Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.6% → 99.8%

Time: 8.5s

Alternatives: 8

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \cosh x\_m \cdot \frac{y\_m}{x\_m}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 4 \cdot 10^{+289}:\\ \;\;\;\;\frac{t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\cosh x\_m}{x\_m}}{z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (cosh x_m) (/ y_m x_m))))
   (*
    y_s
    (* x_s (if (<= t_0 4e+289) (/ t_0 z) (* y_m (/ (/ (cosh x_m) x_m) z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 4e+289) {
		tmp = t_0 / z;
	} else {
		tmp = y_m * ((cosh(x_m) / x_m) / z);
	}
	return 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)
real(8) function code(y_s, x_s, x_m, y_m, z)
    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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x_m) * (y_m / x_m)
    if (t_0 <= 4d+289) then
        tmp = t_0 / z
    else
        tmp = y_m * ((cosh(x_m) / x_m) / z)
    end if
    code = 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);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = Math.cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 4e+289) {
		tmp = t_0 / z;
	} else {
		tmp = y_m * ((Math.cosh(x_m) / x_m) / z);
	}
	return 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)
def code(y_s, x_s, x_m, y_m, z):
	t_0 = math.cosh(x_m) * (y_m / x_m)
	tmp = 0
	if t_0 <= 4e+289:
		tmp = t_0 / z
	else:
		tmp = y_m * ((math.cosh(x_m) / x_m) / z)
	return 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)
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(cosh(x_m) * Float64(y_m / x_m))
	tmp = 0.0
	if (t_0 <= 4e+289)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(y_m * Float64(Float64(cosh(x_m) / x_m) / z));
	end
	return 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);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = cosh(x_m) * (y_m / x_m);
	tmp = 0.0;
	if (t_0 <= 4e+289)
		tmp = t_0 / z;
	else
		tmp = y_m * ((cosh(x_m) / x_m) / z);
	end
	tmp_2 = 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]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 4e+289], N[(t$95$0 / z), $MachinePrecision], N[(y$95$m * N[(N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 4.0000000000000002e289

   1. Initial program 99.3%

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

   if 4.0000000000000002e289 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 67.0%

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

      1. associate-/l* 56.2%

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

      2. associate-/l/ 63.4%

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

   3. Simplified 63.4%

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

      1. clear-num 63.4%

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

      2. un-div-inv 63.4%

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

      3. *-commutative 63.4%

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

      4. associate-/l* 68.8%

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

   6. Applied egg-rr 68.8%

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

      1. associate-*r/ 63.4%

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

      2. associate-/r/ 78.5%

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

      3. associate-/l/ 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-/l/ 78.5%

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

      6. associate-/r* 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 4 \cdot 10^{+289}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 7 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\cosh x\_m}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= x_m 7e-145) (/ (/ y_m z) x_m) (* y_m (/ (/ (cosh x_m) x_m) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 7e-145) {
		tmp = (y_m / z) / x_m;
	} else {
		tmp = y_m * ((cosh(x_m) / x_m) / z);
	}
	return 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)
real(8) function code(y_s, x_s, x_m, y_m, z)
    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
    real(8) :: tmp
    if (x_m <= 7d-145) then
        tmp = (y_m / z) / x_m
    else
        tmp = y_m * ((cosh(x_m) / x_m) / z)
    end if
    code = 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);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 7e-145) {
		tmp = (y_m / z) / x_m;
	} else {
		tmp = y_m * ((Math.cosh(x_m) / x_m) / z);
	}
	return 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)
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if x_m <= 7e-145:
		tmp = (y_m / z) / x_m
	else:
		tmp = y_m * ((math.cosh(x_m) / x_m) / z)
	return 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)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 7e-145)
		tmp = Float64(Float64(y_m / z) / x_m);
	else
		tmp = Float64(y_m * Float64(Float64(cosh(x_m) / x_m) / z));
	end
	return 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);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 7e-145)
		tmp = (y_m / z) / x_m;
	else
		tmp = y_m * ((cosh(x_m) / x_m) / z);
	end
	tmp_2 = 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]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 7e-145], N[(N[(y$95$m / z), $MachinePrecision] / x$95$m), $MachinePrecision], N[(y$95$m * N[(N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 6.99999999999999994e-145

   1. Initial program 87.1%

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

      1. associate-/l* 79.4%

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

      2. associate-/l/ 76.6%

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

   3. Simplified 76.6%

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

      1. associate-*r/ 83.7%

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

      2. frac-times 87.0%

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

      3. *-commutative 87.0%

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

      4. associate-*l/ 97.2%

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

   6. Applied egg-rr 97.2%

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

   7. Taylor expanded in x around 0 62.8%

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

   if 6.99999999999999994e-145 < x

   1. Initial program 88.4%

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

      1. associate-/l* 79.2%

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

      2. associate-/l/ 75.3%

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

   3. Simplified 75.3%

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

      1. clear-num 75.2%

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

      2. un-div-inv 75.2%

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

      3. *-commutative 75.2%

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

      4. associate-/l* 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. associate-*r/ 75.2%

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

      2. associate-/r/ 81.5%

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

      3. associate-/l/ 98.8%

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

      4. *-commutative 98.8%

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

      5. associate-/l/ 81.5%

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

      6. associate-/r* 98.8%

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

   8. Simplified 98.8%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{\cosh x\_m}{x\_m \cdot \frac{z}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\cosh x\_m}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 3.5e-63)
     (/ (cosh x_m) (* x_m (/ z y_m)))
     (* y_m (/ (/ (cosh x_m) x_m) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 3.5e-63) {
		tmp = cosh(x_m) / (x_m * (z / y_m));
	} else {
		tmp = y_m * ((cosh(x_m) / x_m) / z);
	}
	return 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)
real(8) function code(y_s, x_s, x_m, y_m, z)
    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
    real(8) :: tmp
    if (z <= 3.5d-63) then
        tmp = cosh(x_m) / (x_m * (z / y_m))
    else
        tmp = y_m * ((cosh(x_m) / x_m) / z)
    end if
    code = 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);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 3.5e-63) {
		tmp = Math.cosh(x_m) / (x_m * (z / y_m));
	} else {
		tmp = y_m * ((Math.cosh(x_m) / x_m) / z);
	}
	return 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)
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 3.5e-63:
		tmp = math.cosh(x_m) / (x_m * (z / y_m))
	else:
		tmp = y_m * ((math.cosh(x_m) / x_m) / z)
	return 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)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 3.5e-63)
		tmp = Float64(cosh(x_m) / Float64(x_m * Float64(z / y_m)));
	else
		tmp = Float64(y_m * Float64(Float64(cosh(x_m) / x_m) / z));
	end
	return 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);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 3.5e-63)
		tmp = cosh(x_m) / (x_m * (z / y_m));
	else
		tmp = y_m * ((cosh(x_m) / x_m) / z);
	end
	tmp_2 = 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]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 3.5e-63], N[(N[Cosh[x$95$m], $MachinePrecision] / N[(x$95$m * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.50000000000000003e-63

   1. Initial program 87.2%

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

      1. associate-/l* 79.9%

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

      2. associate-/l/ 75.1%

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

   3. Simplified 75.1%

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

      1. clear-num 74.7%

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

      2. un-div-inv 74.7%

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

      3. *-commutative 74.7%

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

      4. associate-/l* 82.3%

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

   6. Applied egg-rr 82.3%

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

   if 3.50000000000000003e-63 < z

   1. Initial program 88.4%

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

      1. associate-/l* 77.9%

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

      2. associate-/l/ 78.8%

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

   3. Simplified 78.8%

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

      1. clear-num 78.7%

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

      2. un-div-inv 78.7%

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

      3. *-commutative 78.7%

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

      4. associate-/l* 80.4%

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

   6. Applied egg-rr 80.4%

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

      1. associate-*r/ 78.7%

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

      2. associate-/r/ 85.2%

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

      3. associate-/l/ 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-/l/ 85.2%

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

      6. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\cosh x\_m}{x\_m}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{y\_m \cdot t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{t\_0}{z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (cosh x_m) x_m)))
   (* y_s (* x_s (if (<= y_m 4.5e+19) (/ (* y_m t_0) z) (* y_m (/ t_0 z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = cosh(x_m) / x_m;
	double tmp;
	if (y_m <= 4.5e+19) {
		tmp = (y_m * t_0) / z;
	} else {
		tmp = y_m * (t_0 / z);
	}
	return 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)
real(8) function code(y_s, x_s, x_m, y_m, z)
    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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x_m) / x_m
    if (y_m <= 4.5d+19) then
        tmp = (y_m * t_0) / z
    else
        tmp = y_m * (t_0 / z)
    end if
    code = 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);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = Math.cosh(x_m) / x_m;
	double tmp;
	if (y_m <= 4.5e+19) {
		tmp = (y_m * t_0) / z;
	} else {
		tmp = y_m * (t_0 / z);
	}
	return 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)
def code(y_s, x_s, x_m, y_m, z):
	t_0 = math.cosh(x_m) / x_m
	tmp = 0
	if y_m <= 4.5e+19:
		tmp = (y_m * t_0) / z
	else:
		tmp = y_m * (t_0 / z)
	return 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)
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(cosh(x_m) / x_m)
	tmp = 0.0
	if (y_m <= 4.5e+19)
		tmp = Float64(Float64(y_m * t_0) / z);
	else
		tmp = Float64(y_m * Float64(t_0 / z));
	end
	return 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);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = cosh(x_m) / x_m;
	tmp = 0.0;
	if (y_m <= 4.5e+19)
		tmp = (y_m * t_0) / z;
	else
		tmp = y_m * (t_0 / z);
	end
	tmp_2 = 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]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 4.5e+19], N[(N[(y$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5e19

   1. Initial program 85.5%

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

      1. associate-/l* 75.3%

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

      2. associate-/l/ 72.9%

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

   3. Simplified 72.9%

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

      1. clear-num 72.5%

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

      2. un-div-inv 72.5%

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

      3. *-commutative 72.5%

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

      4. associate-/l* 77.4%

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

   6. Applied egg-rr 77.4%

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

      1. associate-*r/ 72.5%

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

      2. associate-/r/ 80.9%

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

      3. associate-/l/ 93.0%

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

      4. *-commutative 93.0%

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

      5. associate-/l/ 80.9%

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

      6. associate-/r* 93.0%

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

   8. Simplified 93.0%

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

      1. associate-*r/ 97.5%

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

   10. Applied egg-rr 97.5%

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

   if 4.5e19 < y

   1. Initial program 96.0%

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

      1. associate-/l* 96.0%

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

      2. associate-/l/ 89.8%

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

   3. Simplified 89.8%

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

      1. clear-num 89.8%

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

      2. un-div-inv 89.8%

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

      3. *-commutative 89.8%

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

      4. associate-/l* 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 89.8%

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

      2. associate-/r/ 89.7%

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

      3. associate-/l/ 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-/l/ 89.7%

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

      6. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 7.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\cosh x\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\cosh x\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 7.5e-132)
     (/ (* y_m (/ (cosh x_m) x_m)) z)
     (/ (* y_m (/ (cosh x_m) z)) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 7.5e-132) {
		tmp = (y_m * (cosh(x_m) / x_m)) / z;
	} else {
		tmp = (y_m * (cosh(x_m) / z)) / x_m;
	}
	return 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)
real(8) function code(y_s, x_s, x_m, y_m, z)
    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
    real(8) :: tmp
    if (y_m <= 7.5d-132) then
        tmp = (y_m * (cosh(x_m) / x_m)) / z
    else
        tmp = (y_m * (cosh(x_m) / z)) / x_m
    end if
    code = 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);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 7.5e-132) {
		tmp = (y_m * (Math.cosh(x_m) / x_m)) / z;
	} else {
		tmp = (y_m * (Math.cosh(x_m) / z)) / x_m;
	}
	return 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)
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if y_m <= 7.5e-132:
		tmp = (y_m * (math.cosh(x_m) / x_m)) / z
	else:
		tmp = (y_m * (math.cosh(x_m) / z)) / x_m
	return 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)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 7.5e-132)
		tmp = Float64(Float64(y_m * Float64(cosh(x_m) / x_m)) / z);
	else
		tmp = Float64(Float64(y_m * Float64(cosh(x_m) / z)) / x_m);
	end
	return 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);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (y_m <= 7.5e-132)
		tmp = (y_m * (cosh(x_m) / x_m)) / z;
	else
		tmp = (y_m * (cosh(x_m) / z)) / x_m;
	end
	tmp_2 = 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]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 7.5e-132], N[(N[(y$95$m * N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m * N[(N[Cosh[x$95$m], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 7.49999999999999989e-132

   1. Initial program 84.4%

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

      1. associate-/l* 75.4%

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

      2. associate-/l/ 73.7%

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

   3. Simplified 73.7%

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

      1. clear-num 73.2%

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

      2. un-div-inv 73.2%

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

      3. *-commutative 73.2%

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

      4. associate-/l* 77.8%

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

   6. Applied egg-rr 77.8%

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

      1. associate-*r/ 73.2%

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

      2. associate-/r/ 81.7%

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

      3. associate-/l/ 93.5%

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

      4. *-commutative 93.5%

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

      5. associate-/l/ 81.7%

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

      6. associate-/r* 93.5%

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

   8. Simplified 93.5%

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

      1. associate-*r/ 97.2%

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

   10. Applied egg-rr 97.2%

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

   if 7.49999999999999989e-132 < y

   1. Initial program 94.8%

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

      1. associate-/l* 88.4%

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

      2. associate-/l/ 81.8%

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

   3. Simplified 81.8%

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

      1. associate-*r/ 85.7%

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

      2. frac-times 94.7%

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

      3. *-commutative 94.7%

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

      4. associate-*l/ 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2.16 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (if (<= z 2.16e+77) (/ (/ y_m x_m) z) (/ y_m (* x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 2.16e+77) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = y_m / (x_m * z);
	}
	return 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)
real(8) function code(y_s, x_s, x_m, y_m, z)
    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
    real(8) :: tmp
    if (z <= 2.16d+77) then
        tmp = (y_m / x_m) / z
    else
        tmp = y_m / (x_m * z)
    end if
    code = 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);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 2.16e+77) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = y_m / (x_m * z);
	}
	return 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)
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 2.16e+77:
		tmp = (y_m / x_m) / z
	else:
		tmp = y_m / (x_m * z)
	return 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)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 2.16e+77)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(y_m / Float64(x_m * z));
	end
	return 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);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 2.16e+77)
		tmp = (y_m / x_m) / z;
	else
		tmp = y_m / (x_m * z);
	end
	tmp_2 = 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]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 2.16e+77], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.15999999999999992e77

   1. Initial program 87.9%

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

   3. Taylor expanded in x around 0 51.2%

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

   if 2.15999999999999992e77 < z

   1. Initial program 85.2%

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

      1. associate-/l* 67.3%

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

      2. associate-/l/ 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in x around 0 59.6%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.16 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.8% accurate, 10.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (if (<= y_m 2.4e-132) (/ (/ y_m x_m) z) (/ (/ y_m z) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 2.4e-132) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = (y_m / z) / x_m;
	}
	return 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)
real(8) function code(y_s, x_s, x_m, y_m, z)
    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
    real(8) :: tmp
    if (y_m <= 2.4d-132) then
        tmp = (y_m / x_m) / z
    else
        tmp = (y_m / z) / x_m
    end if
    code = 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);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 2.4e-132) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = (y_m / z) / x_m;
	}
	return 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)
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if y_m <= 2.4e-132:
		tmp = (y_m / x_m) / z
	else:
		tmp = (y_m / z) / x_m
	return 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)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 2.4e-132)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(Float64(y_m / z) / x_m);
	end
	return 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);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (y_m <= 2.4e-132)
		tmp = (y_m / x_m) / z;
	else
		tmp = (y_m / z) / x_m;
	end
	tmp_2 = 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]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 2.4e-132], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.40000000000000015e-132

   1. Initial program 84.4%

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

   3. Taylor expanded in x around 0 49.3%

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

   if 2.40000000000000015e-132 < y

   1. Initial program 94.8%

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

      1. associate-/l* 88.4%

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

      2. associate-/l/ 81.8%

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

   3. Simplified 81.8%

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

      1. associate-*r/ 85.7%

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

      2. frac-times 94.7%

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

      3. *-commutative 94.7%

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

      4. associate-*l/ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 61.1%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.3% accurate, 21.4× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ y_m (* x_m z)))))

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    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
    code = y_s * (x_s * (y_m / (x_m * z)))
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);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (y_m / (x_m * z)));
}

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)
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * (y_m / (x_m * z)))

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (y_m / (x_m * z)));
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]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.5%

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

   1. associate-/l* 79.3%

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

   2. associate-/l/ 76.2%

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

3. Simplified 76.2%

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

5. Taylor expanded in x around 0 48.2%

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

6. Final simplification 48.2%

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

Developer target: 97.3% 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 2024082 
(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))