Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.0% → 99.2%

Time: 9.3s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.7 \cdot 10^{-98}:\\ \;\;\;\;x \cdot \frac{t\_0}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t\_0}{z\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (* z_s (if (<= z_m 2.7e-98) (* x (/ t_0 z_m)) (/ (* x t_0) z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double t_0 = sin(y) / y;
	double tmp;
	if (z_m <= 2.7e-98) {
		tmp = x * (t_0 / z_m);
	} else {
		tmp = (x * t_0) / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (z_m <= 2.7d-98) then
        tmp = x * (t_0 / z_m)
    else
        tmp = (x * t_0) / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (z_m <= 2.7e-98) {
		tmp = x * (t_0 / z_m);
	} else {
		tmp = (x * t_0) / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	t_0 = math.sin(y) / y
	tmp = 0
	if z_m <= 2.7e-98:
		tmp = x * (t_0 / z_m)
	else:
		tmp = (x * t_0) / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (z_m <= 2.7e-98)
		tmp = Float64(x * Float64(t_0 / z_m));
	else
		tmp = Float64(Float64(x * t_0) / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (z_m <= 2.7e-98)
		tmp = x * (t_0 / z_m);
	else
		tmp = (x * t_0) / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(z$95$s * If[LessEqual[z$95$m, 2.7e-98], N[(x * N[(t$95$0 / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.6999999999999999e-98

   1. Initial program 96.5%

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

      1. associate-/l* 96.3%

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

   3. Simplified 96.3%

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

   if 2.6999999999999999e-98 < 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. Add Preprocessing

Alternative 2: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+138}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{\sin y}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 3.6e+138)
    (* x (/ (/ (sin y) y) z_m))
    (* (/ x y) (/ (sin y) z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 3.6e+138) {
		tmp = x * ((sin(y) / y) / z_m);
	} else {
		tmp = (x / y) * (sin(y) / z_m);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 3.6d+138) then
        tmp = x * ((sin(y) / y) / z_m)
    else
        tmp = (x / y) * (sin(y) / z_m)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 3.6e+138) {
		tmp = x * ((Math.sin(y) / y) / z_m);
	} else {
		tmp = (x / y) * (Math.sin(y) / z_m);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 3.6e+138:
		tmp = x * ((math.sin(y) / y) / z_m)
	else:
		tmp = (x / y) * (math.sin(y) / z_m)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 3.6e+138)
		tmp = Float64(x * Float64(Float64(sin(y) / y) / z_m));
	else
		tmp = Float64(Float64(x / y) * Float64(sin(y) / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 3.6e+138)
		tmp = x * ((sin(y) / y) / z_m);
	else
		tmp = (x / y) * (sin(y) / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 3.6e+138], N[(x * N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.6000000000000001e138

   1. Initial program 98.2%

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

      1. associate-/l* 97.1%

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

   3. Simplified 97.1%

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

   if 3.6000000000000001e138 < y

   1. Initial program 93.7%

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

      1. associate-*r/ 93.9%

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

      2. associate-/r* 85.0%

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

      3. times-frac 93.8%

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

   4. Applied egg-rr 93.8%

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

Alternative 3: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{+139}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 1.7e+139)
    (* x (/ (/ (sin y) y) 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 <= 1.7e+139) {
		tmp = x * ((sin(y) / y) / 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 <= 1.7d+139) then
        tmp = x * ((sin(y) / y) / 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 <= 1.7e+139) {
		tmp = x * ((Math.sin(y) / y) / 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 <= 1.7e+139:
		tmp = x * ((math.sin(y) / y) / 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 <= 1.7e+139)
		tmp = Float64(x * Float64(Float64(sin(y) / y) / 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 <= 1.7e+139)
		tmp = x * ((sin(y) / y) / 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, 1.7e+139], N[(x * N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] / z$95$m), $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 < 1.7000000000000001e139

   1. Initial program 98.2%

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

      1. associate-/l* 97.1%

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

   3. Simplified 97.1%

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

   if 1.7000000000000001e139 < y

   1. Initial program 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*l/ 93.9%

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

      3. associate-/l* 93.7%

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

      4. associate-/l* 93.8%

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

   3. Simplified 93.8%

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

Alternative 4: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.32 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{z\_m \cdot y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 2.32e-8) (/ x z_m) (* x (/ (sin y) (* z_m y))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 2.32e-8) {
		tmp = x / z_m;
	} else {
		tmp = x * (sin(y) / (z_m * y));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 2.32d-8) then
        tmp = x / z_m
    else
        tmp = x * (sin(y) / (z_m * y))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 2.32e-8) {
		tmp = x / z_m;
	} else {
		tmp = x * (Math.sin(y) / (z_m * y));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 2.32e-8:
		tmp = x / z_m
	else:
		tmp = x * (math.sin(y) / (z_m * y))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 2.32e-8)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(x * Float64(sin(y) / Float64(z_m * y)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 2.32e-8)
		tmp = x / z_m;
	else
		tmp = x * (sin(y) / (z_m * y));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 2.32e-8], N[(x / z$95$m), $MachinePrecision], N[(x * N[(N[Sin[y], $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.31999999999999993e-8

   1. Initial program 98.5%

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

      1. associate-/l* 96.9%

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

      2. associate-/l/ 89.8%

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

      3. *-commutative 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in y around 0 76.8%

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

   if 2.31999999999999993e-8 < y

   1. Initial program 94.4%

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

      1. associate-/l* 91.0%

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

      2. associate-/l/ 91.1%

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

      3. *-commutative 91.1%

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

   3. Simplified 91.1%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.32 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.3% 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 #s(literal 1 binary64) 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 97.6%

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

   1. associate-/l* 95.6%

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

3. Simplified 95.6%

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

Alternative 6: 62.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{\frac{1}{x} \cdot \frac{y}{\frac{1}{z\_m}}}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 1.35e-7) (/ x z_m) (* y (/ 1.0 (* (/ 1.0 x) (/ y (/ 1.0 z_m))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1.35e-7) {
		tmp = x / z_m;
	} else {
		tmp = y * (1.0 / ((1.0 / x) * (y / (1.0 / z_m))));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 1.35d-7) then
        tmp = x / z_m
    else
        tmp = y * (1.0d0 / ((1.0d0 / x) * (y / (1.0d0 / 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 <= 1.35e-7) {
		tmp = x / z_m;
	} else {
		tmp = y * (1.0 / ((1.0 / x) * (y / (1.0 / z_m))));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 1.35e-7:
		tmp = x / z_m
	else:
		tmp = y * (1.0 / ((1.0 / x) * (y / (1.0 / z_m))))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 1.35e-7)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(y * Float64(1.0 / Float64(Float64(1.0 / x) * Float64(y / Float64(1.0 / z_m)))));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 1.35e-7)
		tmp = x / z_m;
	else
		tmp = y * (1.0 / ((1.0 / x) * (y / (1.0 / z_m))));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 1.35e-7], N[(x / z$95$m), $MachinePrecision], N[(y * N[(1.0 / N[(N[(1.0 / x), $MachinePrecision] * N[(y / N[(1.0 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.35000000000000004e-7

   1. Initial program 98.5%

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

      1. associate-/l* 96.9%

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

      2. associate-/l/ 89.8%

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

      3. *-commutative 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in y around 0 76.8%

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

   if 1.35000000000000004e-7 < y

   1. Initial program 94.4%

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

      1. *-commutative 94.4%

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

      2. associate-*l/ 94.6%

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

      3. associate-/l* 94.5%

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

      4. associate-/l* 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in y around 0 31.1%

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

      1. associate-/r* 31.2%

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

      2. *-un-lft-identity 31.2%

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

      3. times-frac 29.3%

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

   7. Applied egg-rr 29.3%

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

      1. associate-*l/ 29.3%

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

      2. *-un-lft-identity 29.3%

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

      3. clear-num 33.3%

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

   9. Applied egg-rr 33.3%

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

       1. *-un-lft-identity 33.3%

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

       2. div-inv 33.3%

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

       3. times-frac 33.5%

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

   11. Applied egg-rr 33.5%

       \[\leadsto y \cdot \frac{1}{\color{blue}{\frac{1}{x} \cdot \frac{y}{\frac{1}{z}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 62.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{y} \cdot \frac{z\_m \cdot y}{x}}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 5.6e-8) (/ x z_m) (/ 1.0 (* (/ 1.0 y) (/ (* z_m y) x))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 5.6e-8) {
		tmp = x / z_m;
	} else {
		tmp = 1.0 / ((1.0 / y) * ((z_m * y) / x));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 5.6d-8) then
        tmp = x / z_m
    else
        tmp = 1.0d0 / ((1.0d0 / y) * ((z_m * y) / x))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 5.6e-8) {
		tmp = x / z_m;
	} else {
		tmp = 1.0 / ((1.0 / y) * ((z_m * y) / x));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 5.6e-8:
		tmp = x / z_m
	else:
		tmp = 1.0 / ((1.0 / y) * ((z_m * y) / x))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 5.6e-8)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / y) * Float64(Float64(z_m * y) / x)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 5.6e-8)
		tmp = x / z_m;
	else
		tmp = 1.0 / ((1.0 / y) * ((z_m * y) / x));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 5.6e-8], N[(x / z$95$m), $MachinePrecision], N[(1.0 / N[(N[(1.0 / y), $MachinePrecision] * N[(N[(z$95$m * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.5999999999999999e-8

   1. Initial program 98.5%

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

      1. associate-/l* 96.9%

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

      2. associate-/l/ 89.8%

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

      3. *-commutative 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in y around 0 76.8%

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

   if 5.5999999999999999e-8 < y

   1. Initial program 94.4%

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

      1. associate-/l* 91.0%

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

      2. associate-/l/ 91.1%

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

      3. *-commutative 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in y around 0 17.3%

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

      1. div-inv 17.3%

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

      2. clear-num 17.3%

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

   7. Applied egg-rr 17.3%

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

      1. *-un-lft-identity 17.3%

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

      2. rgt-mult-inverse 17.3%

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

      3. un-div-inv 17.3%

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

      4. times-frac 21.9%

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

      5. *-un-lft-identity 21.9%

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

      6. times-frac 33.5%

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

   9. Applied egg-rr 33.5%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{y} \cdot \frac{z \cdot y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z\_m \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 1.35e-16) (/ x z_m) (/ y (* z_m (/ y x))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1.35e-16) {
		tmp = x / z_m;
	} else {
		tmp = y / (z_m * (y / x));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 1.35d-16) then
        tmp = x / z_m
    else
        tmp = y / (z_m * (y / x))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1.35e-16) {
		tmp = x / z_m;
	} else {
		tmp = y / (z_m * (y / x));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 1.35e-16:
		tmp = x / z_m
	else:
		tmp = y / (z_m * (y / x))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 1.35e-16)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(y / Float64(z_m * Float64(y / x)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 1.35e-16)
		tmp = x / z_m;
	else
		tmp = y / (z_m * (y / x));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 1.35e-16], N[(x / z$95$m), $MachinePrecision], N[(y / N[(z$95$m * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.35e-16

   1. Initial program 98.5%

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

      1. associate-/l* 96.8%

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

      2. associate-/l/ 89.7%

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

      3. *-commutative 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in y around 0 76.6%

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

   if 1.35e-16 < y

   1. Initial program 94.6%

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

      1. associate-/l* 91.3%

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

      2. associate-/l/ 91.3%

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

      3. *-commutative 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in y around 0 20.2%

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

      1. *-un-lft-identity 20.2%

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

      2. div-inv 20.2%

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

      3. *-commutative 20.2%

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

      4. *-inverses 20.2%

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

      5. times-frac 24.6%

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

      6. *-commutative 24.6%

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

      7. associate-*l/ 33.5%

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

      8. associate-/r* 33.6%

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

      9. clear-num 35.8%

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

      10. associate-*l/ 35.8%

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

      11. *-un-lft-identity 35.8%

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

      12. div-inv 35.8%

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

      13. clear-num 35.8%

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

   7. Applied egg-rr 35.8%

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

Alternative 9: 62.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{z\_m}{x}}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 1.35e-16) (/ x z_m) (/ y (* y (/ z_m x))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1.35e-16) {
		tmp = x / z_m;
	} else {
		tmp = y / (y * (z_m / x));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 1.35d-16) then
        tmp = x / z_m
    else
        tmp = y / (y * (z_m / x))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1.35e-16) {
		tmp = x / z_m;
	} else {
		tmp = y / (y * (z_m / x));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 1.35e-16:
		tmp = x / z_m
	else:
		tmp = y / (y * (z_m / x))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 1.35e-16)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(y / Float64(y * Float64(z_m / x)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 1.35e-16)
		tmp = x / z_m;
	else
		tmp = y / (y * (z_m / x));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 1.35e-16], N[(x / z$95$m), $MachinePrecision], N[(y / N[(y * N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.35e-16

   1. Initial program 98.5%

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

      1. associate-/l* 96.8%

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

      2. associate-/l/ 89.7%

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

      3. *-commutative 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in y around 0 76.6%

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

   if 1.35e-16 < y

   1. Initial program 94.6%

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

      1. *-commutative 94.6%

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

      2. associate-*l/ 94.8%

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

      3. associate-/l* 94.7%

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

      4. associate-/l* 94.8%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in y around 0 33.6%

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

      1. associate-/r* 33.5%

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

      2. *-un-lft-identity 33.5%

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

      3. times-frac 31.8%

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

   7. Applied egg-rr 31.8%

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

      1. associate-*l/ 31.8%

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

      2. *-un-lft-identity 31.8%

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

      3. clear-num 35.6%

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

   9. Applied egg-rr 35.6%

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

       1. un-div-inv 35.6%

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

       2. div-inv 35.6%

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

       3. clear-num 35.6%

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

   11. Applied egg-rr 35.6%

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

Alternative 10: 62.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 40000000:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{y}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 40000000.0) (/ x z_m) (* 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 <= 40000000.0) {
		tmp = x / z_m;
	} else {
		tmp = 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 <= 40000000.0d0) then
        tmp = x / z_m
    else
        tmp = 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 <= 40000000.0) {
		tmp = x / z_m;
	} else {
		tmp = 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 <= 40000000.0:
		tmp = x / z_m
	else:
		tmp = 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 <= 40000000.0)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(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 <= 40000000.0)
		tmp = x / z_m;
	else
		tmp = 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, 40000000.0], N[(x / z$95$m), $MachinePrecision], N[(y * N[(N[(x / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4e7

   1. Initial program 98.5%

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

      1. associate-/l* 96.9%

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

      2. associate-/l/ 89.9%

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

      3. *-commutative 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in y around 0 76.5%

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

   if 4e7 < y

   1. Initial program 94.2%

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

      1. *-commutative 94.2%

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

      2. associate-*l/ 94.4%

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

      3. associate-/l* 94.3%

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

      4. associate-/l* 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in y around 0 30.8%

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

Alternative 11: 60.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 40000000:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z\_m \cdot y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 40000000.0) (/ x z_m) (* x (/ y (* z_m y))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 40000000.0) {
		tmp = x / z_m;
	} else {
		tmp = x * (y / (z_m * y));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 40000000.0d0) then
        tmp = x / z_m
    else
        tmp = x * (y / (z_m * y))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 40000000.0) {
		tmp = x / z_m;
	} else {
		tmp = x * (y / (z_m * y));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 40000000.0:
		tmp = x / z_m
	else:
		tmp = x * (y / (z_m * y))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 40000000.0)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(x * Float64(y / Float64(z_m * y)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 40000000.0)
		tmp = x / z_m;
	else
		tmp = x * (y / (z_m * y));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 40000000.0], N[(x / z$95$m), $MachinePrecision], N[(x * N[(y / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4e7

   1. Initial program 98.5%

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

      1. associate-/l* 96.9%

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

      2. associate-/l/ 89.9%

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

      3. *-commutative 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in y around 0 76.5%

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

   if 4e7 < y

   1. Initial program 94.2%

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

      1. associate-/l* 90.6%

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

      2. associate-/l/ 90.7%

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

      3. *-commutative 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in y around 0 27.3%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 40000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.7% 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 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m) :precision binary64 (* z_s (/ x z_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

   1. associate-/l* 95.6%

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

   2. associate-/l/ 90.1%

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

   3. *-commutative 90.1%

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

3. Simplified 90.1%

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

5. Taylor expanded in y around 0 64.1%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
6. Add Preprocessing

Developer Target 1: 99.3% 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 2024150 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -42173720203427147/1000000000000000000000000000000000000000000000) (/ (* x (/ 1 (/ y (sin y)))) z) (if (< z 44467023691138110000000000000000000000000000000000000000000000000) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1 (/ y (sin y)))) z))))

  (/ (* x (/ (sin y) y)) z))