Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.7% → 99.8%

Time: 12.7s

Alternatives: 17

Speedup: 1.0×

• 64.3% 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: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2000000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x\_m \cdot y\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x_m) (/ y_m x_m)) z_m)))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= t_0 2000000000.0) t_0 (/ (/ (* (cosh x_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 t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= 2000000000.0) {
		tmp = t_0;
	} else {
		tmp = ((cosh(x_m) * 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) :: t_0
    real(8) :: tmp
    t_0 = (cosh(x_m) * (y_m / x_m)) / z_m
    if (t_0 <= 2000000000.0d0) then
        tmp = t_0
    else
        tmp = ((cosh(x_m) * 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 t_0 = (Math.cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= 2000000000.0) {
		tmp = t_0;
	} else {
		tmp = ((Math.cosh(x_m) * 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):
	t_0 = (math.cosh(x_m) * (y_m / x_m)) / z_m
	tmp = 0
	if t_0 <= 2000000000.0:
		tmp = t_0
	else:
		tmp = ((math.cosh(x_m) * 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)
	t_0 = Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m)
	tmp = 0.0
	if (t_0 <= 2000000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(cosh(x_m) * 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)
	t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	tmp = 0.0;
	if (t_0 <= 2000000000.0)
		tmp = t_0;
	else
		tmp = ((cosh(x_m) * 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_] := 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, 2000000000.0], t$95$0, N[(N[(N[(N[Cosh[x$95$m], $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $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) < 2e9

   1. Initial program 92.7%

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

   if 2e9 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 81.2%

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

      1. *-commutative 81.2%

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

      2. associate-*l/ 99.9%

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

      3. associate-/l* 99.9%

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

      4. associate-/l* 94.6%

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

      5. associate-/r* 76.7%

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

   3. Simplified 76.7%

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

      1. associate-*r/ 77.6%

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

      2. *-commutative 77.6%

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

      3. *-commutative 77.6%

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

      4. associate-/r* 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2000000000:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+172}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x\_m}{x\_m} \cdot \frac{y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

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 5e+172) t_0 (* (/ (cosh x_m) 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 t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= 5e+172) {
		tmp = t_0;
	} else {
		tmp = (cosh(x_m) / 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) :: t_0
    real(8) :: tmp
    t_0 = (cosh(x_m) * (y_m / x_m)) / z_m
    if (t_0 <= 5d+172) then
        tmp = t_0
    else
        tmp = (cosh(x_m) / 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 t_0 = (Math.cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= 5e+172) {
		tmp = t_0;
	} else {
		tmp = (Math.cosh(x_m) / 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):
	t_0 = (math.cosh(x_m) * (y_m / x_m)) / z_m
	tmp = 0
	if t_0 <= 5e+172:
		tmp = t_0
	else:
		tmp = (math.cosh(x_m) / 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)
	t_0 = Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m)
	tmp = 0.0
	if (t_0 <= 5e+172)
		tmp = t_0;
	else
		tmp = Float64(Float64(cosh(x_m) / x_m) * Float64(y_m / z_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	tmp = 0.0;
	if (t_0 <= 5e+172)
		tmp = t_0;
	else
		tmp = (cosh(x_m) / 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_] := 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, 5e+172], t$95$0, N[(N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.0000000000000001e172

   1. Initial program 93.4%

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

   if 5.0000000000000001e172 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 78.5%

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

      1. associate-*r/ 100.0%

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

      2. associate-/r* 77.0%

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

      3. times-frac 91.6%

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

   4. Applied egg-rr 91.6%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 5 \cdot 10^{+172}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.45 \cdot 10^{+171}:\\ \;\;\;\;y\_m \cdot \frac{\cosh x\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x\_m \cdot y\_m}{z\_m} + y\_m \cdot \frac{\frac{1}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= z_m 2.45e+171)
      (* y_m (/ (cosh x_m) (* x_m z_m)))
      (+ (* 0.5 (/ (* x_m y_m) z_m)) (* y_m (/ (/ 1.0 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 (z_m <= 2.45e+171) {
		tmp = y_m * (cosh(x_m) / (x_m * z_m));
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / 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 (z_m <= 2.45d+171) then
        tmp = y_m * (cosh(x_m) / (x_m * z_m))
    else
        tmp = (0.5d0 * ((x_m * y_m) / z_m)) + (y_m * ((1.0d0 / 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 (z_m <= 2.45e+171) {
		tmp = y_m * (Math.cosh(x_m) / (x_m * z_m));
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / 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 z_m <= 2.45e+171:
		tmp = y_m * (math.cosh(x_m) / (x_m * z_m))
	else:
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / 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 (z_m <= 2.45e+171)
		tmp = Float64(y_m * Float64(cosh(x_m) / Float64(x_m * z_m)));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x_m * y_m) / z_m)) + Float64(y_m * Float64(Float64(1.0 / 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 (z_m <= 2.45e+171)
		tmp = y_m * (cosh(x_m) / (x_m * z_m));
	else
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / 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[z$95$m, 2.45e+171], N[(y$95$m * N[(N[Cosh[x$95$m], $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(y$95$m * N[(N[(1.0 / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.4499999999999999e171

   1. Initial program 90.2%

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

      1. *-commutative 90.2%

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

      2. associate-*l/ 97.3%

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

      3. associate-/l* 97.3%

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

      4. associate-/l* 96.0%

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

      5. associate-/r* 89.3%

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

   3. Simplified 89.3%

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

   if 2.4499999999999999e171 < z

   1. Initial program 66.2%

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

      1. *-commutative 66.2%

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

      2. associate-*l/ 88.1%

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

      3. associate-/l* 88.0%

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

      4. associate-/l* 99.8%

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

      5. associate-/r* 46.6%

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

   3. Simplified 46.6%

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

   5. Taylor expanded in x around 0 60.0%

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

      1. clear-num 57.8%

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

      2. associate-/r/ 60.0%

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

      3. *-commutative 60.0%

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

      4. associate-/r* 60.0%

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

   7. Applied egg-rr 60.0%

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{\frac{1}{z}}{x} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.45 \cdot 10^{+171}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + y \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0001:\\ \;\;\;\;0.5 \cdot \left(x\_m \cdot \frac{y\_m}{z\_m}\right) + \frac{y\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x\_m}{x\_m} \cdot \frac{y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 0.0001)
      (+ (* 0.5 (* x_m (/ y_m z_m))) (/ y_m (* x_m z_m)))
      (* (/ (cosh x_m) x_m) (/ y_m z_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.0001) {
		tmp = (0.5 * (x_m * (y_m / z_m))) + (y_m / (x_m * z_m));
	} else {
		tmp = (cosh(x_m) / x_m) * (y_m / z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.0001d0) then
        tmp = (0.5d0 * (x_m * (y_m / z_m))) + (y_m / (x_m * z_m))
    else
        tmp = (cosh(x_m) / x_m) * (y_m / z_m)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.0001) {
		tmp = (0.5 * (x_m * (y_m / z_m))) + (y_m / (x_m * z_m));
	} else {
		tmp = (Math.cosh(x_m) / x_m) * (y_m / z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.0001:
		tmp = (0.5 * (x_m * (y_m / z_m))) + (y_m / (x_m * z_m))
	else:
		tmp = (math.cosh(x_m) / x_m) * (y_m / z_m)
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.0001)
		tmp = Float64(Float64(0.5 * Float64(x_m * Float64(y_m / z_m))) + Float64(y_m / Float64(x_m * z_m)));
	else
		tmp = Float64(Float64(cosh(x_m) / x_m) * Float64(y_m / z_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.0001)
		tmp = (0.5 * (x_m * (y_m / z_m))) + (y_m / (x_m * z_m));
	else
		tmp = (cosh(x_m) / x_m) * (y_m / z_m);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 0.0001], N[(N[(0.5 * N[(x$95$m * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000005e-4

   1. Initial program 88.7%

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

      1. *-commutative 88.7%

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

      2. associate-*l/ 94.7%

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

      3. associate-/l* 94.6%

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

      4. associate-/l* 95.1%

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

      5. associate-/r* 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in x around 0 75.2%

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

      1. associate-/l* 73.2%

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

      2. *-commutative 73.2%

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

   7. Applied egg-rr 73.2%

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

   if 1.00000000000000005e-4 < x

   1. Initial program 83.1%

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

      1. associate-*r/ 100.0%

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

      2. associate-/r* 73.2%

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

      3. times-frac 88.7%

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

   4. Applied egg-rr 88.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0001:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{z}\right) + \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m} + 0.5 \cdot \left(y\_m \cdot \left(x\_m \cdot \frac{1}{z\_m}\right)\right)\\ \mathbf{elif}\;z\_m \leq 1.75 \cdot 10^{+78}:\\ \;\;\;\;\frac{\left(x\_m \cdot z\_m\right) \cdot \left(x\_m \cdot \left(y\_m \cdot 0.5\right)\right) + y\_m \cdot z\_m}{z\_m \cdot \left(x\_m \cdot z\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x\_m \cdot y\_m}{z\_m} + y\_m \cdot \frac{\frac{1}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= z_m 4.8e-63)
      (+ (/ y_m (* x_m z_m)) (* 0.5 (* y_m (* x_m (/ 1.0 z_m)))))
      (if (<= z_m 1.75e+78)
        (/
         (+ (* (* x_m z_m) (* x_m (* y_m 0.5))) (* y_m z_m))
         (* z_m (* x_m z_m)))
        (+ (* 0.5 (/ (* x_m y_m) z_m)) (* y_m (/ (/ 1.0 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 (z_m <= 4.8e-63) {
		tmp = (y_m / (x_m * z_m)) + (0.5 * (y_m * (x_m * (1.0 / z_m))));
	} else if (z_m <= 1.75e+78) {
		tmp = (((x_m * z_m) * (x_m * (y_m * 0.5))) + (y_m * z_m)) / (z_m * (x_m * z_m));
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / 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 (z_m <= 4.8d-63) then
        tmp = (y_m / (x_m * z_m)) + (0.5d0 * (y_m * (x_m * (1.0d0 / z_m))))
    else if (z_m <= 1.75d+78) then
        tmp = (((x_m * z_m) * (x_m * (y_m * 0.5d0))) + (y_m * z_m)) / (z_m * (x_m * z_m))
    else
        tmp = (0.5d0 * ((x_m * y_m) / z_m)) + (y_m * ((1.0d0 / 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 (z_m <= 4.8e-63) {
		tmp = (y_m / (x_m * z_m)) + (0.5 * (y_m * (x_m * (1.0 / z_m))));
	} else if (z_m <= 1.75e+78) {
		tmp = (((x_m * z_m) * (x_m * (y_m * 0.5))) + (y_m * z_m)) / (z_m * (x_m * z_m));
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / 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 z_m <= 4.8e-63:
		tmp = (y_m / (x_m * z_m)) + (0.5 * (y_m * (x_m * (1.0 / z_m))))
	elif z_m <= 1.75e+78:
		tmp = (((x_m * z_m) * (x_m * (y_m * 0.5))) + (y_m * z_m)) / (z_m * (x_m * z_m))
	else:
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / 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 (z_m <= 4.8e-63)
		tmp = Float64(Float64(y_m / Float64(x_m * z_m)) + Float64(0.5 * Float64(y_m * Float64(x_m * Float64(1.0 / z_m)))));
	elseif (z_m <= 1.75e+78)
		tmp = Float64(Float64(Float64(Float64(x_m * z_m) * Float64(x_m * Float64(y_m * 0.5))) + Float64(y_m * z_m)) / Float64(z_m * Float64(x_m * z_m)));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x_m * y_m) / z_m)) + Float64(y_m * Float64(Float64(1.0 / 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 (z_m <= 4.8e-63)
		tmp = (y_m / (x_m * z_m)) + (0.5 * (y_m * (x_m * (1.0 / z_m))));
	elseif (z_m <= 1.75e+78)
		tmp = (((x_m * z_m) * (x_m * (y_m * 0.5))) + (y_m * z_m)) / (z_m * (x_m * z_m));
	else
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / 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[z$95$m, 4.8e-63], N[(N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(y$95$m * N[(x$95$m * N[(1.0 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.75e+78], N[(N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] * N[(x$95$m * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(y$95$m * N[(N[(1.0 / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 4.8000000000000001e-63

   1. Initial program 90.7%

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

      1. *-commutative 90.7%

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

      2. associate-*l/ 97.6%

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

      3. associate-/l* 97.6%

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

      4. associate-/l* 94.9%

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

      5. associate-/r* 86.8%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in x around 0 68.0%

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

      1. div-inv 68.0%

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

      2. *-commutative 68.0%

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

      3. associate-*l* 68.0%

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

   7. Applied egg-rr 68.0%

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

   if 4.8000000000000001e-63 < z < 1.7500000000000001e78

   1. Initial program 93.1%

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

      1. *-commutative 93.1%

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

      2. associate-*l/ 99.8%

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

      3. associate-/l* 99.8%

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

      4. associate-/l* 99.7%

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

      5. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 58.4%

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

      1. associate-*r/ 58.4%

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

      2. frac-add 67.5%

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

      3. *-commutative 67.5%

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

      4. associate-*l* 67.5%

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

   7. Applied egg-rr 67.5%

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

   if 1.7500000000000001e78 < z

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

      2. associate-*l/ 89.4%

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

      3. associate-/l* 89.3%

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

      4. associate-/l* 99.8%

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

      5. associate-/r* 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in x around 0 60.4%

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

      1. clear-num 59.0%

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

      2. associate-/r/ 60.4%

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

      3. *-commutative 60.4%

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

      4. associate-/r* 60.5%

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

   7. Applied egg-rr 60.5%

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{\frac{1}{z}}{x} \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \left(y \cdot \left(x \cdot \frac{1}{z}\right)\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+78}:\\ \;\;\;\;\frac{\left(x \cdot z\right) \cdot \left(x \cdot \left(y \cdot 0.5\right)\right) + y \cdot z}{z \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + y \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.7 \cdot 10^{-128}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m} + 0.5 \cdot \left(y\_m \cdot \left(x\_m \cdot \frac{1}{z\_m}\right)\right)\\ \mathbf{elif}\;z\_m \leq 2.45 \cdot 10^{+110}:\\ \;\;\;\;\frac{z\_m \cdot \frac{y\_m}{z\_m} + x\_m \cdot \left(x\_m \cdot \left(y\_m \cdot 0.5\right)\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x\_m \cdot y\_m}{z\_m} + y\_m \cdot \frac{\frac{1}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= z_m 1.7e-128)
      (+ (/ y_m (* x_m z_m)) (* 0.5 (* y_m (* x_m (/ 1.0 z_m)))))
      (if (<= z_m 2.45e+110)
        (/ (+ (* z_m (/ y_m z_m)) (* x_m (* x_m (* y_m 0.5)))) (* x_m z_m))
        (+ (* 0.5 (/ (* x_m y_m) z_m)) (* y_m (/ (/ 1.0 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 (z_m <= 1.7e-128) {
		tmp = (y_m / (x_m * z_m)) + (0.5 * (y_m * (x_m * (1.0 / z_m))));
	} else if (z_m <= 2.45e+110) {
		tmp = ((z_m * (y_m / z_m)) + (x_m * (x_m * (y_m * 0.5)))) / (x_m * z_m);
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / 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 (z_m <= 1.7d-128) then
        tmp = (y_m / (x_m * z_m)) + (0.5d0 * (y_m * (x_m * (1.0d0 / z_m))))
    else if (z_m <= 2.45d+110) then
        tmp = ((z_m * (y_m / z_m)) + (x_m * (x_m * (y_m * 0.5d0)))) / (x_m * z_m)
    else
        tmp = (0.5d0 * ((x_m * y_m) / z_m)) + (y_m * ((1.0d0 / 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 (z_m <= 1.7e-128) {
		tmp = (y_m / (x_m * z_m)) + (0.5 * (y_m * (x_m * (1.0 / z_m))));
	} else if (z_m <= 2.45e+110) {
		tmp = ((z_m * (y_m / z_m)) + (x_m * (x_m * (y_m * 0.5)))) / (x_m * z_m);
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / 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 z_m <= 1.7e-128:
		tmp = (y_m / (x_m * z_m)) + (0.5 * (y_m * (x_m * (1.0 / z_m))))
	elif z_m <= 2.45e+110:
		tmp = ((z_m * (y_m / z_m)) + (x_m * (x_m * (y_m * 0.5)))) / (x_m * z_m)
	else:
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / 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 (z_m <= 1.7e-128)
		tmp = Float64(Float64(y_m / Float64(x_m * z_m)) + Float64(0.5 * Float64(y_m * Float64(x_m * Float64(1.0 / z_m)))));
	elseif (z_m <= 2.45e+110)
		tmp = Float64(Float64(Float64(z_m * Float64(y_m / z_m)) + Float64(x_m * Float64(x_m * Float64(y_m * 0.5)))) / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x_m * y_m) / z_m)) + Float64(y_m * Float64(Float64(1.0 / 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 (z_m <= 1.7e-128)
		tmp = (y_m / (x_m * z_m)) + (0.5 * (y_m * (x_m * (1.0 / z_m))));
	elseif (z_m <= 2.45e+110)
		tmp = ((z_m * (y_m / z_m)) + (x_m * (x_m * (y_m * 0.5)))) / (x_m * z_m);
	else
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / 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[z$95$m, 1.7e-128], N[(N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(y$95$m * N[(x$95$m * N[(1.0 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2.45e+110], N[(N[(N[(z$95$m * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(x$95$m * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(y$95$m * N[(N[(1.0 / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1.69999999999999987e-128

   1. Initial program 90.5%

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

      1. *-commutative 90.5%

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

      2. associate-*l/ 97.5%

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

      3. associate-/l* 97.4%

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

      4. associate-/l* 94.5%

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

      5. associate-/r* 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in x around 0 67.8%

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

      1. div-inv 67.8%

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

      2. *-commutative 67.8%

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

      3. associate-*l* 67.8%

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

   7. Applied egg-rr 67.8%

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

   if 1.69999999999999987e-128 < z < 2.45000000000000001e110

   1. Initial program 89.0%

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

      1. *-commutative 89.0%

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

      2. associate-*l/ 98.1%

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

      3. associate-/l* 98.1%

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

      4. associate-/l* 99.7%

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

      5. associate-/r* 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in x around 0 61.4%

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

      1. clear-num 61.4%

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

      2. associate-/r/ 61.4%

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

   7. Applied egg-rr 61.4%

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

      1. associate-/l/ 58.0%

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

      2. un-div-inv 57.9%

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

      3. +-commutative 57.9%

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

      4. un-div-inv 58.0%

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

      5. associate-*l/ 58.0%

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

      6. *-un-lft-identity 58.0%

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

      7. associate-*r/ 58.0%

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

      8. frac-add 70.0%

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

      9. associate-*r* 70.0%

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

      10. *-commutative 70.0%

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

      11. associate-*l* 70.0%

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

      12. *-commutative 70.0%

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

   9. Applied egg-rr 70.0%

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

   if 2.45000000000000001e110 < z

   1. Initial program 72.0%

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

      1. *-commutative 72.0%

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

      2. associate-*l/ 88.7%

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

      3. associate-/l* 88.6%

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

      4. associate-/l* 99.7%

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

      5. associate-/r* 59.3%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in x around 0 60.6%

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

      1. clear-num 58.9%

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

      2. associate-/r/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/r* 60.7%

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

   7. Applied egg-rr 60.7%

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{\frac{1}{z}}{x} \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \left(y \cdot \left(x \cdot \frac{1}{z}\right)\right)\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+110}:\\ \;\;\;\;\frac{z \cdot \frac{y}{z} + x \cdot \left(x \cdot \left(y \cdot 0.5\right)\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + y \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{z\_m}{x\_m \cdot y\_m}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8.5 \cdot 10^{-192}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m} \cdot t\_0 + z\_m \cdot 0.5}{z\_m \cdot t\_0}\\ \mathbf{elif}\;z\_m \leq 5.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{z\_m \cdot \frac{y\_m}{z\_m} + x\_m \cdot \left(x\_m \cdot \left(y\_m \cdot 0.5\right)\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x\_m \cdot y\_m}{z\_m} + y\_m \cdot \frac{\frac{1}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ z_m (* x_m y_m))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 8.5e-192)
        (/ (+ (* (/ y_m x_m) t_0) (* z_m 0.5)) (* z_m t_0))
        (if (<= z_m 5.5e+109)
          (/ (+ (* z_m (/ y_m z_m)) (* x_m (* x_m (* y_m 0.5)))) (* x_m z_m))
          (+ (* 0.5 (/ (* x_m y_m) z_m)) (* y_m (/ (/ 1.0 z_m) x_m))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = z_m / (x_m * y_m);
	double tmp;
	if (z_m <= 8.5e-192) {
		tmp = (((y_m / x_m) * t_0) + (z_m * 0.5)) / (z_m * t_0);
	} else if (z_m <= 5.5e+109) {
		tmp = ((z_m * (y_m / z_m)) + (x_m * (x_m * (y_m * 0.5)))) / (x_m * z_m);
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / z_m) / x_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z_m / (x_m * y_m)
    if (z_m <= 8.5d-192) then
        tmp = (((y_m / x_m) * t_0) + (z_m * 0.5d0)) / (z_m * t_0)
    else if (z_m <= 5.5d+109) then
        tmp = ((z_m * (y_m / z_m)) + (x_m * (x_m * (y_m * 0.5d0)))) / (x_m * z_m)
    else
        tmp = (0.5d0 * ((x_m * y_m) / z_m)) + (y_m * ((1.0d0 / z_m) / x_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = z_m / (x_m * y_m);
	double tmp;
	if (z_m <= 8.5e-192) {
		tmp = (((y_m / x_m) * t_0) + (z_m * 0.5)) / (z_m * t_0);
	} else if (z_m <= 5.5e+109) {
		tmp = ((z_m * (y_m / z_m)) + (x_m * (x_m * (y_m * 0.5)))) / (x_m * z_m);
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / z_m) / x_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	t_0 = z_m / (x_m * y_m)
	tmp = 0
	if z_m <= 8.5e-192:
		tmp = (((y_m / x_m) * t_0) + (z_m * 0.5)) / (z_m * t_0)
	elif z_m <= 5.5e+109:
		tmp = ((z_m * (y_m / z_m)) + (x_m * (x_m * (y_m * 0.5)))) / (x_m * z_m)
	else:
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / z_m) / x_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(z_m / Float64(x_m * y_m))
	tmp = 0.0
	if (z_m <= 8.5e-192)
		tmp = Float64(Float64(Float64(Float64(y_m / x_m) * t_0) + Float64(z_m * 0.5)) / Float64(z_m * t_0));
	elseif (z_m <= 5.5e+109)
		tmp = Float64(Float64(Float64(z_m * Float64(y_m / z_m)) + Float64(x_m * Float64(x_m * Float64(y_m * 0.5)))) / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x_m * y_m) / z_m)) + Float64(y_m * Float64(Float64(1.0 / z_m) / x_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	t_0 = z_m / (x_m * y_m);
	tmp = 0.0;
	if (z_m <= 8.5e-192)
		tmp = (((y_m / x_m) * t_0) + (z_m * 0.5)) / (z_m * t_0);
	elseif (z_m <= 5.5e+109)
		tmp = ((z_m * (y_m / z_m)) + (x_m * (x_m * (y_m * 0.5)))) / (x_m * z_m);
	else
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / z_m) / x_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(z$95$m / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 8.5e-192], N[(N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(z$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 5.5e+109], N[(N[(N[(z$95$m * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(x$95$m * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(y$95$m * N[(N[(1.0 / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 8.49999999999999985e-192

   1. Initial program 90.2%

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

      1. *-commutative 90.2%

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

      2. associate-*l/ 97.2%

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

      3. associate-/l* 97.2%

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

      4. associate-/l* 94.5%

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

      5. associate-/r* 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in x around 0 66.0%

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

      1. div-inv 66.0%

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

      2. *-commutative 66.0%

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

      3. associate-*l* 66.0%

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

   7. Applied egg-rr 66.0%

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

      1. fma-define 66.0%

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

      2. associate-*r* 66.0%

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

      3. div-inv 66.0%

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

      4. associate-*l/ 60.7%

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

      5. fma-define 60.7%

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

      6. clear-num 60.7%

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

      7. associate-/r/ 60.7%

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

      8. +-commutative 60.7%

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

      9. associate-/r* 62.2%

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

      10. un-div-inv 62.2%

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

      11. frac-add 53.1%

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

      12. associate-/l/ 53.9%

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

      13. *-commutative 53.9%

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

      14. associate-/l/ 59.1%

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

      15. *-commutative 59.1%

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

   9. Applied egg-rr 59.1%

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

   if 8.49999999999999985e-192 < z < 5.4999999999999998e109

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-*l/ 98.4%

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

      3. associate-/l* 98.5%

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

      4. associate-/l* 98.4%

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

      5. associate-/r* 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 66.5%

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

      1. clear-num 66.5%

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

      2. associate-/r/ 66.5%

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

   7. Applied egg-rr 66.5%

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

      1. associate-/l/ 65.2%

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

      2. un-div-inv 65.1%

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

      3. +-commutative 65.1%

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

      4. un-div-inv 65.2%

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

      5. associate-*l/ 65.2%

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

      6. *-un-lft-identity 65.2%

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

      7. associate-*r/ 65.2%

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

      8. frac-add 73.0%

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

      9. associate-*r* 73.0%

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

      10. *-commutative 73.0%

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

      11. associate-*l* 73.0%

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

      12. *-commutative 73.0%

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

   9. Applied egg-rr 73.0%

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

   if 5.4999999999999998e109 < z

   1. Initial program 72.0%

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

      1. *-commutative 72.0%

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

      2. associate-*l/ 88.7%

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

      3. associate-/l* 88.6%

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

      4. associate-/l* 99.7%

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

      5. associate-/r* 59.3%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in x around 0 60.6%

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

      1. clear-num 58.9%

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

      2. associate-/r/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/r* 60.7%

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

   7. Applied egg-rr 60.7%

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{\frac{1}{z}}{x} \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.5 \cdot 10^{-192}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \frac{z}{x \cdot y} + z \cdot 0.5}{z \cdot \frac{z}{x \cdot y}}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{z \cdot \frac{y}{z} + x \cdot \left(x \cdot \left(y \cdot 0.5\right)\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + y \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{y\_m}{x\_m \cdot z\_m}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{-75}:\\ \;\;\;\;t\_0 + 0.5 \cdot \left(y\_m \cdot \left(x\_m \cdot \frac{1}{z\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + 0.5 \cdot \frac{x\_m \cdot y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

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 (/ y_m (* x_m z_m))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= y_m 1e-75)
        (+ t_0 (* 0.5 (* y_m (* x_m (/ 1.0 z_m)))))
        (+ t_0 (* 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 t_0 = y_m / (x_m * z_m);
	double tmp;
	if (y_m <= 1e-75) {
		tmp = t_0 + (0.5 * (y_m * (x_m * (1.0 / z_m))));
	} else {
		tmp = t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = y_m / (x_m * z_m)
    if (y_m <= 1d-75) then
        tmp = t_0 + (0.5d0 * (y_m * (x_m * (1.0d0 / z_m))))
    else
        tmp = t_0 + (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 t_0 = y_m / (x_m * z_m);
	double tmp;
	if (y_m <= 1e-75) {
		tmp = t_0 + (0.5 * (y_m * (x_m * (1.0 / z_m))));
	} else {
		tmp = t_0 + (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):
	t_0 = y_m / (x_m * z_m)
	tmp = 0
	if y_m <= 1e-75:
		tmp = t_0 + (0.5 * (y_m * (x_m * (1.0 / z_m))))
	else:
		tmp = t_0 + (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)
	t_0 = Float64(y_m / Float64(x_m * z_m))
	tmp = 0.0
	if (y_m <= 1e-75)
		tmp = Float64(t_0 + Float64(0.5 * Float64(y_m * Float64(x_m * Float64(1.0 / z_m)))));
	else
		tmp = Float64(t_0 + 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)
	t_0 = y_m / (x_m * z_m);
	tmp = 0.0;
	if (y_m <= 1e-75)
		tmp = t_0 + (0.5 * (y_m * (x_m * (1.0 / z_m))));
	else
		tmp = t_0 + (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_] := Block[{t$95$0 = N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1e-75], N[(t$95$0 + N[(0.5 * N[(y$95$m * N[(x$95$m * N[(1.0 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + 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 y < 9.9999999999999996e-76

   1. Initial program 85.6%

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

      1. *-commutative 85.6%

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

      2. associate-*l/ 98.3%

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

      3. associate-/l* 98.3%

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

      4. associate-/l* 95.1%

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

      5. associate-/r* 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in x around 0 59.1%

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

      1. div-inv 59.1%

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

      2. *-commutative 59.1%

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

      3. associate-*l* 60.6%

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

   7. Applied egg-rr 60.6%

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

   if 9.9999999999999996e-76 < y

   1. Initial program 91.0%

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

      1. *-commutative 91.0%

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

      2. associate-*l/ 91.0%

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

      3. associate-/l* 90.9%

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

      4. associate-/l* 99.8%

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

      5. associate-/r* 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in x around 0 80.3%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-75}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \left(y \cdot \left(x \cdot \frac{1}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 8.1 \cdot 10^{+29}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m} + 0.5 \cdot \left(y\_m \cdot \left(x\_m \cdot \frac{1}{z\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x\_m \cdot y\_m}{z\_m} + y\_m \cdot \frac{\frac{1}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 8.1e+29)
      (+ (/ y_m (* x_m z_m)) (* 0.5 (* y_m (* x_m (/ 1.0 z_m)))))
      (+ (* 0.5 (/ (* x_m y_m) z_m)) (* y_m (/ (/ 1.0 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 <= 8.1e+29) {
		tmp = (y_m / (x_m * z_m)) + (0.5 * (y_m * (x_m * (1.0 / z_m))));
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / 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 <= 8.1d+29) then
        tmp = (y_m / (x_m * z_m)) + (0.5d0 * (y_m * (x_m * (1.0d0 / z_m))))
    else
        tmp = (0.5d0 * ((x_m * y_m) / z_m)) + (y_m * ((1.0d0 / 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 <= 8.1e+29) {
		tmp = (y_m / (x_m * z_m)) + (0.5 * (y_m * (x_m * (1.0 / z_m))));
	} else {
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / 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 <= 8.1e+29:
		tmp = (y_m / (x_m * z_m)) + (0.5 * (y_m * (x_m * (1.0 / z_m))))
	else:
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / 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 <= 8.1e+29)
		tmp = Float64(Float64(y_m / Float64(x_m * z_m)) + Float64(0.5 * Float64(y_m * Float64(x_m * Float64(1.0 / z_m)))));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x_m * y_m) / z_m)) + Float64(y_m * Float64(Float64(1.0 / 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 <= 8.1e+29)
		tmp = (y_m / (x_m * z_m)) + (0.5 * (y_m * (x_m * (1.0 / z_m))));
	else
		tmp = (0.5 * ((x_m * y_m) / z_m)) + (y_m * ((1.0 / 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, 8.1e+29], N[(N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(y$95$m * N[(x$95$m * N[(1.0 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(y$95$m * N[(N[(1.0 / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 8.1000000000000007e29

   1. Initial program 86.9%

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

      1. *-commutative 86.9%

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

      2. associate-*l/ 98.4%

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

      3. associate-/l* 98.4%

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

      4. associate-/l* 95.5%

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

      5. associate-/r* 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in x around 0 59.4%

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

      1. div-inv 59.4%

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

      2. *-commutative 59.4%

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

      3. associate-*l* 60.8%

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

   7. Applied egg-rr 60.8%

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

   if 8.1000000000000007e29 < y

   1. Initial program 88.1%

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

      1. *-commutative 88.1%

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

      2. associate-*l/ 88.1%

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

      3. associate-/l* 88.0%

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

      4. associate-/l* 99.8%

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

      5. associate-/r* 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in x around 0 86.1%

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

      1. clear-num 85.9%

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

      2. associate-/r/ 86.2%

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

      3. *-commutative 86.2%

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

      4. associate-/r* 86.2%

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

   7. Applied egg-rr 86.2%

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{\frac{1}{z}}{x} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.1 \cdot 10^{+29}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \left(y \cdot \left(x \cdot \frac{1}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + y \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.5 \cdot 10^{-60}:\\ \;\;\;\;y\_m \cdot \left(0.5 \cdot \frac{x\_m}{z\_m} + \frac{1}{x\_m \cdot z\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m} + 0.5 \cdot \frac{x\_m \cdot y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= y_m 3.5e-60)
      (* 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 (y_m <= 3.5e-60) {
		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 (y_m <= 3.5d-60) 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 (y_m <= 3.5e-60) {
		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 y_m <= 3.5e-60:
		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 (y_m <= 3.5e-60)
		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 (y_m <= 3.5e-60)
		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[y$95$m, 3.5e-60], 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 y < 3.49999999999999976e-60

   1. Initial program 85.8%

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

      1. *-commutative 85.8%

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

      2. associate-*l/ 98.3%

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

      3. associate-/l* 98.3%

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

      4. associate-/l* 95.2%

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

      5. associate-/r* 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in x around 0 59.7%

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

   if 3.49999999999999976e-60 < y

   1. Initial program 90.7%

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

      1. *-commutative 90.7%

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

      2. associate-*l/ 90.7%

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

      3. associate-/l* 90.6%

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

      4. associate-/l* 99.8%

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

      5. associate-/r* 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in x around 0 81.1%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-60}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.3% accurate, 8.9× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (* x_s (if (<= x_m 1.4) (/ y_m (* x_m z_m)) (* 0.5 (* 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 (x_m <= 1.4) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = 0.5 * (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 (x_m <= 1.4d0) then
        tmp = y_m / (x_m * z_m)
    else
        tmp = 0.5d0 * (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 (x_m <= 1.4) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = 0.5 * (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 x_m <= 1.4:
		tmp = y_m / (x_m * z_m)
	else:
		tmp = 0.5 * (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 (x_m <= 1.4)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(0.5 * 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 (x_m <= 1.4)
		tmp = y_m / (x_m * z_m);
	else
		tmp = 0.5 * (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[x$95$m, 1.4], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * 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 x < 1.3999999999999999

   1. Initial program 88.7%

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

      1. *-commutative 88.7%

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

      2. associate-*l/ 94.7%

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

      3. associate-/l* 94.6%

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

      4. associate-/l* 95.1%

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

      5. associate-/r* 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in x around 0 65.8%

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

   if 1.3999999999999999 < x

   1. Initial program 83.1%

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

   3. Taylor expanded in x around 0 39.4%

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

   4. Taylor expanded in x around inf 39.3%

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

      1. associate-*r/ 27.3%

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

      2. *-commutative 27.3%

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

      3. associate-*l/ 39.3%

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

      4. associate-*r/ 33.8%

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

   6. Simplified 33.8%

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

4. Final simplification 56.9%

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

Alternative 12: 65.5% accurate, 8.9× speedup

Math

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

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 1.4) (/ 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 <= 1.4) {
		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 <= 1.4d0) 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 <= 1.4) {
		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 <= 1.4:
		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 <= 1.4)
		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 <= 1.4)
		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, 1.4], 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 < 1.3999999999999999

   1. Initial program 88.7%

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

      1. *-commutative 88.7%

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

      2. associate-*l/ 94.7%

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

      3. associate-/l* 94.6%

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

      4. associate-/l* 95.1%

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

      5. associate-/r* 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in x around 0 65.8%

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

   if 1.3999999999999999 < x

   1. Initial program 83.1%

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

   3. Taylor expanded in x around 0 39.4%

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

   4. Taylor expanded in x around inf 39.3%

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

      1. associate-*r* 39.3%

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

      2. *-commutative 39.3%

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

   6. Simplified 39.3%

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

4. Final simplification 58.5%

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

Alternative 13: 65.2% 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 87.2%

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

3. Taylor expanded in x around 0 64.3%

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

4. Taylor expanded in y around 0 64.3%

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

5. Final simplification 64.3%

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

Alternative 14: 65.2% 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 87.2%

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

3. Taylor expanded in x around 0 64.3%

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

4. Final simplification 64.3%

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

Alternative 15: 52.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \end{array}\right)\right) \end{array} \]

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 4.8e+29) (/ (/ 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 <= 4.8e+29) {
		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 <= 4.8d+29) 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 <= 4.8e+29) {
		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 <= 4.8e+29:
		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 <= 4.8e+29)
		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 <= 4.8e+29)
		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, 4.8e+29], 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 < 4.8000000000000002e29

   1. Initial program 86.9%

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

   3. Taylor expanded in x around 0 50.2%

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

   if 4.8000000000000002e29 < y

   1. Initial program 88.1%

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

      1. *-commutative 88.1%

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

      2. associate-*l/ 88.1%

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

      3. associate-/l* 88.0%

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

      4. associate-/l* 99.8%

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

      5. associate-/r* 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in x around 0 52.5%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 150000000:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= z_m 150000000.0) (/ (/ y_m 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 tmp;
	if (z_m <= 150000000.0) {
		tmp = (y_m / z_m) / x_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 (z_m <= 150000000.0d0) then
        tmp = (y_m / z_m) / x_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 (z_m <= 150000000.0) {
		tmp = (y_m / z_m) / x_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 z_m <= 150000000.0:
		tmp = (y_m / z_m) / x_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 (z_m <= 150000000.0)
		tmp = Float64(Float64(y_m / z_m) / x_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 (z_m <= 150000000.0)
		tmp = (y_m / z_m) / x_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[z$95$m, 150000000.0], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$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 z < 1.5e8

   1. Initial program 90.3%

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

      1. *-commutative 90.3%

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

      2. associate-*l/ 97.8%

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

      3. associate-/l* 97.8%

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

      4. associate-/l* 95.3%

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

      5. associate-/r* 87.8%

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

   3. Simplified 87.8%

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

      1. associate-*r/ 88.7%

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

      2. *-commutative 88.7%

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

      3. *-commutative 88.7%

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

      4. associate-/r* 97.4%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in x around 0 53.3%

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

   if 1.5e8 < z

   1. Initial program 78.6%

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

      1. *-commutative 78.6%

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

      2. associate-*l/ 91.6%

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

      3. associate-/l* 91.6%

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

      4. associate-/l* 99.7%

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

      5. associate-/r* 73.7%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in x around 0 51.9%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 150000000:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 48.5% 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 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-*l/ 96.2%

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

   3. associate-/l* 96.1%

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

   4. associate-/l* 96.5%

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

   5. associate-/r* 84.0%

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

3. Simplified 84.0%

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

5. Taylor expanded in x around 0 49.1%

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

6. Final simplification 49.1%

   \[\leadsto \frac{y}{x \cdot z} \]
7. Add Preprocessing

Developer target: 96.9% accurate, 0.9× 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 2024050 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :alt
  (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))