Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.2% → 99.6%

Time: 9.4s

Alternatives: 16

Speedup: 1.0×

• 64.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.2% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x_m \cdot \frac{y_m}{x_m}}{z_m} \leq 10^{+50}:\\ \;\;\;\;\frac{\frac{y_m}{x_m} + 0.5 \cdot \left(x_m \cdot y_m\right)}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m \cdot \frac{\cosh x_m}{z_m}}{x_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 1e+50)
      (/ (+ (/ y_m x_m) (* 0.5 (* x_m y_m))) z_m)
      (/ (* 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 tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 1e+50) {
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / z_m;
	} 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) :: tmp
    if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 1d+50) then
        tmp = ((y_m / x_m) + (0.5d0 * (x_m * y_m))) / z_m
    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 tmp;
	if (((Math.cosh(x_m) * (y_m / x_m)) / z_m) <= 1e+50) {
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / z_m;
	} 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):
	tmp = 0
	if ((math.cosh(x_m) * (y_m / x_m)) / z_m) <= 1e+50:
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / z_m
	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)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 1e+50)
		tmp = Float64(Float64(Float64(y_m / x_m) + Float64(0.5 * Float64(x_m * y_m))) / z_m);
	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)
	tmp = 0.0;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 1e+50)
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / z_m;
	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_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 1e+50], 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[(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) < 1.0000000000000001e50

   1. Initial program 97.7%

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

   2. Taylor expanded in x around 0 78.3%

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

   if 1.0000000000000001e50 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 77.2%

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

      1. associate-*l/ 77.2%

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

   3. Simplified 77.2%

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

      1. associate-*r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 10^{+50}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{z}}{x}\\ \end{array} \]

Alternative 2: 88.4% accurate, 0.5× speedup

Math

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

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 (/ -0.5 (/ (- z_m) x_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 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 * (-0.5 / (-z_m / x_m))) + (y_m / (x_m * 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 * (-0.5 / (-z_m / x_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):
	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 * (-0.5 / (-z_m / x_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)
	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(Float64(y_m * Float64(-0.5 / Float64(Float64(-z_m) / x_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)
	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 * (-0.5 / (-z_m / x_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_] := 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[(N[(y$95$m * N[(-0.5 / N[((-z$95$m) / x$95$m), $MachinePrecision]), $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 (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < +inf.0

   1. Initial program 96.6%

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

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

   1. Initial program 0.0%

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

      1. associate-*l/ 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in x around 0 7.8%

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

      1. *-commutative 7.8%

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

      2. associate-/l* 7.7%

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

      3. associate-*l/ 7.7%

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

   6. Applied egg-rr 7.7%

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

      1. *-commutative 7.7%

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

      2. div-inv 7.7%

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

      3. times-frac 7.8%

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

   8. Applied egg-rr 7.8%

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

      1. div-inv 7.8%

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

      2. inv-pow 7.8%

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

      3. pow-flip 7.8%

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

      4. metadata-eval 7.8%

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

      5. pow1 7.8%

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

      6. associate-/r/ 7.8%

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

      7. frac-2neg 7.8%

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

      8. metadata-eval 7.8%

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

   10. Applied egg-rr 7.8%

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

       1. distribute-neg-frac 7.8%

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

       2. *-lft-identity 7.8%

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

       3. associate-*l/ 7.8%

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

       4. associate-/l/ 7.8%

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

       5. associate-/r* 7.8%

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

       6. associate-/l* 7.8%

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

       7. associate-/l/ 7.7%

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

       8. associate-*r/ 7.7%

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

       9. *-rgt-identity 7.7%

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

       10. associate-/r/ 36.5%

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

   12. Simplified 36.5%

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

4. Final simplification 91.2%

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

Alternative 3: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.2 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{y_m}{z_m}}{x_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x_m} \cdot \frac{\cosh x_m}{z_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= x_m 1.2e-140)
      (/ (/ y_m z_m) x_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 (x_m <= 1.2e-140) {
		tmp = (y_m / z_m) / x_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 (x_m <= 1.2d-140) then
        tmp = (y_m / z_m) / x_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 (x_m <= 1.2e-140) {
		tmp = (y_m / z_m) / x_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 x_m <= 1.2e-140:
		tmp = (y_m / z_m) / x_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 (x_m <= 1.2e-140)
		tmp = Float64(Float64(y_m / z_m) / x_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 (x_m <= 1.2e-140)
		tmp = (y_m / z_m) / x_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[x$95$m, 1.2e-140], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $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 x < 1.19999999999999993e-140

   1. Initial program 88.2%

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

      1. associate-*l/ 88.1%

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

   3. Simplified 88.1%

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

      1. associate-*r/ 98.1%

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

   5. Applied egg-rr 98.1%

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

   6. Taylor expanded in x around 0 62.2%

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

   if 1.19999999999999993e-140 < x

   1. Initial program 87.4%

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

      1. associate-*l/ 87.4%

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

   3. Simplified 87.4%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \end{array} \]

Alternative 4: 72.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 3.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{z_m \cdot \left(x_m \cdot \frac{0.5}{z_m}\right) + \frac{y_m}{x_m} \cdot \frac{1}{y_m}}{\frac{z_m}{y_m}}\\ \mathbf{elif}\;z_m \leq 3.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{y_m}{z_m} \cdot \frac{z_m}{y_m} + x_m \cdot \left(x_m \cdot 0.5\right)}{x_m \cdot \frac{z_m}{y_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x_m \cdot z_m} + 0.5 \cdot \frac{x_m \cdot y_m}{z_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 3.8e-17)
      (/
       (+ (* z_m (* x_m (/ 0.5 z_m))) (* (/ y_m x_m) (/ 1.0 y_m)))
       (/ z_m y_m))
      (if (<= z_m 3.5e+130)
        (/
         (+ (* (/ y_m z_m) (/ z_m y_m)) (* x_m (* x_m 0.5)))
         (* x_m (/ z_m y_m)))
        (+ (/ y_m (* x_m z_m)) (* 0.5 (/ (* x_m y_m) z_m)))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 3.8e-17) {
		tmp = ((z_m * (x_m * (0.5 / z_m))) + ((y_m / x_m) * (1.0 / y_m))) / (z_m / y_m);
	} else if (z_m <= 3.5e+130) {
		tmp = (((y_m / z_m) * (z_m / y_m)) + (x_m * (x_m * 0.5))) / (x_m * (z_m / y_m));
	} else {
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 3.8d-17) then
        tmp = ((z_m * (x_m * (0.5d0 / z_m))) + ((y_m / x_m) * (1.0d0 / y_m))) / (z_m / y_m)
    else if (z_m <= 3.5d+130) then
        tmp = (((y_m / z_m) * (z_m / y_m)) + (x_m * (x_m * 0.5d0))) / (x_m * (z_m / y_m))
    else
        tmp = (y_m / (x_m * z_m)) + (0.5d0 * ((x_m * y_m) / z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 3.8e-17) {
		tmp = ((z_m * (x_m * (0.5 / z_m))) + ((y_m / x_m) * (1.0 / y_m))) / (z_m / y_m);
	} else if (z_m <= 3.5e+130) {
		tmp = (((y_m / z_m) * (z_m / y_m)) + (x_m * (x_m * 0.5))) / (x_m * (z_m / y_m));
	} else {
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 3.8e-17:
		tmp = ((z_m * (x_m * (0.5 / z_m))) + ((y_m / x_m) * (1.0 / y_m))) / (z_m / y_m)
	elif z_m <= 3.5e+130:
		tmp = (((y_m / z_m) * (z_m / y_m)) + (x_m * (x_m * 0.5))) / (x_m * (z_m / y_m))
	else:
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 3.8e-17)
		tmp = Float64(Float64(Float64(z_m * Float64(x_m * Float64(0.5 / z_m))) + Float64(Float64(y_m / x_m) * Float64(1.0 / y_m))) / Float64(z_m / y_m));
	elseif (z_m <= 3.5e+130)
		tmp = Float64(Float64(Float64(Float64(y_m / z_m) * Float64(z_m / y_m)) + Float64(x_m * Float64(x_m * 0.5))) / Float64(x_m * Float64(z_m / y_m)));
	else
		tmp = Float64(Float64(y_m / Float64(x_m * z_m)) + Float64(0.5 * Float64(Float64(x_m * y_m) / z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 3.8e-17)
		tmp = ((z_m * (x_m * (0.5 / z_m))) + ((y_m / x_m) * (1.0 / y_m))) / (z_m / y_m);
	elseif (z_m <= 3.5e+130)
		tmp = (((y_m / z_m) * (z_m / y_m)) + (x_m * (x_m * 0.5))) / (x_m * (z_m / y_m));
	else
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.8e-17], N[(N[(N[(z$95$m * N[(x$95$m * N[(0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 3.5e+130], N[(N[(N[(N[(y$95$m / z$95$m), $MachinePrecision] * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 3.8000000000000001e-17

   1. Initial program 87.8%

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

      1. associate-*l/ 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in x around 0 73.2%

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

      1. *-commutative 73.2%

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

      2. associate-/l* 69.2%

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

      3. associate-*l/ 69.2%

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

   6. Applied egg-rr 69.2%

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

      1. *-commutative 69.2%

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

      2. div-inv 69.2%

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

      3. times-frac 73.2%

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

   8. Applied egg-rr 73.2%

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

      1. +-commutative 73.2%

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

      2. associate-/r* 73.6%

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

      3. associate-*r/ 76.1%

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

      4. frac-add 74.1%

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

      5. div-inv 74.0%

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

   10. Applied egg-rr 74.0%

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

   if 3.8000000000000001e-17 < z < 3.5000000000000001e130

   1. Initial program 93.7%

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

      1. associate-*l/ 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in x around 0 66.9%

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

      1. *-commutative 66.9%

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

      2. associate-/l* 61.1%

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

      3. associate-*l/ 61.1%

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

   6. Applied egg-rr 61.1%

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

      1. +-commutative 61.1%

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

      2. associate-/l/ 61.2%

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

      3. frac-add 78.6%

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

   8. Applied egg-rr 78.6%

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

   if 3.5000000000000001e130 < z

   1. Initial program 83.4%

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

      1. associate-*l/ 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in x around 0 55.3%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot \frac{0.5}{z}\right) + \frac{y}{x} \cdot \frac{1}{y}}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{z}{y} + x \cdot \left(x \cdot 0.5\right)}{x \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]

Alternative 5: 72.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 4.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{z_m \cdot \left(x_m \cdot \frac{0.5}{z_m}\right) + \frac{y_m}{x_m} \cdot \frac{1}{y_m}}{z_m \cdot \frac{1}{y_m}}\\ \mathbf{elif}\;z_m \leq 6 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{y_m}{z_m} \cdot \frac{z_m}{y_m} + x_m \cdot \left(x_m \cdot 0.5\right)}{x_m \cdot \frac{z_m}{y_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x_m \cdot z_m} + 0.5 \cdot \frac{x_m \cdot y_m}{z_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 4.6e-17)
      (/
       (+ (* z_m (* x_m (/ 0.5 z_m))) (* (/ y_m x_m) (/ 1.0 y_m)))
       (* z_m (/ 1.0 y_m)))
      (if (<= z_m 6e+130)
        (/
         (+ (* (/ y_m z_m) (/ z_m y_m)) (* x_m (* x_m 0.5)))
         (* x_m (/ z_m y_m)))
        (+ (/ y_m (* x_m z_m)) (* 0.5 (/ (* x_m y_m) z_m)))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 4.6e-17) {
		tmp = ((z_m * (x_m * (0.5 / z_m))) + ((y_m / x_m) * (1.0 / y_m))) / (z_m * (1.0 / y_m));
	} else if (z_m <= 6e+130) {
		tmp = (((y_m / z_m) * (z_m / y_m)) + (x_m * (x_m * 0.5))) / (x_m * (z_m / y_m));
	} else {
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 4.6d-17) then
        tmp = ((z_m * (x_m * (0.5d0 / z_m))) + ((y_m / x_m) * (1.0d0 / y_m))) / (z_m * (1.0d0 / y_m))
    else if (z_m <= 6d+130) then
        tmp = (((y_m / z_m) * (z_m / y_m)) + (x_m * (x_m * 0.5d0))) / (x_m * (z_m / y_m))
    else
        tmp = (y_m / (x_m * z_m)) + (0.5d0 * ((x_m * y_m) / z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 4.6e-17) {
		tmp = ((z_m * (x_m * (0.5 / z_m))) + ((y_m / x_m) * (1.0 / y_m))) / (z_m * (1.0 / y_m));
	} else if (z_m <= 6e+130) {
		tmp = (((y_m / z_m) * (z_m / y_m)) + (x_m * (x_m * 0.5))) / (x_m * (z_m / y_m));
	} else {
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 4.6e-17:
		tmp = ((z_m * (x_m * (0.5 / z_m))) + ((y_m / x_m) * (1.0 / y_m))) / (z_m * (1.0 / y_m))
	elif z_m <= 6e+130:
		tmp = (((y_m / z_m) * (z_m / y_m)) + (x_m * (x_m * 0.5))) / (x_m * (z_m / y_m))
	else:
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 4.6e-17)
		tmp = Float64(Float64(Float64(z_m * Float64(x_m * Float64(0.5 / z_m))) + Float64(Float64(y_m / x_m) * Float64(1.0 / y_m))) / Float64(z_m * Float64(1.0 / y_m)));
	elseif (z_m <= 6e+130)
		tmp = Float64(Float64(Float64(Float64(y_m / z_m) * Float64(z_m / y_m)) + Float64(x_m * Float64(x_m * 0.5))) / Float64(x_m * Float64(z_m / y_m)));
	else
		tmp = Float64(Float64(y_m / Float64(x_m * z_m)) + Float64(0.5 * Float64(Float64(x_m * y_m) / z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 4.6e-17)
		tmp = ((z_m * (x_m * (0.5 / z_m))) + ((y_m / x_m) * (1.0 / y_m))) / (z_m * (1.0 / y_m));
	elseif (z_m <= 6e+130)
		tmp = (((y_m / z_m) * (z_m / y_m)) + (x_m * (x_m * 0.5))) / (x_m * (z_m / y_m));
	else
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 4.6e-17], N[(N[(N[(z$95$m * N[(x$95$m * N[(0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 6e+130], N[(N[(N[(N[(y$95$m / z$95$m), $MachinePrecision] * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 4.60000000000000018e-17

   1. Initial program 87.8%

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

      1. associate-*l/ 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in x around 0 73.2%

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

      1. *-commutative 73.2%

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

      2. associate-/l* 69.2%

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

      3. associate-*l/ 69.2%

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

   6. Applied egg-rr 69.2%

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

      1. *-commutative 69.2%

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

      2. div-inv 69.2%

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

      3. times-frac 73.2%

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

   8. Applied egg-rr 73.2%

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

      1. associate-*r/ 75.7%

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

      2. associate-/r* 76.1%

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

      3. frac-add 74.1%

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

   10. Applied egg-rr 74.1%

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

   if 4.60000000000000018e-17 < z < 5.9999999999999999e130

   1. Initial program 93.7%

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

      1. associate-*l/ 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in x around 0 66.9%

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

      1. *-commutative 66.9%

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

      2. associate-/l* 61.1%

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

      3. associate-*l/ 61.1%

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

   6. Applied egg-rr 61.1%

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

      1. +-commutative 61.1%

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

      2. associate-/l/ 61.2%

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

      3. frac-add 78.6%

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

   8. Applied egg-rr 78.6%

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

   if 5.9999999999999999e130 < z

   1. Initial program 83.4%

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

      1. associate-*l/ 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in x around 0 55.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot \frac{0.5}{z}\right) + \frac{y}{x} \cdot \frac{1}{y}}{z \cdot \frac{1}{y}}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{z}{y} + x \cdot \left(x \cdot 0.5\right)}{x \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]

Alternative 6: 73.0% accurate, 5.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2.55 \cdot 10^{-17}:\\ \;\;\;\;\frac{z_m \cdot \left(x_m \cdot \frac{0.5}{z_m}\right) + \frac{y_m}{x_m} \cdot \frac{1}{y_m}}{\frac{z_m}{y_m}}\\ \mathbf{elif}\;z_m \leq 6 \cdot 10^{+130}:\\ \;\;\;\;\frac{1 + x_m \cdot \left(x_m \cdot 0.5\right)}{x_m \cdot \frac{z_m}{y_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x_m \cdot z_m} + 0.5 \cdot \frac{x_m \cdot y_m}{z_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 2.55e-17)
      (/
       (+ (* z_m (* x_m (/ 0.5 z_m))) (* (/ y_m x_m) (/ 1.0 y_m)))
       (/ z_m y_m))
      (if (<= z_m 6e+130)
        (/ (+ 1.0 (* x_m (* x_m 0.5))) (* x_m (/ z_m y_m)))
        (+ (/ y_m (* x_m z_m)) (* 0.5 (/ (* x_m y_m) z_m)))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 2.55e-17) {
		tmp = ((z_m * (x_m * (0.5 / z_m))) + ((y_m / x_m) * (1.0 / y_m))) / (z_m / y_m);
	} else if (z_m <= 6e+130) {
		tmp = (1.0 + (x_m * (x_m * 0.5))) / (x_m * (z_m / y_m));
	} else {
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 2.55d-17) then
        tmp = ((z_m * (x_m * (0.5d0 / z_m))) + ((y_m / x_m) * (1.0d0 / y_m))) / (z_m / y_m)
    else if (z_m <= 6d+130) then
        tmp = (1.0d0 + (x_m * (x_m * 0.5d0))) / (x_m * (z_m / y_m))
    else
        tmp = (y_m / (x_m * z_m)) + (0.5d0 * ((x_m * y_m) / z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 2.55e-17) {
		tmp = ((z_m * (x_m * (0.5 / z_m))) + ((y_m / x_m) * (1.0 / y_m))) / (z_m / y_m);
	} else if (z_m <= 6e+130) {
		tmp = (1.0 + (x_m * (x_m * 0.5))) / (x_m * (z_m / y_m));
	} else {
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 2.55e-17:
		tmp = ((z_m * (x_m * (0.5 / z_m))) + ((y_m / x_m) * (1.0 / y_m))) / (z_m / y_m)
	elif z_m <= 6e+130:
		tmp = (1.0 + (x_m * (x_m * 0.5))) / (x_m * (z_m / y_m))
	else:
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 2.55e-17)
		tmp = Float64(Float64(Float64(z_m * Float64(x_m * Float64(0.5 / z_m))) + Float64(Float64(y_m / x_m) * Float64(1.0 / y_m))) / Float64(z_m / y_m));
	elseif (z_m <= 6e+130)
		tmp = Float64(Float64(1.0 + Float64(x_m * Float64(x_m * 0.5))) / Float64(x_m * Float64(z_m / y_m)));
	else
		tmp = Float64(Float64(y_m / Float64(x_m * z_m)) + Float64(0.5 * Float64(Float64(x_m * y_m) / z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 2.55e-17)
		tmp = ((z_m * (x_m * (0.5 / z_m))) + ((y_m / x_m) * (1.0 / y_m))) / (z_m / y_m);
	elseif (z_m <= 6e+130)
		tmp = (1.0 + (x_m * (x_m * 0.5))) / (x_m * (z_m / y_m));
	else
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.55e-17], N[(N[(N[(z$95$m * N[(x$95$m * N[(0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 6e+130], N[(N[(1.0 + N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 2.5500000000000001e-17

   1. Initial program 87.8%

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

      1. associate-*l/ 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in x around 0 73.2%

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

      1. *-commutative 73.2%

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

      2. associate-/l* 69.2%

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

      3. associate-*l/ 69.2%

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

   6. Applied egg-rr 69.2%

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

      1. *-commutative 69.2%

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

      2. div-inv 69.2%

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

      3. times-frac 73.2%

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

   8. Applied egg-rr 73.2%

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

      1. +-commutative 73.2%

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

      2. associate-/r* 73.6%

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

      3. associate-*r/ 76.1%

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

      4. frac-add 74.1%

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

      5. div-inv 74.0%

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

   10. Applied egg-rr 74.0%

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

   if 2.5500000000000001e-17 < z < 5.9999999999999999e130

   1. Initial program 93.7%

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

      1. associate-*l/ 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in x around 0 66.9%

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

      1. *-commutative 66.9%

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

      2. associate-/l* 61.1%

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

      3. associate-*l/ 61.1%

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

   6. Applied egg-rr 61.1%

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

      1. +-commutative 61.1%

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

      2. associate-/l/ 61.2%

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

      3. frac-add 78.6%

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

   8. Applied egg-rr 78.6%

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

   9. Taylor expanded in y around 0 78.5%

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

   if 5.9999999999999999e130 < z

   1. Initial program 83.4%

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

      1. associate-*l/ 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in x around 0 55.3%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.55 \cdot 10^{-17}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot \frac{0.5}{z}\right) + \frac{y}{x} \cdot \frac{1}{y}}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+130}:\\ \;\;\;\;\frac{1 + x \cdot \left(x \cdot 0.5\right)}{x \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]

Alternative 7: 66.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 0.1:\\ \;\;\;\;y_m \cdot \left(0.5 \cdot \frac{x_m}{z_m} + \frac{1}{x_m \cdot z_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y_m}{x_m} + 0.5 \cdot \left(x_m \cdot y_m\right)}{z_m}\\ \end{array}\right)\right) \end{array} \]

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 0.1)
      (* y_m (+ (* 0.5 (/ x_m z_m)) (/ 1.0 (* x_m z_m))))
      (/ (+ (/ 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) {
	double tmp;
	if (y_m <= 0.1) {
		tmp = y_m * ((0.5 * (x_m / z_m)) + (1.0 / (x_m * z_m)));
	} else {
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 0.1d0) then
        tmp = y_m * ((0.5d0 * (x_m / z_m)) + (1.0d0 / (x_m * z_m)))
    else
        tmp = ((y_m / x_m) + (0.5d0 * (x_m * y_m))) / z_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 0.1) {
		tmp = y_m * ((0.5 * (x_m / z_m)) + (1.0 / (x_m * z_m)));
	} else {
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 0.1:
		tmp = y_m * ((0.5 * (x_m / z_m)) + (1.0 / (x_m * z_m)))
	else:
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / z_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 0.1)
		tmp = Float64(y_m * Float64(Float64(0.5 * Float64(x_m / z_m)) + Float64(1.0 / Float64(x_m * z_m))));
	else
		tmp = Float64(Float64(Float64(y_m / x_m) + Float64(0.5 * Float64(x_m * y_m))) / z_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 0.1)
		tmp = y_m * ((0.5 * (x_m / z_m)) + (1.0 / (x_m * z_m)));
	else
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / z_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 0.1], N[(y$95$m * N[(N[(0.5 * N[(x$95$m / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if y < 0.10000000000000001

   1. Initial program 85.1%

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

      1. associate-*l/ 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in x around 0 67.3%

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

   5. Taylor expanded in y around 0 70.2%

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

   if 0.10000000000000001 < y

   1. Initial program 95.8%

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

   2. Taylor expanded in x around 0 73.3%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.1:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]

Alternative 8: 69.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 6.2 \cdot 10^{-46}:\\ \;\;\;\;y_m \cdot \left(0.5 \cdot \frac{x_m}{z_m} + \frac{1}{x_m \cdot z_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x_m \cdot z_m} + 0.5 \cdot \frac{x_m \cdot y_m}{z_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 6.2e-46)
      (* y_m (+ (* 0.5 (/ x_m z_m)) (/ 1.0 (* x_m z_m))))
      (+ (/ y_m (* x_m z_m)) (* 0.5 (/ (* x_m y_m) z_m))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 6.2e-46) {
		tmp = y_m * ((0.5 * (x_m / z_m)) + (1.0 / (x_m * z_m)));
	} else {
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 6.2d-46) then
        tmp = y_m * ((0.5d0 * (x_m / z_m)) + (1.0d0 / (x_m * z_m)))
    else
        tmp = (y_m / (x_m * z_m)) + (0.5d0 * ((x_m * y_m) / z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 6.2e-46) {
		tmp = y_m * ((0.5 * (x_m / z_m)) + (1.0 / (x_m * z_m)));
	} else {
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 6.2e-46:
		tmp = y_m * ((0.5 * (x_m / z_m)) + (1.0 / (x_m * z_m)))
	else:
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 6.2e-46)
		tmp = Float64(y_m * Float64(Float64(0.5 * Float64(x_m / z_m)) + Float64(1.0 / Float64(x_m * z_m))));
	else
		tmp = Float64(Float64(y_m / Float64(x_m * z_m)) + Float64(0.5 * Float64(Float64(x_m * y_m) / z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 6.2e-46)
		tmp = y_m * ((0.5 * (x_m / z_m)) + (1.0 / (x_m * z_m)));
	else
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 6.2e-46], N[(y$95$m * N[(N[(0.5 * N[(x$95$m / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.2000000000000002e-46

   1. Initial program 87.4%

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

      1. associate-*l/ 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in x around 0 73.4%

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

   5. Taylor expanded in y around 0 75.3%

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

   if 6.2000000000000002e-46 < z

   1. Initial program 89.2%

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

      1. associate-*l/ 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in x around 0 61.5%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{-46}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]

Alternative 9: 71.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 3.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{1 + x_m \cdot \left(x_m \cdot 0.5\right)}{x_m \cdot \frac{z_m}{y_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x_m \cdot z_m} + 0.5 \cdot \frac{x_m \cdot y_m}{z_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 3.5e+130)
      (/ (+ 1.0 (* x_m (* x_m 0.5))) (* x_m (/ z_m y_m)))
      (+ (/ y_m (* x_m z_m)) (* 0.5 (/ (* x_m y_m) z_m))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 3.5e+130) {
		tmp = (1.0 + (x_m * (x_m * 0.5))) / (x_m * (z_m / y_m));
	} else {
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 3.5d+130) then
        tmp = (1.0d0 + (x_m * (x_m * 0.5d0))) / (x_m * (z_m / y_m))
    else
        tmp = (y_m / (x_m * z_m)) + (0.5d0 * ((x_m * y_m) / z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 3.5e+130) {
		tmp = (1.0 + (x_m * (x_m * 0.5))) / (x_m * (z_m / y_m));
	} else {
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 3.5e+130:
		tmp = (1.0 + (x_m * (x_m * 0.5))) / (x_m * (z_m / y_m))
	else:
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 3.5e+130)
		tmp = Float64(Float64(1.0 + Float64(x_m * Float64(x_m * 0.5))) / Float64(x_m * Float64(z_m / y_m)));
	else
		tmp = Float64(Float64(y_m / Float64(x_m * z_m)) + Float64(0.5 * Float64(Float64(x_m * y_m) / z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 3.5e+130)
		tmp = (1.0 + (x_m * (x_m * 0.5))) / (x_m * (z_m / y_m));
	else
		tmp = (y_m / (x_m * z_m)) + (0.5 * ((x_m * y_m) / z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.5e+130], N[(N[(1.0 + N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.5000000000000001e130

   1. Initial program 88.7%

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

      1. associate-*l/ 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in x around 0 72.3%

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

      1. *-commutative 72.3%

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

      2. associate-/l* 68.1%

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

      3. associate-*l/ 68.1%

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

   6. Applied egg-rr 68.1%

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

      1. +-commutative 68.1%

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

      2. associate-/l/ 68.4%

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

      3. frac-add 61.5%

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

   8. Applied egg-rr 61.5%

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

   9. Taylor expanded in y around 0 76.1%

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

   if 3.5000000000000001e130 < z

   1. Initial program 83.4%

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

      1. associate-*l/ 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in x around 0 55.3%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{1 + x \cdot \left(x \cdot 0.5\right)}{x \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]

Alternative 10: 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 87.9%

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

2. Taylor expanded in x around 0 68.8%

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

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 11: 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 87.9%

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

2. Taylor expanded in x around 0 68.8%

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

3. Final simplification 68.8%

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

Alternative 12: 65.9% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.47:\\ \;\;\;\;\frac{y_m}{x_m \cdot z_m}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \frac{x_m \cdot 0.5}{z_m}\\ \end{array}\right)\right) \end{array} \]

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 0.47) (/ y_m (* x_m z_m)) (* 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 tmp;
	if (x_m <= 0.47) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = y_m * ((x_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 <= 0.47d0) then
        tmp = y_m / (x_m * z_m)
    else
        tmp = y_m * ((x_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 <= 0.47) {
		tmp = y_m / (x_m * z_m);
	} 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):
	tmp = 0
	if x_m <= 0.47:
		tmp = y_m / (x_m * z_m)
	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)
	tmp = 0.0
	if (x_m <= 0.47)
		tmp = Float64(y_m / Float64(x_m * z_m));
	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)
	tmp = 0.0;
	if (x_m <= 0.47)
		tmp = y_m / (x_m * z_m);
	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_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 0.47], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], 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 x < 0.46999999999999997

   1. Initial program 90.1%

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

      1. associate-*l/ 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in x around 0 65.1%

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

   if 0.46999999999999997 < x

   1. Initial program 81.5%

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

   2. Taylor expanded in x around 0 51.2%

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

   3. Taylor expanded in x around inf 51.2%

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

      1. associate-*l/ 46.8%

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

      2. associate-*r* 46.8%

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

      3. *-commutative 46.8%

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

      4. associate-*r/ 46.8%

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

      5. *-commutative 46.8%

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

   5. Simplified 46.8%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.47:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{z}\\ \end{array} \]

Alternative 13: 66.1% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.47:\\ \;\;\;\;\frac{y_m}{x_m \cdot z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m \cdot \left(x_m \cdot 0.5\right)}{z_m}\\ \end{array}\right)\right) \end{array} \]

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 0.47) (/ y_m (* x_m z_m)) (/ (* 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 tmp;
	if (x_m <= 0.47) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (y_m * (x_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 <= 0.47d0) then
        tmp = y_m / (x_m * z_m)
    else
        tmp = (y_m * (x_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 <= 0.47) {
		tmp = y_m / (x_m * z_m);
	} 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):
	tmp = 0
	if x_m <= 0.47:
		tmp = y_m / (x_m * z_m)
	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)
	tmp = 0.0
	if (x_m <= 0.47)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(y_m * 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)
	tmp = 0.0;
	if (x_m <= 0.47)
		tmp = y_m / (x_m * z_m);
	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_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 0.47], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.46999999999999997

   1. Initial program 90.1%

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

      1. associate-*l/ 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in x around 0 65.1%

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

   if 0.46999999999999997 < x

   1. Initial program 81.5%

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

   2. Taylor expanded in x around 0 51.2%

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

   3. Taylor expanded in x around inf 51.2%

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

      1. *-commutative 51.2%

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

      2. *-commutative 51.2%

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

      3. associate-*r* 51.2%

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

   5. Simplified 51.2%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.47:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5\right)}{z}\\ \end{array} \]

Alternative 14: 52.6% accurate, 15.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x_m \cdot z_m}\\ \end{array}\right)\right) \end{array} \]

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 5e-41) (/ (/ 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 <= 5e-41) {
		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 <= 5d-41) 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 <= 5e-41) {
		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 <= 5e-41:
		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 <= 5e-41)
		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 <= 5e-41)
		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, 5e-41], 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 < 4.9999999999999996e-41

   1. Initial program 84.4%

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

   2. Taylor expanded in x around 0 49.6%

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

   if 4.9999999999999996e-41 < y

   1. Initial program 96.2%

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

      1. associate-*l/ 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in x around 0 52.0%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 15: 56.4% accurate, 15.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 10^{-48}:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y_m}{z_m}}{x_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (* y_s (* x_s (if (<= y_m 1e-48) (/ (/ y_m x_m) z_m) (/ (/ y_m z_m) x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1e-48) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (y_m / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 1d-48) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = (y_m / z_m) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1e-48) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (y_m / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 1e-48:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = (y_m / z_m) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 1e-48)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(Float64(y_m / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 1e-48)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = (y_m / z_m) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1e-48], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.9999999999999997e-49

   1. Initial program 84.2%

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

   2. Taylor expanded in x around 0 49.1%

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

   if 9.9999999999999997e-49 < y

   1. Initial program 96.3%

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

      1. associate-*l/ 96.2%

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

   3. Simplified 96.2%

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

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

   6. Taylor expanded in x around 0 59.2%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-48}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 16: 49.1% 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 87.9%

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

   1. associate-*l/ 87.9%

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

3. Simplified 87.9%

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

4. Taylor expanded in x around 0 50.5%

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

5. Final simplification 50.5%

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

Developer target: 97.0% 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 2023322 
(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))