Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.6% → 99.8%

Time: 12.6s

Alternatives: 16

Speedup: 1.0×

• 64.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (cosh(x) * (y / x)) / 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 = (cosh(x) * (y / x)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (cosh(x) * (y / x)) / 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 = (cosh(x) * (y / x)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\cosh x_m \cdot \frac{y_m}{x_m}}{z_m}\\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y_m}{\frac{z_m}{\cosh x_m}}}{x_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x_m) (/ y_m x_m)) z_m)))
   (*
    z_s
    (*
     y_s
     (* x_s (if (<= t_0 1e+23) t_0 (/ (/ y_m (/ z_m (cosh x_m))) x_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= 1e+23) {
		tmp = t_0;
	} else {
		tmp = (y_m / (z_m / cosh(x_m))) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (cosh(x_m) * (y_m / x_m)) / z_m
    if (t_0 <= 1d+23) then
        tmp = t_0
    else
        tmp = (y_m / (z_m / cosh(x_m))) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = (Math.cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= 1e+23) {
		tmp = t_0;
	} else {
		tmp = (y_m / (z_m / Math.cosh(x_m))) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	t_0 = (math.cosh(x_m) * (y_m / x_m)) / z_m
	tmp = 0
	if t_0 <= 1e+23:
		tmp = t_0
	else:
		tmp = (y_m / (z_m / math.cosh(x_m))) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m)
	tmp = 0.0
	if (t_0 <= 1e+23)
		tmp = t_0;
	else
		tmp = Float64(Float64(y_m / Float64(z_m / cosh(x_m))) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	tmp = 0.0;
	if (t_0 <= 1e+23)
		tmp = t_0;
	else
		tmp = (y_m / (z_m / cosh(x_m))) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 1e+23], t$95$0, N[(N[(y$95$m / N[(z$95$m / N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 9.9999999999999992e22

   1. Initial program 94.4%

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

   if 9.9999999999999992e22 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 74.2%

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

      1. associate-*l/ 74.2%

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

   3. Simplified 74.2%

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

      1. associate-*r/ 100.0%

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

      2. clear-num 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 10^{+23}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{\cosh x}}}{x}\\ \end{array} \]

Alternative 2: 91.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\cosh x_m \cdot \frac{y_m}{x_m}}{z_m}\\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq \infty:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{\frac{x_m \cdot z_m}{\cosh x_m}}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x_m) (/ y_m x_m)) z_m)))
   (*
    z_s
    (*
     y_s
     (* x_s (if (<= t_0 INFINITY) t_0 (/ y_m (/ (* x_m z_m) (cosh x_m)))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = y_m / ((x_m * z_m) / cosh(x_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = (Math.cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = y_m / ((x_m * z_m) / Math.cosh(x_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	t_0 = (math.cosh(x_m) * (y_m / x_m)) / z_m
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = y_m / ((x_m * z_m) / math.cosh(x_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m)
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(y_m / Float64(Float64(x_m * z_m) / cosh(x_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = y_m / ((x_m * z_m) / cosh(x_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, Infinity], t$95$0, N[(y$95$m / N[(N[(x$95$m * z$95$m), $MachinePrecision] / N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < +inf.0

   1. Initial program 95.3%

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

   if +inf.0 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/l/ 25.0%

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

      4. associate-/r/ 71.4%

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

      5. *-commutative 71.4%

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

   3. Simplified 71.4%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq \infty:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{x \cdot z}{\cosh x}}\\ \end{array} \]

Alternative 3: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 1.25 \cdot 10^{-159}:\\ \;\;\;\;\frac{y_m}{\frac{x_m \cdot z_m}{\cosh x_m}}\\ \mathbf{elif}\;y_m \leq 3.2 \cdot 10^{+178}:\\ \;\;\;\;\frac{y_m}{x_m} \cdot \frac{\cosh x_m}{z_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x_m \cdot y_m}{z_m} + \frac{y_m}{x_m \cdot z_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= y_m 1.25e-159)
      (/ y_m (/ (* x_m z_m) (cosh x_m)))
      (if (<= y_m 3.2e+178)
        (* (/ y_m x_m) (/ (cosh x_m) z_m))
        (+ (* 0.5 (/ (* x_m y_m) z_m)) (/ y_m (* x_m z_m)))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1.25e-159) {
		tmp = y_m / ((x_m * z_m) / cosh(x_m));
	} else if (y_m <= 3.2e+178) {
		tmp = (y_m / x_m) * (cosh(x_m) / z_m);
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 1.25d-159) then
        tmp = y_m / ((x_m * z_m) / cosh(x_m))
    else if (y_m <= 3.2d+178) then
        tmp = (y_m / x_m) * (cosh(x_m) / z_m)
    else
        tmp = (0.5d0 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1.25e-159) {
		tmp = y_m / ((x_m * z_m) / Math.cosh(x_m));
	} else if (y_m <= 3.2e+178) {
		tmp = (y_m / x_m) * (Math.cosh(x_m) / z_m);
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 1.25e-159:
		tmp = y_m / ((x_m * z_m) / math.cosh(x_m))
	elif y_m <= 3.2e+178:
		tmp = (y_m / x_m) * (math.cosh(x_m) / z_m)
	else:
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 1.25e-159)
		tmp = Float64(y_m / Float64(Float64(x_m * z_m) / cosh(x_m)));
	elseif (y_m <= 3.2e+178)
		tmp = Float64(Float64(y_m / x_m) * Float64(cosh(x_m) / z_m));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x_m * y_m) / z_m)) + Float64(y_m / Float64(x_m * z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 1.25e-159)
		tmp = y_m / ((x_m * z_m) / cosh(x_m));
	elseif (y_m <= 3.2e+178)
		tmp = (y_m / x_m) * (cosh(x_m) / z_m);
	else
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1.25e-159], N[(y$95$m / N[(N[(x$95$m * z$95$m), $MachinePrecision] / N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 3.2e+178], N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(N[Cosh[x$95$m], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 1.25000000000000008e-159

   1. Initial program 80.2%

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

      1. associate-*r/ 72.1%

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

      2. *-commutative 72.1%

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

      3. associate-/l/ 71.7%

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

      4. associate-/r/ 84.1%

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

      5. *-commutative 84.1%

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

   3. Simplified 84.1%

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

   if 1.25000000000000008e-159 < y < 3.2e178

   1. Initial program 94.3%

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

      1. associate-*l/ 94.2%

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

   3. Simplified 94.2%

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

   if 3.2e178 < y

   1. Initial program 89.7%

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

      1. associate-*l/ 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-159}:\\ \;\;\;\;\frac{y}{\frac{x \cdot z}{\cosh x}}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+178}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 4: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 6.5 \cdot 10^{-191}:\\ \;\;\;\;0.5 \cdot \left(y_m \cdot \frac{x_m}{z_m}\right) + \frac{\frac{y_m}{x_m}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x_m} \cdot \frac{\cosh x_m}{z_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 6.5e-191)
      (+ (* 0.5 (* y_m (/ x_m z_m))) (/ (/ y_m x_m) z_m))
      (* (/ y_m x_m) (/ (cosh x_m) z_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 6.5e-191) {
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	} else {
		tmp = (y_m / x_m) * (cosh(x_m) / z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 6.5d-191) then
        tmp = (0.5d0 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m)
    else
        tmp = (y_m / x_m) * (cosh(x_m) / z_m)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 6.5e-191) {
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	} else {
		tmp = (y_m / x_m) * (Math.cosh(x_m) / z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 6.5e-191:
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m)
	else:
		tmp = (y_m / x_m) * (math.cosh(x_m) / z_m)
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 6.5e-191)
		tmp = Float64(Float64(0.5 * Float64(y_m * Float64(x_m / z_m))) + Float64(Float64(y_m / x_m) / z_m));
	else
		tmp = Float64(Float64(y_m / x_m) * Float64(cosh(x_m) / z_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 6.5e-191)
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	else
		tmp = (y_m / x_m) * (cosh(x_m) / z_m);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 6.5e-191], N[(N[(0.5 * N[(y$95$m * N[(x$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(N[Cosh[x$95$m], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.4999999999999995e-191

   1. Initial program 84.4%

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

      1. associate-*l/ 84.4%

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

   3. Simplified 84.4%

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

   4. Taylor expanded in x around 0 66.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]
   5. Step-by-step derivation

      1. *-commutative 66.3%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{\color{blue}{z \cdot x}} \]

      2. associate-/r* 63.7%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{\frac{y}{z}}{x}} \]

      3. div-inv 63.6%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]

   6. Applied egg-rr 63.6%

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]
   7. Step-by-step derivation

      1. associate-/l* 61.1%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{z}{y}}} + \frac{y}{z} \cdot \frac{1}{x} \]

      2. associate-/r/ 69.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{z} \cdot y\right)} + \frac{y}{z} \cdot \frac{1}{x} \]

   8. Applied egg-rr 69.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{z} \cdot y\right)} + \frac{y}{z} \cdot \frac{1}{x} \]
   9. Step-by-step derivation

      1. associate-*l/ 70.4%

         \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \color{blue}{\frac{y \cdot \frac{1}{x}}{z}} \]

      2. div-inv 70.5%

         \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \frac{\color{blue}{\frac{y}{x}}}{z} \]

   10. Applied egg-rr 70.5%

       \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \color{blue}{\frac{\frac{y}{x}}{z}} \]

   if 6.4999999999999995e-191 < z

   1. Initial program 85.6%

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

      1. associate-*l/ 85.5%

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

   3. Simplified 85.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.5 \cdot 10^{-191}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right) + \frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \end{array} \]

Alternative 5: 73.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.65 \cdot 10^{-34}:\\ \;\;\;\;0.5 \cdot \left(y_m \cdot \frac{x_m}{z_m}\right) + \frac{\frac{y_m}{x_m}}{z_m}\\ \mathbf{elif}\;z_m \leq 2.3 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(y_m \cdot \left(x_m \cdot 0.5\right)\right) \cdot \left(x_m \cdot z_m\right) + y_m \cdot z_m}{z_m \cdot \left(x_m \cdot z_m\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x_m \cdot y_m}{z_m} + \frac{y_m}{x_m \cdot z_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.65e-34)
      (+ (* 0.5 (* y_m (/ x_m z_m))) (/ (/ y_m x_m) z_m))
      (if (<= z_m 2.3e+54)
        (/
         (+ (* (* y_m (* x_m 0.5)) (* x_m z_m)) (* y_m z_m))
         (* z_m (* x_m z_m)))
        (+ (* 0.5 (/ (* x_m y_m) z_m)) (/ y_m (* x_m z_m)))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.65e-34) {
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	} else if (z_m <= 2.3e+54) {
		tmp = (((y_m * (x_m * 0.5)) * (x_m * z_m)) + (y_m * z_m)) / (z_m * (x_m * z_m));
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 1.65d-34) then
        tmp = (0.5d0 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m)
    else if (z_m <= 2.3d+54) then
        tmp = (((y_m * (x_m * 0.5d0)) * (x_m * z_m)) + (y_m * z_m)) / (z_m * (x_m * z_m))
    else
        tmp = (0.5d0 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.65e-34) {
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	} else if (z_m <= 2.3e+54) {
		tmp = (((y_m * (x_m * 0.5)) * (x_m * z_m)) + (y_m * z_m)) / (z_m * (x_m * z_m));
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 1.65e-34:
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m)
	elif z_m <= 2.3e+54:
		tmp = (((y_m * (x_m * 0.5)) * (x_m * z_m)) + (y_m * z_m)) / (z_m * (x_m * z_m))
	else:
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 1.65e-34)
		tmp = Float64(Float64(0.5 * Float64(y_m * Float64(x_m / z_m))) + Float64(Float64(y_m / x_m) / z_m));
	elseif (z_m <= 2.3e+54)
		tmp = Float64(Float64(Float64(Float64(y_m * Float64(x_m * 0.5)) * Float64(x_m * z_m)) + Float64(y_m * z_m)) / Float64(z_m * Float64(x_m * z_m)));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x_m * y_m) / z_m)) + Float64(y_m / Float64(x_m * z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 1.65e-34)
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	elseif (z_m <= 2.3e+54)
		tmp = (((y_m * (x_m * 0.5)) * (x_m * z_m)) + (y_m * z_m)) / (z_m * (x_m * z_m));
	else
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.65e-34], N[(N[(0.5 * N[(y$95$m * N[(x$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2.3e+54], N[(N[(N[(N[(y$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1.64999999999999991e-34

   1. Initial program 86.5%

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

      1. associate-*l/ 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in x around 0 68.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]
   5. Step-by-step derivation

      1. *-commutative 68.0%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{\color{blue}{z \cdot x}} \]

      2. associate-/r* 66.7%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{\frac{y}{z}}{x}} \]

      3. div-inv 66.6%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]

   6. Applied egg-rr 66.6%

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]
   7. Step-by-step derivation

      1. associate-/l* 64.6%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{z}{y}}} + \frac{y}{z} \cdot \frac{1}{x} \]

      2. associate-/r/ 71.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{z} \cdot y\right)} + \frac{y}{z} \cdot \frac{1}{x} \]

   8. Applied egg-rr 71.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{z} \cdot y\right)} + \frac{y}{z} \cdot \frac{1}{x} \]
   9. Step-by-step derivation

      1. associate-*l/ 72.3%

         \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \color{blue}{\frac{y \cdot \frac{1}{x}}{z}} \]

      2. div-inv 72.4%

         \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \frac{\color{blue}{\frac{y}{x}}}{z} \]

   10. Applied egg-rr 72.4%

       \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \color{blue}{\frac{\frac{y}{x}}{z}} \]

   if 1.64999999999999991e-34 < z < 2.29999999999999994e54

   1. Initial program 88.1%

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

      1. associate-*l/ 87.9%

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

   3. Simplified 87.9%

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

   4. Taylor expanded in x around 0 60.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]
   5. Step-by-step derivation

      1. associate-*r/ 60.5%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{z}} + \frac{y}{x \cdot z} \]

      2. frac-add 71.3%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \left(x \cdot y\right)\right) \cdot \left(x \cdot z\right) + z \cdot y}{z \cdot \left(x \cdot z\right)}} \]

      3. associate-*r* 71.3%

         \[\leadsto \frac{\color{blue}{\left(\left(0.5 \cdot x\right) \cdot y\right)} \cdot \left(x \cdot z\right) + z \cdot y}{z \cdot \left(x \cdot z\right)} \]

      4. *-commutative 71.3%

         \[\leadsto \frac{\left(\left(0.5 \cdot x\right) \cdot y\right) \cdot \color{blue}{\left(z \cdot x\right)} + z \cdot y}{z \cdot \left(x \cdot z\right)} \]

      5. *-commutative 71.3%

         \[\leadsto \frac{\left(\left(0.5 \cdot x\right) \cdot y\right) \cdot \left(z \cdot x\right) + z \cdot y}{z \cdot \color{blue}{\left(z \cdot x\right)}} \]

   6. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{\frac{\left(\left(0.5 \cdot x\right) \cdot y\right) \cdot \left(z \cdot x\right) + z \cdot y}{z \cdot \left(z \cdot x\right)}} \]

   if 2.29999999999999994e54 < z

   1. Initial program 78.7%

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

      1. associate-*l/ 78.6%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in x around 0 59.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.65 \cdot 10^{-34}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right) + \frac{\frac{y}{x}}{z}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(x \cdot z\right) + y \cdot z}{z \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 6: 73.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 5.4 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \left(y_m \cdot \frac{x_m}{z_m}\right) + \frac{\frac{y_m}{x_m}}{z_m}\\ \mathbf{elif}\;z_m \leq 1.15 \cdot 10^{+53}:\\ \;\;\;\;\frac{z_m \cdot \frac{y_m}{z_m} + x_m \cdot \left(y_m \cdot \left(x_m \cdot 0.5\right)\right)}{x_m \cdot z_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x_m \cdot y_m}{z_m} + \frac{y_m}{x_m \cdot z_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 5.4e-82)
      (+ (* 0.5 (* y_m (/ x_m z_m))) (/ (/ y_m x_m) z_m))
      (if (<= z_m 1.15e+53)
        (/ (+ (* z_m (/ y_m z_m)) (* x_m (* y_m (* x_m 0.5)))) (* x_m z_m))
        (+ (* 0.5 (/ (* x_m y_m) z_m)) (/ y_m (* x_m z_m)))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 5.4e-82) {
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	} else if (z_m <= 1.15e+53) {
		tmp = ((z_m * (y_m / z_m)) + (x_m * (y_m * (x_m * 0.5)))) / (x_m * z_m);
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 5.4d-82) then
        tmp = (0.5d0 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m)
    else if (z_m <= 1.15d+53) then
        tmp = ((z_m * (y_m / z_m)) + (x_m * (y_m * (x_m * 0.5d0)))) / (x_m * z_m)
    else
        tmp = (0.5d0 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 5.4e-82) {
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	} else if (z_m <= 1.15e+53) {
		tmp = ((z_m * (y_m / z_m)) + (x_m * (y_m * (x_m * 0.5)))) / (x_m * z_m);
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 5.4e-82:
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m)
	elif z_m <= 1.15e+53:
		tmp = ((z_m * (y_m / z_m)) + (x_m * (y_m * (x_m * 0.5)))) / (x_m * z_m)
	else:
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 5.4e-82)
		tmp = Float64(Float64(0.5 * Float64(y_m * Float64(x_m / z_m))) + Float64(Float64(y_m / x_m) / z_m));
	elseif (z_m <= 1.15e+53)
		tmp = Float64(Float64(Float64(z_m * Float64(y_m / z_m)) + Float64(x_m * Float64(y_m * Float64(x_m * 0.5)))) / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x_m * y_m) / z_m)) + Float64(y_m / Float64(x_m * z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 5.4e-82)
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	elseif (z_m <= 1.15e+53)
		tmp = ((z_m * (y_m / z_m)) + (x_m * (y_m * (x_m * 0.5)))) / (x_m * z_m);
	else
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 5.4e-82], N[(N[(0.5 * N[(y$95$m * N[(x$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.15e+53], N[(N[(N[(z$95$m * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(y$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 5.4000000000000003e-82

   1. Initial program 86.4%

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

      1. associate-*l/ 86.3%

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

   3. Simplified 86.3%

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

   4. Taylor expanded in x around 0 68.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]
   5. Step-by-step derivation

      1. *-commutative 68.5%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{\color{blue}{z \cdot x}} \]

      2. associate-/r* 67.2%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{\frac{y}{z}}{x}} \]

      3. div-inv 67.1%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]

   6. Applied egg-rr 67.1%

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]
   7. Step-by-step derivation

      1. associate-/l* 64.9%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{z}{y}}} + \frac{y}{z} \cdot \frac{1}{x} \]

      2. associate-/r/ 71.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{z} \cdot y\right)} + \frac{y}{z} \cdot \frac{1}{x} \]

   8. Applied egg-rr 71.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{z} \cdot y\right)} + \frac{y}{z} \cdot \frac{1}{x} \]
   9. Step-by-step derivation

      1. associate-*l/ 73.1%

         \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \color{blue}{\frac{y \cdot \frac{1}{x}}{z}} \]

      2. div-inv 73.1%

         \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \frac{\color{blue}{\frac{y}{x}}}{z} \]

   10. Applied egg-rr 73.1%

       \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \color{blue}{\frac{\frac{y}{x}}{z}} \]

   if 5.4000000000000003e-82 < z < 1.1500000000000001e53

   1. Initial program 88.3%

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

      1. associate-*l/ 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in x around 0 59.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]
   5. Step-by-step derivation

      1. +-commutative 59.5%

         \[\leadsto \color{blue}{\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}} \]

      2. *-commutative 59.5%

         \[\leadsto \frac{y}{\color{blue}{z \cdot x}} + 0.5 \cdot \frac{x \cdot y}{z} \]

      3. associate-/r* 59.2%

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} + 0.5 \cdot \frac{x \cdot y}{z} \]

      4. associate-*r/ 59.2%

         \[\leadsto \frac{\frac{y}{z}}{x} + \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{z}} \]

      5. frac-add 70.2%

         \[\leadsto \color{blue}{\frac{\frac{y}{z} \cdot z + x \cdot \left(0.5 \cdot \left(x \cdot y\right)\right)}{x \cdot z}} \]

      6. associate-*r* 70.2%

         \[\leadsto \frac{\frac{y}{z} \cdot z + x \cdot \color{blue}{\left(\left(0.5 \cdot x\right) \cdot y\right)}}{x \cdot z} \]

      7. *-commutative 70.2%

         \[\leadsto \frac{\frac{y}{z} \cdot z + x \cdot \left(\left(0.5 \cdot x\right) \cdot y\right)}{\color{blue}{z \cdot x}} \]

   6. Applied egg-rr 70.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{z} \cdot z + x \cdot \left(\left(0.5 \cdot x\right) \cdot y\right)}{z \cdot x}} \]

   if 1.1500000000000001e53 < z

   1. Initial program 78.7%

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

      1. associate-*l/ 78.6%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in x around 0 59.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right) + \frac{\frac{y}{x}}{z}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+53}:\\ \;\;\;\;\frac{z \cdot \frac{y}{z} + x \cdot \left(y \cdot \left(x \cdot 0.5\right)\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 7: 65.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 6.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{y_m}{x_m \cdot z_m}\\ \mathbf{elif}\;x_m \leq 4.3 \cdot 10^{+248}:\\ \;\;\;\;\frac{\frac{y_m}{x_m} + 0.5 \cdot \left(x_m \cdot y_m\right)}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z_m} \cdot \left(y_m \cdot 0.5\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 6.8e-12)
      (/ y_m (* x_m z_m))
      (if (<= x_m 4.3e+248)
        (/ (+ (/ y_m x_m) (* 0.5 (* x_m y_m))) z_m)
        (* (/ x_m z_m) (* y_m 0.5))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 6.8e-12) {
		tmp = y_m / (x_m * z_m);
	} else if (x_m <= 4.3e+248) {
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / z_m;
	} else {
		tmp = (x_m / z_m) * (y_m * 0.5);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 6.8d-12) then
        tmp = y_m / (x_m * z_m)
    else if (x_m <= 4.3d+248) then
        tmp = ((y_m / x_m) + (0.5d0 * (x_m * y_m))) / z_m
    else
        tmp = (x_m / z_m) * (y_m * 0.5d0)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 6.8e-12) {
		tmp = y_m / (x_m * z_m);
	} else if (x_m <= 4.3e+248) {
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / z_m;
	} else {
		tmp = (x_m / z_m) * (y_m * 0.5);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 6.8e-12:
		tmp = y_m / (x_m * z_m)
	elif x_m <= 4.3e+248:
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / z_m
	else:
		tmp = (x_m / z_m) * (y_m * 0.5)
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 6.8e-12)
		tmp = Float64(y_m / Float64(x_m * z_m));
	elseif (x_m <= 4.3e+248)
		tmp = Float64(Float64(Float64(y_m / x_m) + Float64(0.5 * Float64(x_m * y_m))) / z_m);
	else
		tmp = Float64(Float64(x_m / z_m) * Float64(y_m * 0.5));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 6.8e-12)
		tmp = y_m / (x_m * z_m);
	elseif (x_m <= 4.3e+248)
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / z_m;
	else
		tmp = (x_m / z_m) * (y_m * 0.5);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 6.8e-12], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 4.3e+248], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] + N[(0.5 * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 6.8000000000000001e-12

   1. Initial program 87.2%

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

      1. associate-*l/ 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in x around 0 62.4%

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

   if 6.8000000000000001e-12 < x < 4.3000000000000001e248

   1. Initial program 88.1%

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

      1. associate-*l/ 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in x around 0 37.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]

   5. Taylor expanded in z around 0 37.9%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}{z}} \]

   if 4.3000000000000001e248 < x

   1. Initial program 38.5%

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

      1. associate-*l/ 38.5%

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

   3. Simplified 38.5%

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

   4. Taylor expanded in x around 0 34.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]

   5. Taylor expanded in x around inf 34.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   6. Step-by-step derivation

      1. *-commutative 34.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{z} \cdot 0.5} \]

      2. associate-*l/ 70.5%

         \[\leadsto \color{blue}{\left(\frac{x}{z} \cdot y\right)} \cdot 0.5 \]

      3. associate-*l* 70.5%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(y \cdot 0.5\right)} \]

   7. Simplified 70.5%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(y \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+248}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 64.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.5 \cdot 10^{+92}:\\ \;\;\;\;y_m \cdot \left(0.5 \cdot \frac{x_m}{z_m} + \frac{1}{x_m \cdot z_m}\right)\\ \mathbf{elif}\;x_m \leq 3 \cdot 10^{+249}:\\ \;\;\;\;\frac{\frac{y_m}{x_m} + 0.5 \cdot \left(x_m \cdot y_m\right)}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z_m} \cdot \left(y_m \cdot 0.5\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 1.5e+92)
      (* y_m (+ (* 0.5 (/ x_m z_m)) (/ 1.0 (* x_m z_m))))
      (if (<= x_m 3e+249)
        (/ (+ (/ y_m x_m) (* 0.5 (* x_m y_m))) z_m)
        (* (/ x_m z_m) (* y_m 0.5))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.5e+92) {
		tmp = y_m * ((0.5 * (x_m / z_m)) + (1.0 / (x_m * z_m)));
	} else if (x_m <= 3e+249) {
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / z_m;
	} else {
		tmp = (x_m / z_m) * (y_m * 0.5);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 1.5d+92) then
        tmp = y_m * ((0.5d0 * (x_m / z_m)) + (1.0d0 / (x_m * z_m)))
    else if (x_m <= 3d+249) then
        tmp = ((y_m / x_m) + (0.5d0 * (x_m * y_m))) / z_m
    else
        tmp = (x_m / z_m) * (y_m * 0.5d0)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.5e+92) {
		tmp = y_m * ((0.5 * (x_m / z_m)) + (1.0 / (x_m * z_m)));
	} else if (x_m <= 3e+249) {
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / z_m;
	} else {
		tmp = (x_m / z_m) * (y_m * 0.5);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 1.5e+92:
		tmp = y_m * ((0.5 * (x_m / z_m)) + (1.0 / (x_m * z_m)))
	elif x_m <= 3e+249:
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / z_m
	else:
		tmp = (x_m / z_m) * (y_m * 0.5)
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 1.5e+92)
		tmp = Float64(y_m * Float64(Float64(0.5 * Float64(x_m / z_m)) + Float64(1.0 / Float64(x_m * z_m))));
	elseif (x_m <= 3e+249)
		tmp = Float64(Float64(Float64(y_m / x_m) + Float64(0.5 * Float64(x_m * y_m))) / z_m);
	else
		tmp = Float64(Float64(x_m / z_m) * Float64(y_m * 0.5));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 1.5e+92)
		tmp = y_m * ((0.5 * (x_m / z_m)) + (1.0 / (x_m * z_m)));
	elseif (x_m <= 3e+249)
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / z_m;
	else
		tmp = (x_m / z_m) * (y_m * 0.5);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 1.5e+92], N[(y$95$m * N[(N[(0.5 * N[(x$95$m / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 3e+249], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] + N[(0.5 * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.50000000000000007e92

   1. Initial program 88.8%

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

      1. associate-*l/ 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in x around 0 69.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]

   5. Taylor expanded in y around 0 68.3%

      \[\leadsto \color{blue}{y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)} \]

   if 1.50000000000000007e92 < x < 3.00000000000000016e249

   1. Initial program 78.1%

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

      1. associate-*l/ 78.1%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in x around 0 55.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]

   5. Taylor expanded in z around 0 55.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}{z}} \]

   if 3.00000000000000016e249 < x

   1. Initial program 38.5%

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

      1. associate-*l/ 38.5%

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

   3. Simplified 38.5%

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

   4. Taylor expanded in x around 0 34.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]

   5. Taylor expanded in x around inf 34.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   6. Step-by-step derivation

      1. *-commutative 34.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{z} \cdot 0.5} \]

      2. associate-*l/ 70.5%

         \[\leadsto \color{blue}{\left(\frac{x}{z} \cdot y\right)} \cdot 0.5 \]

      3. associate-*l* 70.5%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(y \cdot 0.5\right)} \]

   7. Simplified 70.5%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(y \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+92}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+249}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y \cdot 0.5\right)\\ \end{array} \]

Alternative 9: 68.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 2.55 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \left(y_m \cdot \frac{x_m}{z_m}\right) + \frac{\frac{y_m}{x_m}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x_m \cdot z_m} + 0.5 \cdot \left(x_m \cdot \frac{y_m}{z_m}\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= y_m 2.55e-20)
      (+ (* 0.5 (* y_m (/ x_m z_m))) (/ (/ y_m x_m) z_m))
      (+ (/ y_m (* x_m z_m)) (* 0.5 (* x_m (/ y_m z_m)))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 2.55e-20) {
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	} else {
		tmp = (y_m / (x_m * z_m)) + (0.5 * (x_m * (y_m / z_m)));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 2.55d-20) then
        tmp = (0.5d0 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m)
    else
        tmp = (y_m / (x_m * z_m)) + (0.5d0 * (x_m * (y_m / z_m)))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 2.55e-20) {
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	} else {
		tmp = (y_m / (x_m * z_m)) + (0.5 * (x_m * (y_m / z_m)));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 2.55e-20:
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m)
	else:
		tmp = (y_m / (x_m * z_m)) + (0.5 * (x_m * (y_m / z_m)))
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 2.55e-20)
		tmp = Float64(Float64(0.5 * Float64(y_m * Float64(x_m / z_m))) + Float64(Float64(y_m / x_m) / z_m));
	else
		tmp = Float64(Float64(y_m / Float64(x_m * z_m)) + Float64(0.5 * Float64(x_m * Float64(y_m / z_m))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 2.55e-20)
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	else
		tmp = (y_m / (x_m * z_m)) + (0.5 * (x_m * (y_m / z_m)));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 2.55e-20], N[(N[(0.5 * N[(y$95$m * N[(x$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x$95$m * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.55000000000000009e-20

   1. Initial program 81.9%

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

      1. associate-*l/ 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in x around 0 58.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]
   5. Step-by-step derivation

      1. *-commutative 58.4%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{\color{blue}{z \cdot x}} \]

      2. associate-/r* 56.5%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{\frac{y}{z}}{x}} \]

      3. div-inv 56.4%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]

   6. Applied egg-rr 56.4%

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]
   7. Step-by-step derivation

      1. associate-/l* 53.3%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{z}{y}}} + \frac{y}{z} \cdot \frac{1}{x} \]

      2. associate-/r/ 60.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{z} \cdot y\right)} + \frac{y}{z} \cdot \frac{1}{x} \]

   8. Applied egg-rr 60.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{z} \cdot y\right)} + \frac{y}{z} \cdot \frac{1}{x} \]
   9. Step-by-step derivation

      1. associate-*l/ 62.3%

         \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \color{blue}{\frac{y \cdot \frac{1}{x}}{z}} \]

      2. div-inv 62.4%

         \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \frac{\color{blue}{\frac{y}{x}}}{z} \]

   10. Applied egg-rr 62.4%

       \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \color{blue}{\frac{\frac{y}{x}}{z}} \]

   if 2.55000000000000009e-20 < y

   1. Initial program 92.3%

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

      1. associate-*l/ 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in x around 0 83.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]
   5. Step-by-step derivation

      1. associate-/l* 76.8%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{z}{y}}} + \frac{y}{x \cdot z} \]

      2. div-inv 76.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \frac{1}{\frac{z}{y}}\right)} + \frac{y}{x \cdot z} \]

      3. clear-num 76.8%

         \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\frac{y}{z}}\right) + \frac{y}{x \cdot z} \]

   6. Applied egg-rr 76.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \frac{y}{z}\right)} + \frac{y}{x \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.55 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right) + \frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \left(x \cdot \frac{y}{z}\right)\\ \end{array} \]

Alternative 10: 68.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 10^{-20}:\\ \;\;\;\;0.5 \cdot \left(y_m \cdot \frac{x_m}{z_m}\right) + \frac{\frac{y_m}{x_m}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x_m \cdot z_m} + 0.5 \cdot \frac{y_m}{\frac{z_m}{x_m}}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= y_m 1e-20)
      (+ (* 0.5 (* y_m (/ x_m z_m))) (/ (/ y_m x_m) z_m))
      (+ (/ y_m (* x_m z_m)) (* 0.5 (/ y_m (/ z_m x_m)))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1e-20) {
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	} else {
		tmp = (y_m / (x_m * z_m)) + (0.5 * (y_m / (z_m / x_m)));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 1d-20) then
        tmp = (0.5d0 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m)
    else
        tmp = (y_m / (x_m * z_m)) + (0.5d0 * (y_m / (z_m / x_m)))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1e-20) {
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	} else {
		tmp = (y_m / (x_m * z_m)) + (0.5 * (y_m / (z_m / x_m)));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 1e-20:
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m)
	else:
		tmp = (y_m / (x_m * z_m)) + (0.5 * (y_m / (z_m / x_m)))
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 1e-20)
		tmp = Float64(Float64(0.5 * Float64(y_m * Float64(x_m / z_m))) + Float64(Float64(y_m / x_m) / z_m));
	else
		tmp = Float64(Float64(y_m / Float64(x_m * z_m)) + Float64(0.5 * Float64(y_m / Float64(z_m / x_m))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 1e-20)
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	else
		tmp = (y_m / (x_m * z_m)) + (0.5 * (y_m / (z_m / x_m)));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1e-20], N[(N[(0.5 * N[(y$95$m * N[(x$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(y$95$m / N[(z$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999945e-21

   1. Initial program 81.9%

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

      1. associate-*l/ 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in x around 0 58.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]
   5. Step-by-step derivation

      1. *-commutative 58.4%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{\color{blue}{z \cdot x}} \]

      2. associate-/r* 56.5%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{\frac{y}{z}}{x}} \]

      3. div-inv 56.4%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]

   6. Applied egg-rr 56.4%

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]
   7. Step-by-step derivation

      1. associate-/l* 53.3%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{z}{y}}} + \frac{y}{z} \cdot \frac{1}{x} \]

      2. associate-/r/ 60.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{z} \cdot y\right)} + \frac{y}{z} \cdot \frac{1}{x} \]

   8. Applied egg-rr 60.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{z} \cdot y\right)} + \frac{y}{z} \cdot \frac{1}{x} \]
   9. Step-by-step derivation

      1. associate-*l/ 62.3%

         \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \color{blue}{\frac{y \cdot \frac{1}{x}}{z}} \]

      2. div-inv 62.4%

         \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \frac{\color{blue}{\frac{y}{x}}}{z} \]

   10. Applied egg-rr 62.4%

       \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \color{blue}{\frac{\frac{y}{x}}{z}} \]

   if 9.99999999999999945e-21 < y

   1. Initial program 92.3%

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

      1. associate-*l/ 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in x around 0 83.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]
   5. Step-by-step derivation

      1. div-inv 83.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(x \cdot y\right) \cdot \frac{1}{z}\right)} + \frac{y}{x \cdot z} \]

      2. *-commutative 83.2%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(y \cdot x\right)} \cdot \frac{1}{z}\right) + \frac{y}{x \cdot z} \]

      3. associate-*l* 76.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y \cdot \left(x \cdot \frac{1}{z}\right)\right)} + \frac{y}{x \cdot z} \]

   6. Applied egg-rr 76.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y \cdot \left(x \cdot \frac{1}{z}\right)\right)} + \frac{y}{x \cdot z} \]
   7. Step-by-step derivation

      1. un-div-inv 76.8%

         \[\leadsto 0.5 \cdot \left(y \cdot \color{blue}{\frac{x}{z}}\right) + \frac{y}{x \cdot z} \]

      2. associate-*r/ 83.2%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{y \cdot x}{z}} + \frac{y}{x \cdot z} \]

      3. associate-/l* 76.8%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{y}{\frac{z}{x}}} + \frac{y}{x \cdot z} \]

   8. Applied egg-rr 76.8%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{y}{\frac{z}{x}}} + \frac{y}{x \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-20}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right) + \frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 11: 72.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 2 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(y_m \cdot \frac{x_m}{z_m}\right) + \frac{\frac{y_m}{x_m}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x_m \cdot y_m}{z_m} + \frac{y_m}{x_m \cdot z_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= y_m 2e-21)
      (+ (* 0.5 (* y_m (/ x_m z_m))) (/ (/ y_m x_m) z_m))
      (+ (* 0.5 (/ (* x_m y_m) z_m)) (/ y_m (* x_m z_m))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 2e-21) {
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 2d-21) then
        tmp = (0.5d0 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m)
    else
        tmp = (0.5d0 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 2e-21) {
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 2e-21:
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m)
	else:
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 2e-21)
		tmp = Float64(Float64(0.5 * Float64(y_m * Float64(x_m / z_m))) + Float64(Float64(y_m / x_m) / z_m));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x_m * y_m) / z_m)) + Float64(y_m / Float64(x_m * z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 2e-21)
		tmp = (0.5 * (y_m * (x_m / z_m))) + ((y_m / x_m) / z_m);
	else
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m / (x_m * z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 2e-21], N[(N[(0.5 * N[(y$95$m * N[(x$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999982e-21

   1. Initial program 81.9%

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

      1. associate-*l/ 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in x around 0 58.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]
   5. Step-by-step derivation

      1. *-commutative 58.4%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{\color{blue}{z \cdot x}} \]

      2. associate-/r* 56.5%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{\frac{y}{z}}{x}} \]

      3. div-inv 56.4%

         \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]

   6. Applied egg-rr 56.4%

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]
   7. Step-by-step derivation

      1. associate-/l* 53.3%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{z}{y}}} + \frac{y}{z} \cdot \frac{1}{x} \]

      2. associate-/r/ 60.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{z} \cdot y\right)} + \frac{y}{z} \cdot \frac{1}{x} \]

   8. Applied egg-rr 60.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{z} \cdot y\right)} + \frac{y}{z} \cdot \frac{1}{x} \]
   9. Step-by-step derivation

      1. associate-*l/ 62.3%

         \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \color{blue}{\frac{y \cdot \frac{1}{x}}{z}} \]

      2. div-inv 62.4%

         \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \frac{\color{blue}{\frac{y}{x}}}{z} \]

   10. Applied egg-rr 62.4%

       \[\leadsto 0.5 \cdot \left(\frac{x}{z} \cdot y\right) + \color{blue}{\frac{\frac{y}{x}}{z}} \]

   if 1.99999999999999982e-21 < y

   1. Initial program 92.3%

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

      1. associate-*l/ 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in x around 0 83.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right) + \frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 12: 65.2% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.4:\\ \;\;\;\;\frac{y_m}{x_m \cdot z_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x_m \cdot y_m}{z_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (* x_s (if (<= x_m 1.4) (/ y_m (* x_m z_m)) (* 0.5 (/ (* x_m y_m) z_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = 0.5 * ((x_m * y_m) / z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 1.4d0) then
        tmp = y_m / (x_m * z_m)
    else
        tmp = 0.5d0 * ((x_m * y_m) / z_m)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = 0.5 * ((x_m * y_m) / z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 1.4:
		tmp = y_m / (x_m * z_m)
	else:
		tmp = 0.5 * ((x_m * y_m) / z_m)
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 1.4)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(0.5 * Float64(Float64(x_m * y_m) / z_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 1.4)
		tmp = y_m / (x_m * z_m);
	else
		tmp = 0.5 * ((x_m * y_m) / z_m);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 1.4], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 87.4%

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

      1. associate-*l/ 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in x around 0 62.9%

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

   if 1.3999999999999999 < x

   1. Initial program 78.3%

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

      1. associate-*l/ 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in x around 0 34.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]

   5. Taylor expanded in x around inf 34.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]

Alternative 13: 65.2% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.4:\\ \;\;\;\;\frac{y_m}{x_m \cdot z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z_m} \cdot \left(y_m \cdot 0.5\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (* x_s (if (<= x_m 1.4) (/ y_m (* x_m z_m)) (* (/ x_m z_m) (* y_m 0.5)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (x_m / z_m) * (y_m * 0.5);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 1.4d0) then
        tmp = y_m / (x_m * z_m)
    else
        tmp = (x_m / z_m) * (y_m * 0.5d0)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (x_m / z_m) * (y_m * 0.5);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 1.4:
		tmp = y_m / (x_m * z_m)
	else:
		tmp = (x_m / z_m) * (y_m * 0.5)
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 1.4)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(x_m / z_m) * Float64(y_m * 0.5));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 1.4)
		tmp = y_m / (x_m * z_m);
	else
		tmp = (x_m / z_m) * (y_m * 0.5);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 1.4], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 87.4%

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

      1. associate-*l/ 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in x around 0 62.9%

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

   if 1.3999999999999999 < x

   1. Initial program 78.3%

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

      1. associate-*l/ 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in x around 0 34.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]

   5. Taylor expanded in x around inf 34.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   6. Step-by-step derivation

      1. *-commutative 34.6%

         \[\leadsto \color{blue}{\frac{x \cdot y}{z} \cdot 0.5} \]

      2. associate-*l/ 41.4%

         \[\leadsto \color{blue}{\left(\frac{x}{z} \cdot y\right)} \cdot 0.5 \]

      3. associate-*l* 41.4%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(y \cdot 0.5\right)} \]

   7. Simplified 41.4%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(y \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y \cdot 0.5\right)\\ \end{array} \]

Alternative 14: 52.3% accurate, 15.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x_m \cdot z_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (* y_s (* x_s (if (<= y_m 2e-21) (/ (/ y_m x_m) z_m) (/ y_m (* x_m z_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 2e-21) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = y_m / (x_m * z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 2d-21) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = y_m / (x_m * z_m)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 2e-21) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = y_m / (x_m * z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 2e-21:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = y_m / (x_m * z_m)
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 2e-21)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(y_m / Float64(x_m * z_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 2e-21)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = y_m / (x_m * z_m);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 2e-21], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999982e-21

   1. Initial program 81.9%

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

      1. associate-*l/ 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in x around 0 45.6%

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

      1. *-commutative 45.6%

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

      2. div-inv 45.7%

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

   6. Applied egg-rr 45.7%

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

   if 1.99999999999999982e-21 < y

   1. Initial program 92.3%

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

      1. associate-*l/ 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in x around 0 51.0%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 15: 56.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 10^{-19}:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y_m}{z_m}}{x_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (* y_s (* x_s (if (<= y_m 1e-19) (/ (/ y_m x_m) z_m) (/ (/ y_m z_m) x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1e-19) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (y_m / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 1d-19) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = (y_m / z_m) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1e-19) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (y_m / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 1e-19:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = (y_m / z_m) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 1e-19)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(Float64(y_m / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 1e-19)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = (y_m / z_m) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1e-19], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.9999999999999998e-20

   1. Initial program 82.0%

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

      1. associate-*l/ 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in x around 0 45.9%

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

      1. *-commutative 45.9%

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

      2. div-inv 46.0%

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

   6. Applied egg-rr 46.0%

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

   if 9.9999999999999998e-20 < y

   1. Initial program 92.3%

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

      1. associate-*l/ 92.2%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in x around 0 42.6%

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

      1. associate-*r/ 54.2%

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

      2. associate-*l/ 54.2%

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

      3. *-un-lft-identity 54.2%

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

   6. Applied egg-rr 54.2%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-19}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 16: 48.7% accurate, 21.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \frac{y_m}{x_m \cdot z_m}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (* z_s (* y_s (* x_s (/ y_m (* x_m z_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	return z_s * (y_s * (x_s * (y_m / (x_m * z_m))));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    code = z_s * (y_s * (x_s * (y_m / (x_m * z_m))))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	return z_s * (y_s * (x_s * (y_m / (x_m * z_m))));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	return z_s * (y_s * (x_s * (y_m / (x_m * z_m))))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	return Float64(z_s * Float64(y_s * Float64(x_s * Float64(y_m / Float64(x_m * z_m)))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = z_s * (y_s * (x_s * (y_m / (x_m * z_m))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.9%

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

   1. associate-*l/ 84.8%

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

3. Simplified 84.8%

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

4. Taylor expanded in x around 0 47.0%

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

5. Final simplification 47.0%

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

Developer target: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y z) x) (cosh x))))
   (if (< y -4.618902267687042e-52)
     t_0
     (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = ((y / z) / x) * cosh(x)
    if (y < (-4.618902267687042d-52)) then
        tmp = t_0
    else if (y < 1.038530535935153d-39) then
        tmp = ((cosh(x) * y) / x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * Math.cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((Math.cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / z) / x) * math.cosh(x)
	tmp = 0
	if y < -4.618902267687042e-52:
		tmp = t_0
	elif y < 1.038530535935153e-39:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) / x) * cosh(x))
	tmp = 0.0
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / z) / x) * cosh(x);
	tmp = 0.0;
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -4.618902267687042e-52], t$95$0, If[Less[y, 1.038530535935153e-39], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2023336 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))