Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.0% → 99.7%

Time: 9.9s

Alternatives: 8

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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 7e-10) (/ x (/ y (/ (sin y) z_m))) (/ (* 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 <= 7e-10) {
		tmp = x / (y / (sin(y) / z_m));
	} 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 <= 7d-10) then
        tmp = x / (y / (sin(y) / z_m))
    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 <= 7e-10) {
		tmp = x / (y / (Math.sin(y) / z_m));
	} 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 <= 7e-10:
		tmp = x / (y / (math.sin(y) / z_m))
	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 <= 7e-10)
		tmp = Float64(x / Float64(y / Float64(sin(y) / z_m)));
	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 <= 7e-10)
		tmp = x / (y / (sin(y) / z_m));
	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, 7e-10], N[(x / N[(y / N[(N[Sin[y], $MachinePrecision] / z$95$m), $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 < 6.99999999999999961e-10

   1. Initial program 96.6%

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. associate-*l/ N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z \cdot y}\right), \color{blue}{\sin y}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(z \cdot y\right)\right), \sin \color{blue}{y}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y \cdot z\right)\right), \sin y\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \sin y\right) \]

      8. sin-lowering-sin.f64 86.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sin.f64}\left(y\right)\right) \]

   4. Applied egg-rr 86.0%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. associate-/l/ N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{y}{\frac{\sin y}{z}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{\sin y}{z}\right)}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\sin y, \color{blue}{z}\right)\right)\right) \]

      9. sin-lowering-sin.f64 91.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), z\right)\right)\right) \]

   6. Applied egg-rr 91.7%

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

   if 6.99999999999999961e-10 < 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: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{\sin y}{z\_m}}}\\ \end{array} \end{array} \]

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 1.8e-26) (/ x 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 <= 1.8e-26) {
		tmp = x / 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 <= 1.8d-26) then
        tmp = x / 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 <= 1.8e-26) {
		tmp = x / 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 <= 1.8e-26:
		tmp = x / 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 <= 1.8e-26)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(x / Float64(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 <= 1.8e-26)
		tmp = x / 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, 1.8e-26], N[(x / z$95$m), $MachinePrecision], N[(x / N[(y / N[(N[Sin[y], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.8000000000000001e-26

   1. Initial program 98.4%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 76.1%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 76.1%

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

   if 1.8000000000000001e-26 < y

   1. Initial program 95.0%

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. associate-*l/ N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z \cdot y}\right), \color{blue}{\sin y}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(z \cdot y\right)\right), \sin \color{blue}{y}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y \cdot z\right)\right), \sin y\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \sin y\right) \]

      8. sin-lowering-sin.f64 91.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sin.f64}\left(y\right)\right) \]

   4. Applied egg-rr 91.9%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. associate-/l/ N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{y}{\frac{\sin y}{z}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{\sin y}{z}\right)}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\sin y, \color{blue}{z}\right)\right)\right) \]

      9. sin-lowering-sin.f64 92.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), z\right)\right)\right) \]

   6. Applied egg-rr 92.0%

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

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 10^{-5}:\\ \;\;\;\;\frac{x}{z\_m \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{z\_m \cdot y}\\ \end{array} \end{array} \]

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 1e-5)
    (/ x (* z_m (+ 1.0 (* (* y y) 0.16666666666666666))))
    (* (sin y) (/ x (* z_m y))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1e-5) {
		tmp = x / (z_m * (1.0 + ((y * y) * 0.16666666666666666)));
	} else {
		tmp = sin(y) * (x / (z_m * y));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 1d-5) then
        tmp = x / (z_m * (1.0d0 + ((y * y) * 0.16666666666666666d0)))
    else
        tmp = sin(y) * (x / (z_m * y))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1e-5) {
		tmp = x / (z_m * (1.0 + ((y * y) * 0.16666666666666666)));
	} else {
		tmp = Math.sin(y) * (x / (z_m * y));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 1e-5:
		tmp = x / (z_m * (1.0 + ((y * y) * 0.16666666666666666)))
	else:
		tmp = math.sin(y) * (x / (z_m * y))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 1e-5)
		tmp = Float64(x / Float64(z_m * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))));
	else
		tmp = Float64(sin(y) * Float64(x / Float64(z_m * y)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 1e-5)
		tmp = x / (z_m * (1.0 + ((y * y) * 0.16666666666666666)));
	else
		tmp = sin(y) * (x / (z_m * y));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 1e-5], N[(x / N[(z$95$m * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000008e-5

   1. Initial program 98.5%

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. associate-*l/ N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z \cdot y}\right), \color{blue}{\sin y}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(z \cdot y\right)\right), \sin \color{blue}{y}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y \cdot z\right)\right), \sin y\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \sin y\right) \]

      8. sin-lowering-sin.f64 84.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sin.f64}\left(y\right)\right) \]

   4. Applied egg-rr 84.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. associate-/l/ N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{y}{\frac{\sin y}{z}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{\sin y}{z}\right)}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\sin y, \color{blue}{z}\right)\right)\right) \]

      9. sin-lowering-sin.f64 84.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), z\right)\right)\right) \]

   6. Applied egg-rr 84.4%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(z + \frac{1}{6} \cdot \left({y}^{2} \cdot z\right)\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(z + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{z}\right)\right) \]

      2. distribute-rgt1-in N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{z}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(z \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \left(1 + \color{blue}{\frac{1}{6} \cdot {y}^{2}}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 81.9%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right)\right) \]

   9. Simplified 81.9%

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

   if 1.00000000000000008e-5 < y

   1. Initial program 94.3%

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. associate-*l/ N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z \cdot y}\right), \color{blue}{\sin y}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(z \cdot y\right)\right), \sin \color{blue}{y}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y \cdot z\right)\right), \sin y\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \sin y\right) \]

      8. sin-lowering-sin.f64 90.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sin.f64}\left(y\right)\right) \]

   4. Applied egg-rr 90.9%

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

4. Final simplification 83.8%

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

Alternative 4: 62.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z\_m}{\frac{x}{y}} \cdot \frac{1}{y}}\\ \end{array} \end{array} \]

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 1.55e-15) (/ x z_m) (/ 1.0 (* (/ z_m (/ x y)) (/ 1.0 y))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1.55e-15) {
		tmp = x / z_m;
	} else {
		tmp = 1.0 / ((z_m / (x / y)) * (1.0 / y));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 1.55d-15) then
        tmp = x / z_m
    else
        tmp = 1.0d0 / ((z_m / (x / y)) * (1.0d0 / y))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1.55e-15) {
		tmp = x / z_m;
	} else {
		tmp = 1.0 / ((z_m / (x / y)) * (1.0 / y));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 1.55e-15:
		tmp = x / z_m
	else:
		tmp = 1.0 / ((z_m / (x / y)) * (1.0 / y))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 1.55e-15)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(1.0 / Float64(Float64(z_m / Float64(x / y)) * Float64(1.0 / y)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 1.55e-15)
		tmp = x / z_m;
	else
		tmp = 1.0 / ((z_m / (x / y)) * (1.0 / y));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 1.55e-15], N[(x / z$95$m), $MachinePrecision], N[(1.0 / N[(N[(z$95$m / N[(x / y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.5499999999999999e-15

   1. Initial program 98.5%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 76.6%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 76.6%

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

   if 1.5499999999999999e-15 < y

   1. Initial program 94.6%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 20.2%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 20.2%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{z}{x}\right)}\right) \]

      3. /-lowering-/.f64 20.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   7. Applied egg-rr 20.2%

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

      1. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{z}{x}}{\color{blue}{1}}\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{z}{\color{blue}{1 \cdot x}}\right)\right) \]

      3. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{z}{\frac{y}{y} \cdot x}\right)\right) \]

      4. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{z}{\frac{y}{\color{blue}{\frac{y}{x}}}}\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\frac{y}{\frac{y}{x}}}{z}}}\right)\right) \]

      6. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\frac{y}{\color{blue}{z \cdot \frac{y}{x}}}}\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{z \cdot \frac{y}{x}}{\color{blue}{y}}\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(z \cdot \frac{y}{x}\right) \cdot \color{blue}{\frac{1}{y}}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(z \cdot \frac{y}{x}\right), \color{blue}{\left(\frac{1}{y}\right)}\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(z \cdot \frac{1}{\frac{x}{y}}\right), \left(\frac{1}{y}\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{z}{\frac{x}{y}}\right), \left(\frac{\color{blue}{1}}{y}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \left(\frac{x}{y}\right)\right), \left(\frac{\color{blue}{1}}{y}\right)\right)\right) \]

      13. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \left(x \cdot \frac{1}{y}\right)\right), \left(\frac{1}{y}\right)\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \left(\frac{x \cdot 1}{y}\right)\right), \left(\frac{1}{y}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \left(\frac{1 \cdot x}{y}\right)\right), \left(\frac{1}{y}\right)\right)\right) \]

      16. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \left(\frac{\frac{y}{y} \cdot x}{y}\right)\right), \left(\frac{1}{y}\right)\right)\right) \]

      17. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \left(\frac{\frac{y}{\frac{y}{x}}}{y}\right)\right), \left(\frac{1}{y}\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{/.f64}\left(\left(\frac{y}{\frac{y}{x}}\right), y\right)\right), \left(\frac{1}{y}\right)\right)\right) \]

      19. frac-2neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{/.f64}\left(\left(\frac{y}{\frac{\mathsf{neg}\left(y\right)}{\mathsf{neg}\left(x\right)}}\right), y\right)\right), \left(\frac{1}{y}\right)\right)\right) \]

      20. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{/.f64}\left(\left(\frac{y}{\mathsf{neg}\left(y\right)} \cdot \left(\mathsf{neg}\left(x\right)\right)\right), y\right)\right), \left(\frac{1}{y}\right)\right)\right) \]

      21. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{y}{y}\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right), y\right)\right), \left(\frac{1}{y}\right)\right)\right) \]

      22. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right), y\right)\right), \left(\frac{1}{y}\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{/.f64}\left(\left(-1 \cdot \left(\mathsf{neg}\left(x\right)\right)\right), y\right)\right), \left(\frac{1}{y}\right)\right)\right) \]

      24. neg-mul-1 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right), y\right)\right), \left(\frac{1}{y}\right)\right)\right) \]

      25. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{/.f64}\left(x, y\right)\right), \left(\frac{1}{y}\right)\right)\right) \]

      26. /-lowering-/.f64 35.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{/.f64}\left(x, y\right)\right), \mathsf{/.f64}\left(1, \color{blue}{y}\right)\right)\right) \]

   9. Applied egg-rr 35.8%

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

Alternative 5: 62.5% accurate, 8.9× speedup

Math

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

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 1e-16) (/ x z_m) (/ y (/ y (/ x z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1e-16) {
		tmp = x / z_m;
	} else {
		tmp = y / (y / (x / z_m));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 1d-16) then
        tmp = x / z_m
    else
        tmp = y / (y / (x / z_m))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1e-16) {
		tmp = x / z_m;
	} else {
		tmp = y / (y / (x / z_m));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 1e-16:
		tmp = x / z_m
	else:
		tmp = y / (y / (x / z_m))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 1e-16)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(y / Float64(y / Float64(x / z_m)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 1e-16)
		tmp = x / z_m;
	else
		tmp = y / (y / (x / z_m));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 9.9999999999999998e-17

   1. Initial program 98.5%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 76.6%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 76.6%

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

   if 9.9999999999999998e-17 < y

   1. Initial program 94.6%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 20.2%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 20.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{z}\right), \color{blue}{x}\right) \]

      4. /-lowering-/.f64 20.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, z\right), x\right) \]

   7. Applied egg-rr 20.2%

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. clear-num N/A

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

      4. *-inverses N/A

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

      5. associate-/r* N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{y}{x} \cdot z\right)}\right) \]

      11. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(\left(y \cdot \frac{1}{x}\right) \cdot z\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(y \cdot \color{blue}{\left(\frac{1}{x} \cdot z\right)}\right)\right) \]

      13. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(y \cdot \frac{1}{\color{blue}{\frac{x}{z}}}\right)\right) \]

      14. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(\frac{y}{\color{blue}{\frac{x}{z}}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

      16. /-lowering-/.f64 35.6%

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right)\right) \]

   9. Applied egg-rr 35.6%

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

Alternative 6: 62.4% accurate, 8.9× speedup

Math

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

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) (* y (/ x (* z_m y))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 4e7

   1. Initial program 98.5%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 76.5%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. associate-*l/ N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z \cdot y}\right), \color{blue}{\sin y}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(z \cdot y\right)\right), \sin \color{blue}{y}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y \cdot z\right)\right), \sin y\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \sin y\right) \]

      8. sin-lowering-sin.f64 90.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sin.f64}\left(y\right)\right) \]

   4. Applied egg-rr 90.7%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \color{blue}{y}\right) \]
   6. Step-by-step derivation

   7. Simplified 30.9%

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

4. Final simplification 67.0%

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

Alternative 7: 66.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \frac{x}{z\_m \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)} \end{array} \]

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 (* z_m (+ 1.0 (* (* y 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) {
	return z_s * (x / (z_m * (1.0 + ((y * y) * 0.16666666666666666))));
}

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 * (1.0d0 + ((y * y) * 0.16666666666666666d0))))
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 * (1.0 + ((y * y) * 0.16666666666666666))));
}

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 * (1.0 + ((y * y) * 0.16666666666666666))))

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(z_m * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666)))))
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 * (1.0 + ((y * y) * 0.16666666666666666))));
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[(z$95$m * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.6%

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

   1. associate-*r/ N/A

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

   2. associate-/l/ N/A

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

   3. associate-*l/ N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z \cdot y}\right), \color{blue}{\sin y}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(z \cdot y\right)\right), \sin \color{blue}{y}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y \cdot z\right)\right), \sin y\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \sin y\right) \]

   8. sin-lowering-sin.f64 85.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sin.f64}\left(y\right)\right) \]

4. Applied egg-rr 85.8%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. associate-/l/ N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{y}{\frac{\sin y}{z}}\right)}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{\sin y}{z}\right)}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\sin y, \color{blue}{z}\right)\right)\right) \]

   9. sin-lowering-sin.f64 85.8%

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), z\right)\right)\right) \]

6. Applied egg-rr 85.8%

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

7. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(z + \frac{1}{6} \cdot \left({y}^{2} \cdot z\right)\right)}\right) \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(z + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{z}\right)\right) \]

   2. distribute-rgt1-in N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{z}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(z \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \left(1 + \color{blue}{\frac{1}{6} \cdot {y}^{2}}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 71.7%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right)\right) \]

9. Simplified 71.7%

   \[\leadsto \frac{x}{\color{blue}{z \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}} \]
10. Add Preprocessing

Alternative 8: 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. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. /-lowering-/.f64 64.1%

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

5. Simplified 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))