Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.4% → 99.8%

Time: 13.9s

Alternatives: 23

Speedup: 3.4×

• Could not converge on a ground truth

  1. x = 2.806749526593545e+113
  2. y = 6.455207398756919e-49
  3. z = 2.914989531696698e+38

• 64.0% 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: 84.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \cosh x\_m \cdot y\_m\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 6 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{t\_0}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \frac{z}{t\_0}}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (cosh x_m) y_m)))
   (*
    x_s
    (*
     y_s
     (if (<= (* (cosh x_m) (/ y_m x_m)) 6e+307)
       (/ (/ t_0 x_m) z)
       (/ 1.0 (* x_m (/ z t_0))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = cosh(x_m) * y_m;
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 6e+307) {
		tmp = (t_0 / x_m) / z;
	} else {
		tmp = 1.0 / (x_m * (z / t_0));
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_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
    if ((cosh(x_m) * (y_m / x_m)) <= 6d+307) then
        tmp = (t_0 / x_m) / z
    else
        tmp = 1.0d0 / (x_m * (z / t_0))
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = Math.cosh(x_m) * y_m;
	double tmp;
	if ((Math.cosh(x_m) * (y_m / x_m)) <= 6e+307) {
		tmp = (t_0 / x_m) / z;
	} else {
		tmp = 1.0 / (x_m * (z / t_0));
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	t_0 = math.cosh(x_m) * y_m
	tmp = 0
	if (math.cosh(x_m) * (y_m / x_m)) <= 6e+307:
		tmp = (t_0 / x_m) / z
	else:
		tmp = 1.0 / (x_m * (z / t_0))
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(cosh(x_m) * y_m)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 6e+307)
		tmp = Float64(Float64(t_0 / x_m) / z);
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(z / t_0)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = cosh(x_m) * y_m;
	tmp = 0.0;
	if ((cosh(x_m) * (y_m / x_m)) <= 6e+307)
		tmp = (t_0 / x_m) / z;
	else
		tmp = 1.0 / (x_m * (z / t_0));
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Cosh[x$95$m], $MachinePrecision] * y$95$m), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 6e+307], N[(N[(t$95$0 / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.9999999999999997e307

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 92.8

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

   4. Applied rewrites 92.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. times-frac N/A

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

      7. associate-/l/ N/A

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

      8. associate-*r/ N/A

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

      9. lift-cosh.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-cosh.f64 N/A

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

      14. associate-*r/ N/A

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

      15. associate-/l/ N/A

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

      16. times-frac N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/l* N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. clear-num N/A

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

      22. div-inv N/A

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

      23. lift-*.f64 N/A

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

   6. Applied rewrites 92.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cosh.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. *-commutative N/A

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

      14. lift-cosh.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 99.7

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

   8. Applied rewrites 99.7%

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

   if 5.9999999999999997e307 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 79.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. times-frac N/A

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

      7. associate-/l/ N/A

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

      8. associate-*r/ N/A

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

      9. lift-cosh.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-cosh.f64 N/A

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

      14. associate-*r/ N/A

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

      15. associate-/l/ N/A

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

      16. times-frac N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/l* N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. clear-num N/A

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

      22. div-inv N/A

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

      23. lift-*.f64 N/A

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

   6. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 6 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{z}{\cosh x \cdot y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\cosh x\_m \cdot y\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\cosh x\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 2e-22)
     (/ (* (cosh x_m) y_m) (* x_m z))
     (/ (* y_m (/ (cosh x_m) z)) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 2e-22) {
		tmp = (cosh(x_m) * y_m) / (x_m * z);
	} else {
		tmp = (y_m * (cosh(x_m) / z)) / x_m;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((cosh(x_m) * (y_m / x_m)) / z) <= 2d-22) then
        tmp = (cosh(x_m) * y_m) / (x_m * z)
    else
        tmp = (y_m * (cosh(x_m) / z)) / x_m
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((Math.cosh(x_m) * (y_m / x_m)) / z) <= 2e-22) {
		tmp = (Math.cosh(x_m) * y_m) / (x_m * z);
	} else {
		tmp = (y_m * (Math.cosh(x_m) / z)) / x_m;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if ((math.cosh(x_m) * (y_m / x_m)) / z) <= 2e-22:
		tmp = (math.cosh(x_m) * y_m) / (x_m * z)
	else:
		tmp = (y_m * (math.cosh(x_m) / z)) / x_m
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 2e-22)
		tmp = Float64(Float64(cosh(x_m) * y_m) / Float64(x_m * z));
	else
		tmp = Float64(Float64(y_m * Float64(cosh(x_m) / z)) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 2e-22)
		tmp = (cosh(x_m) * y_m) / (x_m * z);
	else
		tmp = (y_m * (cosh(x_m) / z)) / x_m;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e-22], N[(N[(N[Cosh[x$95$m], $MachinePrecision] * y$95$m), $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $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 (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 2.0000000000000001e-22

   1. Initial program 95.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 92.9

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

   4. Applied rewrites 92.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 87.9

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

   6. Applied rewrites 87.9%

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

   if 2.0000000000000001e-22 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 72.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 97.1% 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 2024222 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -2309451133843521/5000000000000000000000000000000000000000000000000000000000000000000) (* (/ (/ y z) x) (cosh x)) (if (< y 1038530535935153/1000000000000000000000000000000000000000000000000000000) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x)))))

  (/ (* (cosh x) (/ y x)) z))