Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 85.4% → 99.8%

Time: 9.1s

Alternatives: 16

Speedup: 1.0×

• 63.9% 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: 85.4% 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

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\cosh x_m \cdot \frac{y_m}{x_m}}{z_m}\\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 2 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m \cdot \frac{\cosh x_m}{z_m}}{x_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

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 2e-24) t_0 (/ (* y_m (/ (cosh x_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 t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= 2e-24) {
		tmp = t_0;
	} else {
		tmp = (y_m * (cosh(x_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) :: t_0
    real(8) :: tmp
    t_0 = (cosh(x_m) * (y_m / x_m)) / z_m
    if (t_0 <= 2d-24) then
        tmp = t_0
    else
        tmp = (y_m * (cosh(x_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 t_0 = (Math.cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= 2e-24) {
		tmp = t_0;
	} else {
		tmp = (y_m * (Math.cosh(x_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):
	t_0 = (math.cosh(x_m) * (y_m / x_m)) / z_m
	tmp = 0
	if t_0 <= 2e-24:
		tmp = t_0
	else:
		tmp = (y_m * (math.cosh(x_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)
	t_0 = Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m)
	tmp = 0.0
	if (t_0 <= 2e-24)
		tmp = t_0;
	else
		tmp = Float64(Float64(y_m * Float64(cosh(x_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)
	t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	tmp = 0.0;
	if (t_0 <= 2e-24)
		tmp = t_0;
	else
		tmp = (y_m * (cosh(x_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_] := 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, 2e-24], t$95$0, N[(N[(y$95$m * N[(N[Cosh[x$95$m], $MachinePrecision] / z$95$m), $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) < 1.99999999999999985e-24

   1. Initial program 96.4%

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

   if 1.99999999999999985e-24 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 72.1%

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

      1. associate-*l/ 72.1%

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

   3. Simplified 72.1%

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

      1. associate-*r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{z}}{x}\\ \end{array} \]

Alternative 2: 89.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\cosh x_m \cdot \frac{y_m}{x_m}}{z_m}\\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq \infty:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \frac{x_m \cdot 0.5}{z_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

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 0.5) 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 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 * 0.5) / z_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 * 0.5) / 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):
	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 * 0.5) / 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)
	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 * 0.5) / 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)
	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 * 0.5) / 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_] := 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 * 0.5), $MachinePrecision] / z$95$m), $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 96.5%

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

   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. Taylor expanded in x around 0 12.5%

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

   3. Taylor expanded in x around inf 12.5%

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

      1. associate-*r/ 12.5%

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

      2. associate-/l* 12.5%

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

      3. *-commutative 12.5%

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

   5. Simplified 12.5%

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

      1. associate-/r* 12.3%

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

      2. associate-/l* 12.3%

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

      3. associate-/r/ 33.9%

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

      4. *-commutative 33.9%

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

   7. Applied egg-rr 33.9%

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

4. Final simplification 88.9%

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

Alternative 3: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\cosh x_m}{x_m \cdot \frac{z_m}{y_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x_m} \cdot \frac{\cosh x_m}{z_m}\\ \end{array}\right)\right) \end{array} \]

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 9.5e-7)
      (/ (cosh x_m) (* x_m (/ z_m y_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 <= 9.5e-7) {
		tmp = cosh(x_m) / (x_m * (z_m / y_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 <= 9.5d-7) then
        tmp = cosh(x_m) / (x_m * (z_m / y_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 <= 9.5e-7) {
		tmp = Math.cosh(x_m) / (x_m * (z_m / y_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 <= 9.5e-7:
		tmp = math.cosh(x_m) / (x_m * (z_m / y_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 <= 9.5e-7)
		tmp = Float64(cosh(x_m) / Float64(x_m * Float64(z_m / y_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 <= 9.5e-7)
		tmp = cosh(x_m) / (x_m * (z_m / y_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, 9.5e-7], N[(N[Cosh[x$95$m], $MachinePrecision] / N[(x$95$m * N[(z$95$m / y$95$m), $MachinePrecision]), $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 < 9.5000000000000001e-7

   1. Initial program 89.7%

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

      1. associate-/l* 86.0%

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

   3. Simplified 86.0%

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

      1. associate-/r/ 89.8%

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

   5. Applied egg-rr 89.8%

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

   if 9.5000000000000001e-7 < z

   1. Initial program 69.3%

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

      1. associate-*l/ 69.2%

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

   3. Simplified 69.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \end{array} \]

Alternative 4: 85.4% accurate, 1.0× speedup

Math

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

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) (/ (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) {
	return z_s * (y_s * (x_s * ((y_m / x_m) * (cosh(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) * (cosh(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) * (Math.cosh(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) * (math.cosh(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(Float64(y_m / x_m) * Float64(cosh(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) * (cosh(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[(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. Initial program 84.8%

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

   1. associate-*l/ 84.7%

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

3. Simplified 84.7%

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

4. Final simplification 84.7%

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

Alternative 5: 69.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 9.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{y_m \cdot \left(x_m \cdot 0.5 + \frac{1}{x_m}\right)}{z_m}\\ \mathbf{elif}\;y_m \leq 900000000:\\ \;\;\;\;\frac{\frac{0.25 \cdot \left(\left(x_m \cdot y_m\right) \cdot \left(x_m \cdot y_m\right)\right) - \frac{y_m}{x_m} \cdot \frac{y_m}{x_m}}{0.5 \cdot \left(x_m \cdot y_m\right) - \frac{y_m}{x_m}}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x_m \cdot y_m}{z_m} + \frac{y_m}{x_m \cdot z_m}\\ \end{array}\right)\right) \end{array} \]

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 9.5e-127)
      (/ (* y_m (+ (* x_m 0.5) (/ 1.0 x_m))) z_m)
      (if (<= y_m 900000000.0)
        (/
         (/
          (- (* 0.25 (* (* x_m y_m) (* x_m y_m))) (* (/ y_m x_m) (/ y_m x_m)))
          (- (* 0.5 (* x_m y_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 <= 9.5e-127) {
		tmp = (y_m * ((x_m * 0.5) + (1.0 / x_m))) / z_m;
	} else if (y_m <= 900000000.0) {
		tmp = (((0.25 * ((x_m * y_m) * (x_m * y_m))) - ((y_m / x_m) * (y_m / x_m))) / ((0.5 * (x_m * y_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 <= 9.5d-127) then
        tmp = (y_m * ((x_m * 0.5d0) + (1.0d0 / x_m))) / z_m
    else if (y_m <= 900000000.0d0) then
        tmp = (((0.25d0 * ((x_m * y_m) * (x_m * y_m))) - ((y_m / x_m) * (y_m / x_m))) / ((0.5d0 * (x_m * y_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 <= 9.5e-127) {
		tmp = (y_m * ((x_m * 0.5) + (1.0 / x_m))) / z_m;
	} else if (y_m <= 900000000.0) {
		tmp = (((0.25 * ((x_m * y_m) * (x_m * y_m))) - ((y_m / x_m) * (y_m / x_m))) / ((0.5 * (x_m * y_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 <= 9.5e-127:
		tmp = (y_m * ((x_m * 0.5) + (1.0 / x_m))) / z_m
	elif y_m <= 900000000.0:
		tmp = (((0.25 * ((x_m * y_m) * (x_m * y_m))) - ((y_m / x_m) * (y_m / x_m))) / ((0.5 * (x_m * y_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 <= 9.5e-127)
		tmp = Float64(Float64(y_m * Float64(Float64(x_m * 0.5) + Float64(1.0 / x_m))) / z_m);
	elseif (y_m <= 900000000.0)
		tmp = Float64(Float64(Float64(Float64(0.25 * Float64(Float64(x_m * y_m) * Float64(x_m * y_m))) - Float64(Float64(y_m / x_m) * Float64(y_m / x_m))) / Float64(Float64(0.5 * Float64(x_m * y_m)) - 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 <= 9.5e-127)
		tmp = (y_m * ((x_m * 0.5) + (1.0 / x_m))) / z_m;
	elseif (y_m <= 900000000.0)
		tmp = (((0.25 * ((x_m * y_m) * (x_m * y_m))) - ((y_m / x_m) * (y_m / x_m))) / ((0.5 * (x_m * y_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, 9.5e-127], N[(N[(y$95$m * N[(N[(x$95$m * 0.5), $MachinePrecision] + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], If[LessEqual[y$95$m, 900000000.0], N[(N[(N[(N[(0.25 * N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $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 < 9.4999999999999997e-127

   1. Initial program 79.0%

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

   2. Taylor expanded in x around 0 64.6%

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

   3. Taylor expanded in y around 0 64.6%

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

   if 9.4999999999999997e-127 < y < 9e8

   1. Initial program 93.1%

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

   2. Taylor expanded in x around 0 64.4%

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

   3. Taylor expanded in y around 0 64.3%

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

      1. distribute-lft-in 64.3%

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

      2. flip-+ 70.6%

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

      3. *-commutative 70.6%

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

      4. *-commutative 70.6%

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

      5. associate-*l* 70.6%

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

      6. associate-*l* 70.6%

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

      7. div-inv 70.7%

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

      8. div-inv 70.4%

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

      9. *-commutative 70.4%

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

      10. associate-*l* 70.4%

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

      11. div-inv 70.6%

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

   5. Applied egg-rr 70.6%

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

      1. swap-sqr 73.8%

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

      2. metadata-eval 73.8%

         \[\leadsto \frac{\frac{\color{blue}{0.25} \cdot \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right)\right) - \frac{y}{x} \cdot \frac{y}{x}}{0.5 \cdot \left(x \cdot y\right) - \frac{y}{x}}}{z} \]

      3. *-commutative 73.8%

         \[\leadsto \frac{\frac{0.25 \cdot \left(\color{blue}{\left(y \cdot x\right)} \cdot \left(x \cdot y\right)\right) - \frac{y}{x} \cdot \frac{y}{x}}{0.5 \cdot \left(x \cdot y\right) - \frac{y}{x}}}{z} \]

      4. *-commutative 73.8%

         \[\leadsto \frac{\frac{0.25 \cdot \left(\left(y \cdot x\right) \cdot \color{blue}{\left(y \cdot x\right)}\right) - \frac{y}{x} \cdot \frac{y}{x}}{0.5 \cdot \left(x \cdot y\right) - \frac{y}{x}}}{z} \]

      5. *-commutative 73.8%

         \[\leadsto \frac{\frac{0.25 \cdot \left(\left(y \cdot x\right) \cdot \left(y \cdot x\right)\right) - \frac{y}{x} \cdot \frac{y}{x}}{0.5 \cdot \color{blue}{\left(y \cdot x\right)} - \frac{y}{x}}}{z} \]

   7. Simplified 73.8%

      \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot \left(\left(y \cdot x\right) \cdot \left(y \cdot x\right)\right) - \frac{y}{x} \cdot \frac{y}{x}}{0.5 \cdot \left(y \cdot x\right) - \frac{y}{x}}}}{z} \]

   if 9e8 < y

   1. Initial program 98.3%

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

      1. associate-*l/ 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in x around 0 87.8%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \mathbf{elif}\;y \leq 900000000:\\ \;\;\;\;\frac{\frac{0.25 \cdot \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right)\right) - \frac{y}{x} \cdot \frac{y}{x}}{0.5 \cdot \left(x \cdot y\right) - \frac{y}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 6: 70.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.65 \cdot 10^{-26}:\\ \;\;\;\;\frac{y_m \cdot \left(x_m \cdot 0.5 + \frac{1}{x_m}\right)}{z_m}\\ \mathbf{elif}\;z_m \leq 5.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{\left(x_m \cdot z_m\right) \cdot \left(x_m \cdot \left(y_m \cdot 0.5\right)\right) + y_m \cdot z_m}{z_m \cdot \left(x_m \cdot z_m\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x_m \cdot y_m}{z_m} + \frac{y_m}{x_m \cdot z_m}\\ \end{array}\right)\right) \end{array} \]

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-26)
      (/ (* y_m (+ (* x_m 0.5) (/ 1.0 x_m))) z_m)
      (if (<= z_m 5.5e+47)
        (/
         (+ (* (* x_m z_m) (* x_m (* y_m 0.5))) (* 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-26) {
		tmp = (y_m * ((x_m * 0.5) + (1.0 / x_m))) / z_m;
	} else if (z_m <= 5.5e+47) {
		tmp = (((x_m * z_m) * (x_m * (y_m * 0.5))) + (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-26) then
        tmp = (y_m * ((x_m * 0.5d0) + (1.0d0 / x_m))) / z_m
    else if (z_m <= 5.5d+47) then
        tmp = (((x_m * z_m) * (x_m * (y_m * 0.5d0))) + (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-26) {
		tmp = (y_m * ((x_m * 0.5) + (1.0 / x_m))) / z_m;
	} else if (z_m <= 5.5e+47) {
		tmp = (((x_m * z_m) * (x_m * (y_m * 0.5))) + (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-26:
		tmp = (y_m * ((x_m * 0.5) + (1.0 / x_m))) / z_m
	elif z_m <= 5.5e+47:
		tmp = (((x_m * z_m) * (x_m * (y_m * 0.5))) + (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-26)
		tmp = Float64(Float64(y_m * Float64(Float64(x_m * 0.5) + Float64(1.0 / x_m))) / z_m);
	elseif (z_m <= 5.5e+47)
		tmp = Float64(Float64(Float64(Float64(x_m * z_m) * Float64(x_m * Float64(y_m * 0.5))) + 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-26)
		tmp = (y_m * ((x_m * 0.5) + (1.0 / x_m))) / z_m;
	elseif (z_m <= 5.5e+47)
		tmp = (((x_m * z_m) * (x_m * (y_m * 0.5))) + (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-26], N[(N[(y$95$m * N[(N[(x$95$m * 0.5), $MachinePrecision] + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], If[LessEqual[z$95$m, 5.5e+47], N[(N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] * N[(x$95$m * N[(y$95$m * 0.5), $MachinePrecision]), $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 < 1.6499999999999999e-26

   1. Initial program 89.4%

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

   2. Taylor expanded in x around 0 76.8%

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

   3. Taylor expanded in y around 0 76.8%

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

   if 1.6499999999999999e-26 < z < 5.4999999999999998e47

   1. Initial program 78.9%

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

      1. associate-*l/ 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in x around 0 34.8%

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

      1. associate-*r/ 34.8%

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

      2. *-commutative 34.8%

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

      3. frac-add 53.6%

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

      4. *-commutative 53.6%

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

      5. *-commutative 53.6%

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

      6. associate-*l* 53.6%

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

   6. Applied egg-rr 53.6%

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

   if 5.4999999999999998e47 < z

   1. Initial program 68.7%

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

      1. associate-*l/ 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in x around 0 64.5%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.65 \cdot 10^{-26}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{\left(x \cdot z\right) \cdot \left(x \cdot \left(y \cdot 0.5\right)\right) + y \cdot z}{z \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 7: 69.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{y_m}{x_m} + 0.5 \cdot \left(x_m \cdot y_m\right)}{z_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x_m \cdot y_m}{z_m} + \frac{y_m}{x_m \cdot z_m}\\ \end{array}\right)\right) \end{array} \]

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+83)
      (/ (+ (/ y_m x_m) (* 0.5 (* x_m y_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+83) {
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_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+83) then
        tmp = ((y_m / x_m) + (0.5d0 * (x_m * y_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+83) {
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_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+83:
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_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+83)
		tmp = Float64(Float64(Float64(y_m / x_m) + Float64(0.5 * Float64(x_m * y_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+83)
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_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+83], 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[(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 < 2.00000000000000006e83

   1. Initial program 82.1%

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

   2. Taylor expanded in x around 0 65.5%

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

   if 2.00000000000000006e83 < y

   1. Initial program 97.9%

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

      1. associate-*l/ 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in x around 0 89.3%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 8: 66.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \frac{y_m \cdot \left(x_m \cdot 0.5 + \frac{1}{x_m}\right)}{z_m}\right)\right) \end{array} \]

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 0.5) (/ 1.0 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 * 0.5) + (1.0 / 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 * 0.5d0) + (1.0d0 / 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 * 0.5) + (1.0 / 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 * 0.5) + (1.0 / 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(Float64(y_m * Float64(Float64(x_m * 0.5) + Float64(1.0 / 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 * 0.5) + (1.0 / 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[(N[(y$95$m * N[(N[(x$95$m * 0.5), $MachinePrecision] + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

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

2. Taylor expanded in x around 0 69.2%

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

3. Taylor expanded in y around 0 68.8%

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

4. Final simplification 68.8%

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

Alternative 9: 66.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \frac{\frac{y_m}{x_m} + 0.5 \cdot \left(x_m \cdot y_m\right)}{z_m}\right)\right) \end{array} \]

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) (* 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) {
	return z_s * (y_s * (x_s * (((y_m / x_m) + (0.5 * (x_m * y_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) + (0.5d0 * (x_m * y_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) + (0.5 * (x_m * y_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) + (0.5 * (x_m * y_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(Float64(Float64(y_m / x_m) + Float64(0.5 * Float64(x_m * y_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) + (0.5 * (x_m * y_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[(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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

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

2. Taylor expanded in x around 0 69.2%

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

3. Final simplification 69.2%

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

Alternative 10: 62.0% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.42:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x_m}{\frac{z_m}{y_m}}\\ \end{array}\right)\right) \end{array} \]

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 1.42) (/ (/ y_m x_m) z_m) (* 0.5 (/ x_m (/ z_m y_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 <= 1.42) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = 0.5 * (x_m / (z_m / y_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 <= 1.42d0) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = 0.5d0 * (x_m / (z_m / y_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 <= 1.42) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = 0.5 * (x_m / (z_m / y_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 <= 1.42:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = 0.5 * (x_m / (z_m / y_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 <= 1.42)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(0.5 * Float64(x_m / Float64(z_m / y_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 <= 1.42)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = 0.5 * (x_m / (z_m / y_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, 1.42], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(0.5 * N[(x$95$m / N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.4199999999999999

   1. Initial program 88.2%

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

   2. Taylor expanded in x around 0 65.6%

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

   if 1.4199999999999999 < x

   1. Initial program 76.0%

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

   2. Taylor expanded in x around 0 45.8%

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

   3. Taylor expanded in x around inf 45.8%

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

      1. associate-/l* 36.5%

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

   5. Simplified 36.5%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 11: 62.0% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.42:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \left(0.5 \cdot \frac{y_m}{z_m}\right)\\ \end{array}\right)\right) \end{array} \]

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 (<= x_m 1.42) (/ (/ y_m x_m) z_m) (* x_m (* 0.5 (/ 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 <= 1.42) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = x_m * (0.5 * (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 <= 1.42d0) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = x_m * (0.5d0 * (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 <= 1.42) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = x_m * (0.5 * (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 <= 1.42:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = x_m * (0.5 * (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 <= 1.42)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(x_m * Float64(0.5 * 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 (x_m <= 1.42)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = x_m * (0.5 * (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, 1.42], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(x$95$m * N[(0.5 * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.4199999999999999

   1. Initial program 88.2%

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

   2. Taylor expanded in x around 0 65.6%

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

   if 1.4199999999999999 < x

   1. Initial program 76.0%

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

   2. Taylor expanded in x around 0 45.8%

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

   3. Taylor expanded in x around inf 45.8%

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

      1. associate-*r/ 45.8%

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

      2. associate-/l* 45.8%

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

      3. *-commutative 45.8%

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

   5. Simplified 45.8%

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

      1. associate-/l* 45.8%

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

      2. *-commutative 45.8%

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

      3. associate-*r/ 45.8%

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

      4. *-commutative 45.8%

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

      5. div-inv 45.8%

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

      6. associate-*l* 36.5%

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

      7. div-inv 36.5%

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

      8. associate-*l* 36.5%

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

   7. Applied egg-rr 36.5%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{z}\right)\\ \end{array} \]

Alternative 12: 65.8% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.42:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\left(x_m \cdot y_m\right) \cdot \frac{0.5}{z_m}\\ \end{array}\right)\right) \end{array} \]

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 1.42) (/ (/ y_m x_m) z_m) (* (* x_m y_m) (/ 0.5 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 <= 1.42) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (x_m * y_m) * (0.5 / 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 <= 1.42d0) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = (x_m * y_m) * (0.5d0 / 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 <= 1.42) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (x_m * y_m) * (0.5 / 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 <= 1.42:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = (x_m * y_m) * (0.5 / 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 <= 1.42)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(Float64(x_m * y_m) * Float64(0.5 / 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 <= 1.42)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = (x_m * y_m) * (0.5 / 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, 1.42], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.4199999999999999

   1. Initial program 88.2%

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

   2. Taylor expanded in x around 0 65.6%

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

   if 1.4199999999999999 < x

   1. Initial program 76.0%

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

   2. Taylor expanded in x around 0 45.8%

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

   3. Taylor expanded in x around inf 45.8%

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

      1. associate-*r/ 45.8%

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

      2. associate-/l* 45.8%

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

      3. *-commutative 45.8%

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

   5. Simplified 45.8%

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

      1. associate-/r/ 45.8%

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

      2. *-commutative 45.8%

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

   7. Applied egg-rr 45.8%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{z}\\ \end{array} \]

Alternative 13: 65.8% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.45:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{z_m}{x_m \cdot y_m}}\\ \end{array}\right)\right) \end{array} \]

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 1.45) (/ (/ y_m x_m) z_m) (/ 0.5 (/ z_m (* x_m y_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 <= 1.45) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = 0.5 / (z_m / (x_m * y_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 <= 1.45d0) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = 0.5d0 / (z_m / (x_m * y_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 <= 1.45) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = 0.5 / (z_m / (x_m * y_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 <= 1.45:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = 0.5 / (z_m / (x_m * y_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 <= 1.45)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(0.5 / Float64(z_m / Float64(x_m * y_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 <= 1.45)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = 0.5 / (z_m / (x_m * y_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, 1.45], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(0.5 / N[(z$95$m / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 88.2%

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

   2. Taylor expanded in x around 0 65.6%

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

   if 1.44999999999999996 < x

   1. Initial program 76.0%

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

   2. Taylor expanded in x around 0 45.8%

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

   3. Taylor expanded in x around inf 45.8%

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

      1. associate-*r/ 45.8%

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

      2. associate-/l* 45.8%

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

      3. *-commutative 45.8%

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

   5. Simplified 45.8%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{z}{x \cdot y}}\\ \end{array} \]

Alternative 14: 52.8% accurate, 15.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 1.32 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x_m \cdot z_m}\\ \end{array}\right)\right) \end{array} \]

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 1.32e+83) (/ (/ 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 <= 1.32e+83) {
		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 <= 1.32d+83) 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 <= 1.32e+83) {
		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 <= 1.32e+83:
		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 <= 1.32e+83)
		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 <= 1.32e+83)
		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, 1.32e+83], 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 < 1.3199999999999999e83

   1. Initial program 82.1%

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

   2. Taylor expanded in x around 0 50.2%

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

   if 1.3199999999999999e83 < y

   1. Initial program 97.9%

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

      1. associate-*l/ 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in x around 0 54.5%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 15: 56.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 4.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y_m}{z_m}}{x_m}\\ \end{array}\right)\right) \end{array} \]

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 4.2e-105) (/ (/ 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 <= 4.2e-105) {
		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 <= 4.2d-105) 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 <= 4.2e-105) {
		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 <= 4.2e-105:
		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 <= 4.2e-105)
		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 <= 4.2e-105)
		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, 4.2e-105], 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 < 4.2e-105

   1. Initial program 78.8%

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

   2. Taylor expanded in x around 0 48.9%

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

   if 4.2e-105 < y

   1. Initial program 97.5%

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

      1. associate-*l/ 97.5%

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

   3. Simplified 97.5%

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

      1. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 64.4%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 16: 49.7% 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 84.8%

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

   1. associate-*l/ 84.7%

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

3. Simplified 84.7%

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

4. Taylor expanded in x around 0 49.6%

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

5. Final simplification 49.6%

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

Developer target: 97.2% 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 2023334 
(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))