e-21) {
		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 <= 5e-21:
		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 <= 5e-21)
		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 <= 5e-21)
		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, 5e-21], 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 < 4.99999999999999973e-21

   1. Initial program 97.9%

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

   2. Taylor expanded in y around 0 72.9%

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

   if 4.99999999999999973e-21 < y

   1. Initial program 93.4%

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

      1. associate-*r/ 88.4%

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

      2. associate-/l/ 86.0%

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

      3. *-commutative 86.0%

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

   3. Simplified 86.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \end{array} \]

Alternative 3: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z_m}\\ \end{array} \end{array} \]

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 2.7e-14) (/ 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 <= 2.7e-14) {
		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 <= 2.7d-14) 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 <= 2.7e-14) {
		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 <= 2.7e-14:
		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 <= 2.7e-14)
		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 <= 2.7e-14)
		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, 2.7e-14], 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 < 2.6999999999999999e-14

   1. Initial program 97.9%

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

   2. Taylor expanded in y around 0 73.0%

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

   if 2.6999999999999999e-14 < y

   1. Initial program 93.3%

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

      1. *-lft-identity 93.3%

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

      2. metadata-eval 93.3%

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

      3. times-frac 93.3%

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

      4. neg-mul-1 93.3%

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

      5. distribute-lft-neg-out 93.3%

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

      6. associate-*r/ 93.4%

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

      7. associate-*l/ 93.3%

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

      8. *-commutative 93.3%

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

      9. times-frac 93.5%

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

      10. remove-double-neg 93.5%

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

      11. distribute-frac-neg 93.5%

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

      12. sin-neg 93.5%

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

      13. sin-neg 93.5%

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

      14. neg-mul-1 93.5%

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

      15. associate-/l* 93.4%

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

      16. associate-/r/ 93.5%

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

      17. distribute-lft-neg-in 93.5%

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

      18. metadata-eval 93.5%

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

      19. metadata-eval 93.5%

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

      20. neg-mul-1 93.5%

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

      21. sin-neg 93.5%

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

      22. *-commutative 93.5%

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

   3. Simplified 93.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]

Alternative 4: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+167}:\\ \;\;\;\;\frac{x}{z_m} \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z_m}\\ \end{array} \end{array} \]

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 1.25e+167)
    (* (/ x z_m) (/ (sin y) 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 <= 1.25e+167) {
		tmp = (x / z_m) * (sin(y) / 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 <= 1.25d+167) then
        tmp = (x / z_m) * (sin(y) / 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 <= 1.25e+167) {
		tmp = (x / z_m) * (Math.sin(y) / 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 <= 1.25e+167:
		tmp = (x / z_m) * (math.sin(y) / 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 <= 1.25e+167)
		tmp = Float64(Float64(x / z_m) * Float64(sin(y) / 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 <= 1.25e+167)
		tmp = (x / z_m) * (sin(y) / 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, 1.25e+167], N[(N[(x / z$95$m), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $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 < 1.2499999999999999e167

   1. Initial program 96.9%

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

      1. *-commutative 96.9%

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

      2. associate-*r/ 96.5%

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

   3. Simplified 96.5%

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

   if 1.2499999999999999e167 < y

   1. Initial program 95.3%

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

      1. *-lft-identity 95.3%

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

      2. metadata-eval 95.3%

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

      3. times-frac 95.3%

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

      4. neg-mul-1 95.3%

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

      5. distribute-lft-neg-out 95.3%

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

      6. associate-*r/ 95.3%

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

      7. associate-*l/ 95.4%

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

      8. *-commutative 95.4%

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

      9. times-frac 95.5%

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

      10. remove-double-neg 95.5%

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

      11. distribute-frac-neg 95.5%

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

      12. sin-neg 95.5%

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

      13. sin-neg 95.5%

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

      14. neg-mul-1 95.5%

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

      15. associate-/l* 95.5%

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

      16. associate-/r/ 95.5%

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

      17. distribute-lft-neg-in 95.5%

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

      18. metadata-eval 95.5%

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

      19. metadata-eval 95.5%

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

      20. neg-mul-1 95.5%

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

      21. sin-neg 95.5%

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

      22. *-commutative 95.5%

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

   3. Simplified 95.5%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+167}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]

Alternative 5: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2.45 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{\sin y}{z_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z_m} \cdot \frac{\sin y}{y}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2.45e-20)
    (/ x (/ y (/ (sin y) z_m)))
    (* (/ x z_m) (/ (sin 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 (z_m <= 2.45e-20) {
		tmp = x / (y / (sin(y) / z_m));
	} else {
		tmp = (x / z_m) * (sin(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 (z_m <= 2.45d-20) then
        tmp = x / (y / (sin(y) / z_m))
    else
        tmp = (x / z_m) * (sin(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 (z_m <= 2.45e-20) {
		tmp = x / (y / (Math.sin(y) / z_m));
	} else {
		tmp = (x / z_m) * (Math.sin(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 z_m <= 2.45e-20:
		tmp = x / (y / (math.sin(y) / z_m))
	else:
		tmp = (x / z_m) * (math.sin(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 (z_m <= 2.45e-20)
		tmp = Float64(x / Float64(y / Float64(sin(y) / z_m)));
	else
		tmp = Float64(Float64(x / z_m) * Float64(sin(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 (z_m <= 2.45e-20)
		tmp = x / (y / (sin(y) / z_m));
	else
		tmp = (x / z_m) * (sin(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[z$95$m, 2.45e-20], N[(x / N[(y / N[(N[Sin[y], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.4500000000000001e-20

   1. Initial program 95.7%

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

      1. associate-*r/ 97.1%

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

      2. associate-/l/ 84.0%

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

      3. *-commutative 84.0%

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

   3. Simplified 84.0%

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

      1. clear-num 83.9%

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

      2. un-div-inv 84.0%

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

      3. associate-/l* 93.9%

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

   5. Applied egg-rr 93.9%

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

   if 2.4500000000000001e-20 < z

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.45 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{\sin y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \end{array} \]

Alternative 6: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 10^{+16}:\\ \;\;\;\;\frac{x}{\frac{z_m}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z_m} \cdot t_0\\ \end{array} \end{array} \end{array} \]

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+16) (/ x (/ z_m t_0)) (* (/ x z_m) t_0)))))

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+16) {
		tmp = x / (z_m / t_0);
	} else {
		tmp = (x / z_m) * t_0;
	}
	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+16) then
        tmp = x / (z_m / t_0)
    else
        tmp = (x / z_m) * t_0
    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+16) {
		tmp = x / (z_m / t_0);
	} else {
		tmp = (x / z_m) * t_0;
	}
	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+16:
		tmp = x / (z_m / t_0)
	else:
		tmp = (x / z_m) * t_0
	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+16)
		tmp = Float64(x / Float64(z_m / t_0));
	else
		tmp = Float64(Float64(x / z_m) * t_0);
	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+16)
		tmp = x / (z_m / t_0);
	else
		tmp = (x / z_m) * t_0;
	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+16], N[(x / N[(z$95$m / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1e16

   1. Initial program 95.8%

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

      1. associate-/l* 97.0%

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

   3. Simplified 97.0%

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

   if 1e16 < z

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+16}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \end{array} \]

Alternative 7: 60.9% accurate, 10.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{+74}:\\ \;\;\;\;\frac{1}{\frac{z_m}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z_m \cdot \left(-y\right)}{y}}\\ \end{array} \end{array} \]

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 4.6e+74) (/ 1.0 (/ z_m x)) (/ 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 <= 4.6e+74) {
		tmp = 1.0 / (z_m / x);
	} else {
		tmp = 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 <= 4.6d+74) then
        tmp = 1.0d0 / (z_m / x)
    else
        tmp = 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 <= 4.6e+74) {
		tmp = 1.0 / (z_m / x);
	} else {
		tmp = 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 <= 4.6e+74:
		tmp = 1.0 / (z_m / x)
	else:
		tmp = 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 <= 4.6e+74)
		tmp = Float64(1.0 / Float64(z_m / x));
	else
		tmp = Float64(x / Float64(Float64(z_m * Float64(-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 <= 4.6e+74)
		tmp = 1.0 / (z_m / x);
	else
		tmp = 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, 4.6e+74], N[(1.0 / N[(z$95$m / x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(z$95$m * (-y)), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.5999999999999997e74

   1. Initial program 98.1%

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

      1. associate-*r/ 98.8%

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

      2. associate-/l/ 86.8%

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

      3. *-commutative 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in y around 0 70.4%

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

      1. div-inv 70.7%

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

      2. clear-num 71.0%

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

   6. Applied egg-rr 71.0%

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

   if 4.5999999999999997e74 < y

   1. Initial program 90.9%

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

      1. associate-*r/ 83.9%

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

      2. associate-/l/ 80.5%

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

      3. *-commutative 80.5%

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

   3. Simplified 80.5%

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

      1. clear-num 80.4%

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

      2. un-div-inv 80.5%

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

      3. associate-/l* 80.5%

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

   5. Applied egg-rr 80.5%

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

   6. Taylor expanded in y around 0 21.3%

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

      1. frac-2neg 21.3%

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

      2. div-inv 21.3%

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

      3. distribute-neg-frac 21.3%

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

   8. Applied egg-rr 21.3%

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

      1. un-div-inv 21.3%

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

      2. neg-mul-1 21.3%

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

      3. *-commutative 21.3%

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

      4. clear-num 21.3%

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

      5. associate-/r/ 21.3%

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

      6. frac-times 33.6%

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

      7. associate-*r/ 33.6%

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

      8. associate-/r/ 33.6%

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

      9. /-rgt-identity 33.6%

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

      10. *-commutative 33.6%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 3.7%

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

      13. sqr-neg 3.7%

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

      14. sqrt-unprod 31.2%

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

      15. add-sqr-sqrt 31.2%

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

   10. Applied egg-rr 31.2%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{+74}:\\ \;\;\;\;\frac{1}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(-y\right)}{y}}\\ \end{array} \]

Alternative 8: 61.1% accurate, 11.8× speedup

Math

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

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 1e-6) (/ x z_m) (/ 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 <= 1e-6) {
		tmp = x / z_m;
	} else {
		tmp = 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 <= 1d-6) then
        tmp = x / z_m
    else
        tmp = 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 <= 1e-6) {
		tmp = x / z_m;
	} else {
		tmp = 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 <= 1e-6:
		tmp = x / z_m
	else:
		tmp = 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 <= 1e-6)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(x / Float64(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 <= 1e-6)
		tmp = x / z_m;
	else
		tmp = 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, 1e-6], N[(x / z$95$m), $MachinePrecision], N[(x / N[(N[(z$95$m * y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999955e-7

   1. Initial program 98.0%

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

   2. Taylor expanded in y around 0 73.4%

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

   if 9.99999999999999955e-7 < y

   1. Initial program 93.0%

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

      1. associate-*r/ 87.7%

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

      2. associate-/l/ 85.2%

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

      3. *-commutative 85.2%

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

   3. Simplified 85.2%

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

      1. clear-num 85.1%

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

      2. un-div-inv 85.2%

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

      3. associate-/l* 85.3%

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

   5. Applied egg-rr 85.3%

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

   6. Taylor expanded in y around 0 25.4%

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

      1. frac-2neg 25.4%

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

      2. div-inv 25.4%

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

      3. distribute-neg-frac 25.4%

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

   8. Applied egg-rr 25.4%

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

      1. distribute-lft-neg-out 25.4%

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

      2. distribute-rgt-neg-in 25.4%

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

      3. clear-num 25.5%

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

      4. distribute-frac-neg 25.5%

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

      5. frac-2neg 25.5%

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

      6. associate-*r/ 35.2%

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

      7. *-commutative 35.2%

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

   10. Applied egg-rr 35.2%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-6}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot y}{y}}\\ \end{array} \]

Alternative 9: 58.8% 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 96.7%

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

   1. associate-*r/ 95.9%

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

   2. associate-/l/ 85.6%

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

   3. *-commutative 85.6%

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

3. Simplified 85.6%

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

4. Taylor expanded in y around 0 61.1%

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

   1. div-inv 61.3%

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

   2. clear-num 61.5%

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

6. Applied egg-rr 61.5%

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

7. Final simplification 61.5%

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

Alternative 10: 59.0% 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 96.7%

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

2. Taylor expanded in y around 0 61.3%

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

3. Final simplification 61.3%

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

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 2023334 
(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))