Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 99.6%

Time: 10.5s

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: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z\_m}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2e-51) (/ x (* y (/ z_m (sin y)))) (/ (* x (/ (sin y) 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 (z_m <= 2e-51) {
		tmp = x / (y * (z_m / sin(y)));
	} else {
		tmp = (x * (sin(y) / 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 (z_m <= 2d-51) then
        tmp = x / (y * (z_m / sin(y)))
    else
        tmp = (x * (sin(y) / 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 (z_m <= 2e-51) {
		tmp = x / (y * (z_m / Math.sin(y)));
	} else {
		tmp = (x * (Math.sin(y) / 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 z_m <= 2e-51:
		tmp = x / (y * (z_m / math.sin(y)))
	else:
		tmp = (x * (math.sin(y) / 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 (z_m <= 2e-51)
		tmp = Float64(x / Float64(y * Float64(z_m / sin(y))));
	else
		tmp = Float64(Float64(x * Float64(sin(y) / 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 (z_m <= 2e-51)
		tmp = x / (y * (z_m / sin(y)));
	else
		tmp = (x * (sin(y) / 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[z$95$m, 2e-51], N[(x / N[(y * N[(z$95$m / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2e-51

   1. Initial program 94.3%

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

      1. associate-/l* 95.9%

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

      2. associate-/l/ 88.3%

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

      3. *-commutative 88.3%

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

   3. Simplified 88.3%

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

      1. associate-/r* 95.9%

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

      2. clear-num 95.8%

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

      3. un-div-inv 96.0%

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

      4. clear-num 95.9%

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

      5. associate-/r* 88.2%

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

      6. clear-num 88.3%

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

      7. associate-/l* 92.4%

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

   6. Applied egg-rr 92.4%

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

   if 2e-51 < 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 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{z\_m \cdot y}\\ \end{array} \end{array} \]

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 1.25e-30) (/ 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 <= 1.25e-30) {
		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 <= 1.25d-30) 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 <= 1.25e-30) {
		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 <= 1.25e-30:
		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 <= 1.25e-30)
		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 <= 1.25e-30)
		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, 1.25e-30], 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 < 1.24999999999999993e-30

   1. Initial program 96.3%

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

      1. associate-/l* 95.2%

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

      2. associate-/l/ 87.4%

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

      3. *-commutative 87.4%

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

   3. Simplified 87.4%

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

   5. Taylor expanded in y around 0 65.1%

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

   if 1.24999999999999993e-30 < y

   1. Initial program 94.5%

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

      1. associate-/l* 97.9%

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

      2. associate-/l/ 97.2%

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

      3. *-commutative 97.2%

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

   3. Simplified 97.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 95.8%

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

   1. associate-/l* 96.0%

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

3. Simplified 96.0%

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

5. Final simplification 96.0%

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

Alternative 4: 62.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\left(z\_m \cdot y\right) \cdot 0.16666666666666666 + \frac{z\_m}{y}}}{y}\\ \end{array} \end{array} \]

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 1.65e-16)
    (/ 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 <= 1.65e-16) {
		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 <= 1.65d-16) 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 <= 1.65e-16) {
		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 <= 1.65e-16:
		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 <= 1.65e-16)
		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 <= 1.65e-16)
		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, 1.65e-16], 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 < 1.64999999999999994e-16

   1. Initial program 96.3%

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

      1. associate-/l* 95.3%

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

      2. associate-/l/ 87.5%

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

      3. *-commutative 87.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in y around 0 65.4%

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

   if 1.64999999999999994e-16 < y

   1. Initial program 94.3%

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

      1. *-commutative 94.3%

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

      2. associate-*l/ 94.3%

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

      3. associate-/l* 94.3%

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

      4. associate-/l* 94.3%

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

   3. Simplified 94.3%

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

      1. associate-/r* 96.9%

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

      2. *-un-lft-identity 96.9%

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

      3. times-frac 91.7%

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

   6. Applied egg-rr 91.7%

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

      1. *-commutative 91.7%

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

      2. frac-times 96.9%

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

      3. *-un-lft-identity 96.9%

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

      4. associate-/r/ 97.0%

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

      5. associate-*r/ 97.0%

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

      6. *-commutative 97.0%

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

      7. associate-/r* 91.8%

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

   8. Applied egg-rr 91.8%

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

   9. Taylor expanded in y around 0 43.6%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\left(z \cdot y\right) \cdot 0.16666666666666666 + \frac{z}{y}}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y \cdot \left(z\_m \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

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 2.4)
    (/ x z_m)
    (* (/ 1.0 y) (/ x (* y (* z_m 0.16666666666666666)))))))

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 <= 2.4) {
		tmp = x / z_m;
	} else {
		tmp = (1.0 / y) * (x / (y * (z_m * 0.16666666666666666)));
	}
	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 <= 2.4d0) then
        tmp = x / z_m
    else
        tmp = (1.0d0 / y) * (x / (y * (z_m * 0.16666666666666666d0)))
    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 <= 2.4) {
		tmp = x / z_m;
	} else {
		tmp = (1.0 / y) * (x / (y * (z_m * 0.16666666666666666)));
	}
	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 <= 2.4:
		tmp = x / z_m
	else:
		tmp = (1.0 / y) * (x / (y * (z_m * 0.16666666666666666)))
	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 <= 2.4)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / Float64(y * Float64(z_m * 0.16666666666666666))));
	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 <= 2.4)
		tmp = x / z_m;
	else
		tmp = (1.0 / y) * (x / (y * (z_m * 0.16666666666666666)));
	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, 2.4], N[(x / z$95$m), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / N[(y * N[(z$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 96.3%

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

      1. associate-/l* 95.3%

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

      2. associate-/l/ 87.7%

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

      3. *-commutative 87.7%

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

   3. Simplified 87.7%

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

   5. Taylor expanded in y around 0 65.7%

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

   if 2.39999999999999991 < y

   1. Initial program 94.1%

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

      1. associate-/l* 97.8%

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

      2. associate-/l/ 97.0%

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

      3. *-commutative 97.0%

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

   3. Simplified 97.0%

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

      1. associate-/r* 97.8%

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

      2. clear-num 96.9%

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

      3. un-div-inv 96.9%

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

      4. clear-num 96.9%

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

      5. associate-/r* 96.9%

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

      6. clear-num 96.9%

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

      7. associate-/l* 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in y around 0 39.7%

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

   8. Taylor expanded in y around inf 39.7%

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(y \cdot z\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 39.7%

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

      2. associate-*r* 39.7%

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

   10. Simplified 39.7%

       \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y \cdot \left(z \cdot 0.16666666666666666\right)\right)}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 39.7%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot \left(y \cdot \left(z \cdot 0.16666666666666666\right)\right)} \]

       2. times-frac 41.2%

          \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y \cdot \left(z \cdot 0.16666666666666666\right)}} \]

   12. Applied egg-rr 41.2%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y \cdot \left(z \cdot 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(\left(z\_m \cdot y\right) \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

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 2.4) (/ x z_m) (/ x (* y (* (* z_m y) 0.16666666666666666))))))

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 <= 2.4) {
		tmp = x / z_m;
	} else {
		tmp = x / (y * ((z_m * y) * 0.16666666666666666));
	}
	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 <= 2.4d0) then
        tmp = x / z_m
    else
        tmp = x / (y * ((z_m * y) * 0.16666666666666666d0))
    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 <= 2.4) {
		tmp = x / z_m;
	} else {
		tmp = x / (y * ((z_m * y) * 0.16666666666666666));
	}
	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 <= 2.4:
		tmp = x / z_m
	else:
		tmp = x / (y * ((z_m * y) * 0.16666666666666666))
	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 <= 2.4)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(x / Float64(y * Float64(Float64(z_m * y) * 0.16666666666666666)));
	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 <= 2.4)
		tmp = x / z_m;
	else
		tmp = x / (y * ((z_m * y) * 0.16666666666666666));
	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, 2.4], N[(x / z$95$m), $MachinePrecision], N[(x / N[(y * N[(N[(z$95$m * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 96.3%

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

      1. associate-/l* 95.3%

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

      2. associate-/l/ 87.7%

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

      3. *-commutative 87.7%

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

   3. Simplified 87.7%

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

   5. Taylor expanded in y around 0 65.7%

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

   if 2.39999999999999991 < y

   1. Initial program 94.1%

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

      1. associate-/l* 97.8%

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

      2. associate-/l/ 97.0%

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

      3. *-commutative 97.0%

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

   3. Simplified 97.0%

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

      1. associate-/r* 97.8%

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

      2. clear-num 96.9%

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

      3. un-div-inv 96.9%

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

      4. clear-num 96.9%

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

      5. associate-/r* 96.9%

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

      6. clear-num 96.9%

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

      7. associate-/l* 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in y around 0 39.7%

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

   8. Taylor expanded in y around inf 39.7%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(\left(z \cdot y\right) \cdot 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y \cdot \left(z\_m \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

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 2.4) (/ x z_m) (/ (/ x y) (* y (* z_m 0.16666666666666666))))))

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 <= 2.4) {
		tmp = x / z_m;
	} else {
		tmp = (x / y) / (y * (z_m * 0.16666666666666666));
	}
	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 <= 2.4d0) then
        tmp = x / z_m
    else
        tmp = (x / y) / (y * (z_m * 0.16666666666666666d0))
    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 <= 2.4) {
		tmp = x / z_m;
	} else {
		tmp = (x / y) / (y * (z_m * 0.16666666666666666));
	}
	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 <= 2.4:
		tmp = x / z_m
	else:
		tmp = (x / y) / (y * (z_m * 0.16666666666666666))
	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 <= 2.4)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(Float64(x / y) / Float64(y * Float64(z_m * 0.16666666666666666)));
	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 <= 2.4)
		tmp = x / z_m;
	else
		tmp = (x / y) / (y * (z_m * 0.16666666666666666));
	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, 2.4], N[(x / z$95$m), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y * N[(z$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 96.3%

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

      1. associate-/l* 95.3%

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

      2. associate-/l/ 87.7%

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

      3. *-commutative 87.7%

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

   3. Simplified 87.7%

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

   5. Taylor expanded in y around 0 65.7%

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

   if 2.39999999999999991 < y

   1. Initial program 94.1%

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

      1. associate-/l* 97.8%

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

      2. associate-/l/ 97.0%

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

      3. *-commutative 97.0%

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

   3. Simplified 97.0%

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

      1. associate-/r* 97.8%

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

      2. clear-num 96.9%

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

      3. un-div-inv 96.9%

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

      4. clear-num 96.9%

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

      5. associate-/r* 96.9%

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

      6. clear-num 96.9%

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

      7. associate-/l* 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in y around 0 39.7%

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

   8. Taylor expanded in y around inf 39.7%

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(y \cdot z\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 39.7%

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

      2. associate-*r* 39.7%

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

   10. Simplified 39.7%

       \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y \cdot \left(z \cdot 0.16666666666666666\right)\right)}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 39.7%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot \left(y \cdot \left(z \cdot 0.16666666666666666\right)\right)} \]

       2. times-frac 41.2%

          \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y \cdot \left(z \cdot 0.16666666666666666\right)}} \]

   12. Applied egg-rr 41.2%

       \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y \cdot \left(z \cdot 0.16666666666666666\right)}} \]
   13. Step-by-step derivation

       1. *-commutative 41.2%

          \[\leadsto \color{blue}{\frac{x}{y \cdot \left(z \cdot 0.16666666666666666\right)} \cdot \frac{1}{y}} \]

       2. associate-*l/ 39.8%

          \[\leadsto \color{blue}{\frac{x \cdot \frac{1}{y}}{y \cdot \left(z \cdot 0.16666666666666666\right)}} \]

       3. associate-/l* 39.8%

          \[\leadsto \frac{\color{blue}{\frac{x \cdot 1}{y}}}{y \cdot \left(z \cdot 0.16666666666666666\right)} \]

       4. *-rgt-identity 39.8%

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

   14. Simplified 39.8%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y \cdot \left(z \cdot 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 50000000:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z\_m \cdot y}{y}}\\ \end{array} \end{array} \]

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 50000000.0) (/ 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 <= 50000000.0) {
		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 <= 50000000.0d0) 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 <= 50000000.0) {
		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 <= 50000000.0:
		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 <= 50000000.0)
		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 <= 50000000.0)
		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, 50000000.0], 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 < 5e7

   1. Initial program 96.3%

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

      1. associate-/l* 95.3%

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

      2. associate-/l/ 87.7%

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

      3. *-commutative 87.7%

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

   3. Simplified 87.7%

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

   5. Taylor expanded in y around 0 65.7%

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

   if 5e7 < y

   1. Initial program 94.1%

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

      1. associate-/l* 97.8%

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

      2. associate-/l/ 97.0%

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

      3. *-commutative 97.0%

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

   3. Simplified 97.0%

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

      1. associate-/r* 97.8%

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

      2. clear-num 96.9%

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

      3. un-div-inv 96.9%

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

      4. clear-num 96.9%

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

      5. associate-/r* 96.9%

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

      6. clear-num 96.9%

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

      7. associate-/l* 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in y around 0 24.6%

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

      1. associate-*r/ 32.5%

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

   9. Applied egg-rr 32.5%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 50000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot y}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.6% 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 95.8%

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

   1. associate-/l* 96.0%

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

   2. associate-/l/ 90.2%

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

   3. *-commutative 90.2%

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

3. Simplified 90.2%

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

5. Taylor expanded in y around 0 54.7%

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

   1. div-inv 54.9%

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

   2. clear-num 55.2%

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

7. Applied egg-rr 55.2%

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

8. Final simplification 55.2%

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

Alternative 10: 57.7% 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 95.8%

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

   1. associate-/l* 96.0%

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

   2. associate-/l/ 90.2%

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

   3. *-commutative 90.2%

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

3. Simplified 90.2%

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

5. Taylor expanded in y around 0 54.9%

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

6. Final simplification 54.9%

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

Developer target: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\sin y}\\ t_1 := \frac{x \cdot \frac{1}{t\_0}}{z}\\ \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (sin y))) (t_1 (/ (* x (/ 1.0 t_0)) z)))
   (if (< z -4.2173720203427147e-29)
     t_1
     (if (< z 4.446702369113811e+64) (/ x (* z t_0)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = y / sin(y);
	double t_1 = (x * (1.0 / t_0)) / z;
	double tmp;
	if (z < -4.2173720203427147e-29) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x / (z * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y / sin(y)
    t_1 = (x * (1.0d0 / t_0)) / z
    if (z < (-4.2173720203427147d-29)) then
        tmp = t_1
    else if (z < 4.446702369113811d+64) then
        tmp = x / (z * t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / Math.sin(y);
	double t_1 = (x * (1.0 / t_0)) / z;
	double tmp;
	if (z < -4.2173720203427147e-29) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x / (z * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / math.sin(y)
	t_1 = (x * (1.0 / t_0)) / z
	tmp = 0
	if z < -4.2173720203427147e-29:
		tmp = t_1
	elif z < 4.446702369113811e+64:
		tmp = x / (z * t_0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / sin(y))
	t_1 = Float64(Float64(x * Float64(1.0 / t_0)) / z)
	tmp = 0.0
	if (z < -4.2173720203427147e-29)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = Float64(x / Float64(z * t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / sin(y);
	t_1 = (x * (1.0 / t_0)) / z;
	tmp = 0.0;
	if (z < -4.2173720203427147e-29)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = x / (z * t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Less[z, -4.2173720203427147e-29], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x / N[(z * t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024040 
(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))