Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.5% → 99.7%

Time: 14.4s

Alternatives: 6

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: 84.5% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.1 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{y}{z\_m} \cdot \cosh x}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 3.1e+40)
    (/ (* (/ y z_m) (cosh x)) x)
    (* y (/ (/ (cosh x) x) z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 3.1e+40) {
		tmp = ((y / z_m) * cosh(x)) / x;
	} else {
		tmp = y * ((cosh(x) / x) / z_m);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 3.1d+40) then
        tmp = ((y / z_m) * cosh(x)) / x
    else
        tmp = y * ((cosh(x) / x) / z_m)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 3.1e+40) {
		tmp = ((y / z_m) * Math.cosh(x)) / x;
	} else {
		tmp = y * ((Math.cosh(x) / x) / z_m);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 3.1e+40:
		tmp = ((y / z_m) * math.cosh(x)) / x
	else:
		tmp = y * ((math.cosh(x) / x) / z_m)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 3.1e+40)
		tmp = Float64(Float64(Float64(y / z_m) * cosh(x)) / x);
	else
		tmp = Float64(y * Float64(Float64(cosh(x) / x) / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 3.1e+40)
		tmp = ((y / z_m) * cosh(x)) / x;
	else
		tmp = y * ((cosh(x) / x) / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 3.1e+40], N[(N[(N[(y / z$95$m), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.0999999999999998e40

   1. Initial program 85.2%

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

      1. associate-/l* 80.0%

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

      2. associate-/l/ 78.2%

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

   3. Simplified 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-/r* 84.4%

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

      3. associate-*l/ 92.5%

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

   6. Applied egg-rr 92.5%

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

   if 3.0999999999999998e40 < z

   1. Initial program 77.2%

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

      1. associate-/l* 58.0%

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

      2. associate-/l/ 55.2%

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

   3. Simplified 55.2%

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

      1. clear-num 55.1%

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

      2. un-div-inv 55.1%

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

      3. *-commutative 55.1%

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

      4. associate-/l* 61.9%

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

   6. Applied egg-rr 61.9%

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

      1. associate-*r/ 55.1%

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

      2. associate-/r/ 61.5%

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

      3. associate-/l/ 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/l/ 61.5%

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

      6. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.1 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \cosh x}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+115}:\\ \;\;\;\;\frac{t\_0}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y x))))
   (* z_s (if (<= t_0 1e+115) (/ t_0 z_m) (* y (/ (/ (cosh x) x) z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double t_0 = cosh(x) * (y / x);
	double tmp;
	if (t_0 <= 1e+115) {
		tmp = t_0 / z_m;
	} else {
		tmp = y * ((cosh(x) / x) / z_m);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x) * (y / x)
    if (t_0 <= 1d+115) then
        tmp = t_0 / z_m
    else
        tmp = y * ((cosh(x) / x) / z_m)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double t_0 = Math.cosh(x) * (y / x);
	double tmp;
	if (t_0 <= 1e+115) {
		tmp = t_0 / z_m;
	} else {
		tmp = y * ((Math.cosh(x) / x) / z_m);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	t_0 = math.cosh(x) * (y / x)
	tmp = 0
	if t_0 <= 1e+115:
		tmp = t_0 / z_m
	else:
		tmp = y * ((math.cosh(x) / x) / z_m)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	t_0 = Float64(cosh(x) * Float64(y / x))
	tmp = 0.0
	if (t_0 <= 1e+115)
		tmp = Float64(t_0 / z_m);
	else
		tmp = Float64(y * Float64(Float64(cosh(x) / x) / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	t_0 = cosh(x) * (y / x);
	tmp = 0.0;
	if (t_0 <= 1e+115)
		tmp = t_0 / z_m;
	else
		tmp = y * ((cosh(x) / x) / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t$95$0, 1e+115], N[(t$95$0 / z$95$m), $MachinePrecision], N[(y * N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1e115

   1. Initial program 96.5%

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

   if 1e115 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.9%

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

      1. associate-/l* 62.9%

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

      2. associate-/l/ 67.8%

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

   3. Simplified 67.8%

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

      1. clear-num 67.7%

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

      2. un-div-inv 67.7%

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

      3. *-commutative 67.7%

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

      4. associate-/l* 74.5%

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

   6. Applied egg-rr 74.5%

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

      1. associate-*r/ 67.7%

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

      2. associate-/r/ 77.1%

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

      3. associate-/l/ 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/l/ 77.1%

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

      6. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 10^{+115}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 1.0× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m) :precision binary64 (* z_s (* y (/ (/ (cosh x) x) z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * (y * ((cosh(x) / x) / z_m));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = z_s * (y * ((cosh(x) / x) / z_m))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	return z_s * (y * ((Math.cosh(x) / x) / z_m));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	return z_s * (y * ((math.cosh(x) / x) / z_m))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(y * Float64(Float64(cosh(x) / x) / z_m)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m)
	tmp = z_s * (y * ((cosh(x) / x) / z_m));
end

Wolfram

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_, x_, y_, z$95$m_] := N[(z$95$s * N[(y * N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.7%

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

   1. associate-/l* 75.9%

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

   2. associate-/l/ 74.0%

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

3. Simplified 74.0%

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

   1. clear-num 73.9%

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

   2. un-div-inv 73.9%

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

   3. *-commutative 73.9%

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

   4. associate-/l* 80.1%

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

6. Applied egg-rr 80.1%

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

   1. associate-*r/ 73.9%

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

   2. associate-/r/ 79.0%

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

   3. associate-/l/ 96.5%

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

   4. *-commutative 96.5%

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

   5. associate-/l/ 79.0%

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

   6. associate-/r* 96.5%

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

8. Simplified 96.5%

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

9. Final simplification 96.5%

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

Alternative 4: 50.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 9.6 \cdot 10^{-166}:\\ \;\;\;\;\frac{\frac{y}{x}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z\_m \cdot x}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 9.6e-166) (/ (/ y x) z_m) (/ y (* z_m x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 9.6e-166) {
		tmp = (y / x) / z_m;
	} else {
		tmp = y / (z_m * x);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 9.6d-166) then
        tmp = (y / x) / z_m
    else
        tmp = y / (z_m * x)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 9.6e-166) {
		tmp = (y / x) / z_m;
	} else {
		tmp = y / (z_m * x);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 9.6e-166:
		tmp = (y / x) / z_m
	else:
		tmp = y / (z_m * x)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 9.6e-166)
		tmp = Float64(Float64(y / x) / z_m);
	else
		tmp = Float64(y / Float64(z_m * x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 9.6e-166)
		tmp = (y / x) / z_m;
	else
		tmp = y / (z_m * x);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 9.6e-166], N[(N[(y / x), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.5999999999999994e-166

   1. Initial program 80.5%

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

   3. Taylor expanded in x around 0 45.4%

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

   if 9.5999999999999994e-166 < y

   1. Initial program 89.5%

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

      1. associate-/l* 80.9%

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

      2. associate-/l/ 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in x around 0 57.7%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.6 \cdot 10^{-166}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.9% accurate, 10.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{y}{z\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z\_m \cdot x}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= z_m 5e-30) (/ (/ y z_m) x) (/ y (* z_m x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5e-30) {
		tmp = (y / z_m) / x;
	} else {
		tmp = y / (z_m * x);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 5d-30) then
        tmp = (y / z_m) / x
    else
        tmp = y / (z_m * x)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5e-30) {
		tmp = (y / z_m) / x;
	} else {
		tmp = y / (z_m * x);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 5e-30:
		tmp = (y / z_m) / x
	else:
		tmp = y / (z_m * x)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 5e-30)
		tmp = Float64(Float64(y / z_m) / x);
	else
		tmp = Float64(y / Float64(z_m * x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 5e-30)
		tmp = (y / z_m) / x;
	else
		tmp = y / (z_m * x);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 5e-30], N[(N[(y / z$95$m), $MachinePrecision] / x), $MachinePrecision], N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.99999999999999972e-30

   1. Initial program 85.0%

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

      1. associate-/l* 79.5%

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

      2. associate-/l/ 77.6%

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

   3. Simplified 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-/r* 84.1%

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

      3. associate-*l/ 92.2%

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

   6. Applied egg-rr 92.2%

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

   7. Taylor expanded in x around 0 50.4%

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

   if 4.99999999999999972e-30 < z

   1. Initial program 79.4%

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

      1. associate-/l* 63.6%

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

      2. associate-/l/ 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in x around 0 48.2%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.4% accurate, 21.4× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m) :precision binary64 (* z_s (/ y (* z_m x))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * (y / (z_m * x));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = z_s * (y / (z_m * x))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	return z_s * (y / (z_m * x));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	return z_s * (y / (z_m * x))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(y / Float64(z_m * x)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m)
	tmp = z_s * (y / (z_m * x));
end

Wolfram

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_, x_, y_, z$95$m_] := N[(z$95$s * N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.7%

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

   1. associate-/l* 75.9%

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

   2. associate-/l/ 74.0%

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

3. Simplified 74.0%

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

5. Taylor expanded in x around 0 47.6%

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

6. Final simplification 47.6%

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

Developer target: 97.0% 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 2024076 
(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))