Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 83.9% → 99.5%

Time: 10.7s

Alternatives: 21

Speedup: 2.8×

• 64.8% 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: 83.9% 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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{y\_m}{x\_m} \cdot \cosh x\_m}{z\_m} \leq 2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x\_m \cdot x\_m}{z\_m}, \mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), \frac{0.5}{z\_m}\right), x\_m \cdot x\_m, \frac{1}{z\_m}\right)}{x\_m} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \cosh x\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (/ y_m x_m) (cosh x_m)) z_m) 2e-75)
      (*
       (/
        (fma
         (fma
          (/ (* x_m x_m) z_m)
          (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
          (/ 0.5 z_m))
         (* x_m x_m)
         (/ 1.0 z_m))
        x_m)
       y_m)
      (/ (/ (* y_m (cosh x_m)) z_m) x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((((y_m / x_m) * cosh(x_m)) / z_m) <= 2e-75) {
		tmp = (fma(fma(((x_m * x_m) / z_m), fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (0.5 / z_m)), (x_m * x_m), (1.0 / z_m)) / x_m) * y_m;
	} else {
		tmp = ((y_m * cosh(x_m)) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(y_m / x_m) * cosh(x_m)) / z_m) <= 2e-75)
		tmp = Float64(Float64(fma(fma(Float64(Float64(x_m * x_m) / z_m), fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664), Float64(0.5 / z_m)), Float64(x_m * x_m), Float64(1.0 / z_m)) / x_m) * y_m);
	else
		tmp = Float64(Float64(Float64(y_m * cosh(x_m)) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 2e-75], N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + N[(0.5 / z$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(1.0 / z$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(y$95$m * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.9999999999999999e-75

   1. Initial program 94.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 97.0

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{\frac{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} \cdot \frac{{x}^{2}}{z} + \frac{1}{24} \cdot \frac{1}{z}\right) + \frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}}{x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} \cdot \frac{{x}^{2}}{z} + \frac{1}{24} \cdot \frac{1}{z}\right) + \frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}}{x}} \]

   7. Applied rewrites 86.5%

      \[\leadsto y \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x \cdot x}{z}, \mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), \frac{0.5}{z}\right), x \cdot x, \frac{1}{z}\right)}{x}} \]

   if 1.9999999999999999e-75 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 74.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x \cdot x}{z}, \mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), \frac{0.5}{z}\right), x \cdot x, \frac{1}{z}\right)}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \cosh x}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, 0.5, \frac{1}{x\_m}\right) \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x\_m}{x\_m}}{z\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+260)
      (/ (* (fma x_m 0.5 (/ 1.0 x_m)) y_m) z_m)
      (* (/ (/ (cosh x_m) x_m) z_m) y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+260) {
		tmp = (fma(x_m, 0.5, (1.0 / x_m)) * y_m) / z_m;
	} else {
		tmp = ((cosh(x_m) / x_m) / z_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+260)
		tmp = Float64(Float64(fma(x_m, 0.5, Float64(1.0 / x_m)) * y_m) / z_m);
	else
		tmp = Float64(Float64(Float64(cosh(x_m) / x_m) / z_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+260], N[(N[(N[(x$95$m * 0.5 + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000013e260

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot y} + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{1 \cdot y + \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x}}{z} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{x}}{z} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)}}{z} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \frac{y}{x} \cdot 1}}{z} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{\frac{y}{x}}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{x}} + \frac{y}{x}}{z} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x}} + \frac{y}{x}}{z} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{\color{blue}{y \cdot 1}}{x}}{z} \]

      11. associate-/l* N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \color{blue}{y \cdot \frac{1}{x}}}{z} \]

      12. distribute-lft-out N/A

          \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right)}}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

   5. Applied rewrites 84.2%

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

   if 2.00000000000000013e260 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 66.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 0.5, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x}{x}}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{y\_m}{x\_m} \cdot \cosh x\_m}{z\_m} \leq 2 \cdot 10^{-75}:\\ \;\;\;\;\frac{y\_m}{z\_m \cdot x\_m} \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (/ y_m x_m) (cosh x_m)) z_m) 2e-75)
      (* (/ y_m (* z_m x_m)) (fma (* x_m x_m) 0.5 1.0))
      (/
       (/
        (*
         (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)
         y_m)
        z_m)
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((((y_m / x_m) * cosh(x_m)) / z_m) <= 2e-75) {
		tmp = (y_m / (z_m * x_m)) * fma((x_m * x_m), 0.5, 1.0);
	} else {
		tmp = ((fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) * y_m) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(y_m / x_m) * cosh(x_m)) / z_m) <= 2e-75)
		tmp = Float64(Float64(y_m / Float64(z_m * x_m)) * fma(Float64(x_m * x_m), 0.5, 1.0));
	else
		tmp = Float64(Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * y_m) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 2e-75], N[(N[(y$95$m / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.9999999999999999e-75

   1. Initial program 94.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{x}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{y}{z}\right)} + \frac{y}{z}}{x} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{z}} + \frac{y}{z}}{x} \]

      3. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{y}{z}}}{x} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{z}}{x} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{\frac{y}{z}}{x}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{x \cdot z}} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x \cdot z} \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x \cdot z} \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x \cdot z} \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x \cdot z} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x \cdot z} \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{\color{blue}{z \cdot x}} \]

      15. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if 1.9999999999999999e-75 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 74.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{y \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}}{z}}{x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)}}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right)}{z}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)}}{z}}{x} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right)}{z}}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)}{z}}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)}{z}}{x} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)}{z}}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)}{z}}{x} \]

      9. lower-*.f64 92.6

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)}{z}}{x} \]

   7. Applied rewrites 92.6%

      \[\leadsto \frac{\frac{y \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}}{z}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 2 \cdot 10^{-75}:\\ \;\;\;\;\frac{y}{z \cdot x} \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, 0.5, \frac{1}{x\_m}\right) \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m}}{z\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+260)
      (/ (* (fma x_m 0.5 (/ 1.0 x_m)) y_m) z_m)
      (*
       (/
        (/
         (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)
         x_m)
        z_m)
       y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+260) {
		tmp = (fma(x_m, 0.5, (1.0 / x_m)) * y_m) / z_m;
	} else {
		tmp = ((fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) / z_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+260)
		tmp = Float64(Float64(fma(x_m, 0.5, Float64(1.0 / x_m)) * y_m) / z_m);
	else
		tmp = Float64(Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) / z_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+260], N[(N[(N[(x$95$m * 0.5 + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000013e260

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot y} + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{1 \cdot y + \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x}}{z} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{x}}{z} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)}}{z} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \frac{y}{x} \cdot 1}}{z} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{\frac{y}{x}}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{x}} + \frac{y}{x}}{z} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x}} + \frac{y}{x}}{z} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{\color{blue}{y \cdot 1}}{x}}{z} \]

      11. associate-/l* N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \color{blue}{y \cdot \frac{1}{x}}}{z} \]

      12. distribute-lft-out N/A

          \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right)}}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

   5. Applied rewrites 84.2%

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

   if 2.00000000000000013e260 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 66.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\frac{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}}{x}}{z} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)}}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right)}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)}{x}}{z} \]

      9. lower-*.f64 85.7

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)}{x}}{z} \]

   7. Applied rewrites 85.7%

      \[\leadsto y \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}}{x}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 0.5, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{y\_m}{x\_m} \cdot \cosh x\_m}{z\_m} \leq 2 \cdot 10^{-75}:\\ \;\;\;\;\frac{y\_m}{z\_m \cdot x\_m} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot y\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (fma (* x_m x_m) 0.5 1.0)))
   (*
    x_s
    (*
     y_s
     (*
      z_s
      (if (<= (/ (* (/ y_m x_m) (cosh x_m)) z_m) 2e-75)
        (* (/ y_m (* z_m x_m)) t_0)
        (/ (/ (* t_0 y_m) z_m) x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double t_0 = fma((x_m * x_m), 0.5, 1.0);
	double tmp;
	if ((((y_m / x_m) * cosh(x_m)) / z_m) <= 2e-75) {
		tmp = (y_m / (z_m * x_m)) * t_0;
	} else {
		tmp = ((t_0 * y_m) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	t_0 = fma(Float64(x_m * x_m), 0.5, 1.0)
	tmp = 0.0
	if (Float64(Float64(Float64(y_m / x_m) * cosh(x_m)) / z_m) <= 2e-75)
		tmp = Float64(Float64(y_m / Float64(z_m * x_m)) * t_0);
	else
		tmp = Float64(Float64(Float64(t_0 * y_m) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 2e-75], N[(N[(y$95$m / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(t$95$0 * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.9999999999999999e-75

   1. Initial program 94.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{x}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{y}{z}\right)} + \frac{y}{z}}{x} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{z}} + \frac{y}{z}}{x} \]

      3. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{y}{z}}}{x} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{z}}{x} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{\frac{y}{z}}{x}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{x \cdot z}} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x \cdot z} \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x \cdot z} \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x \cdot z} \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x \cdot z} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x \cdot z} \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{\color{blue}{z \cdot x}} \]

      15. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if 1.9999999999999999e-75 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 74.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{y \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{z}}{x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)}}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right)}{z}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)}}{z}}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right)}{z}}{x} \]

      5. lower-*.f64 83.6

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

   7. Applied rewrites 83.6%

      \[\leadsto \frac{\frac{y \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}}{z}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 2 \cdot 10^{-75}:\\ \;\;\;\;\frac{y}{z \cdot x} \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{y\_m}{x\_m} \cdot \cosh x\_m}{z\_m} \leq 10^{+291}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)}{z\_m \cdot x\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (/ y_m x_m) (cosh x_m)) z_m) 1e+291)
      (/ (/ y_m x_m) z_m)
      (* (/ (fma 0.5 (* x_m x_m) 1.0) (* z_m x_m)) y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((((y_m / x_m) * cosh(x_m)) / z_m) <= 1e+291) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (fma(0.5, (x_m * x_m), 1.0) / (z_m * x_m)) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(y_m / x_m) * cosh(x_m)) / z_m) <= 1e+291)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) / Float64(z_m * x_m)) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 1e+291], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 9.9999999999999996e290

   1. Initial program 95.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 64.6

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

   5. Applied rewrites 64.6%

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

   if 9.9999999999999996e290 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 69.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{y \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{z}}{x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)}}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right)}{z}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)}}{z}}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right)}{z}}{x} \]

      5. lower-*.f64 79.7

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

   7. Applied rewrites 79.7%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}}{x} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z \cdot x}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}}{z \cdot x} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{\color{blue}{z \cdot x}} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z \cdot x}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z \cdot x}} \]

      8. lower-/.f64 58.8

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

   9. Applied rewrites 58.8%

      \[\leadsto \color{blue}{y \cdot \frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 10^{+291}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z \cdot x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{y\_m}{x\_m} \cdot \cosh x\_m}{z\_m} \leq 10^{+291}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z\_m \cdot x\_m} \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (/ y_m x_m) (cosh x_m)) z_m) 1e+291)
      (/ (/ y_m x_m) z_m)
      (* (/ y_m (* z_m x_m)) (fma (* x_m x_m) 0.5 1.0)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((((y_m / x_m) * cosh(x_m)) / z_m) <= 1e+291) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (y_m / (z_m * x_m)) * fma((x_m * x_m), 0.5, 1.0);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(y_m / x_m) * cosh(x_m)) / z_m) <= 1e+291)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(Float64(y_m / Float64(z_m * x_m)) * fma(Float64(x_m * x_m), 0.5, 1.0));
	end
	return Float64(x_s * Float64(y_s * Float64(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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 1e+291], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y$95$m / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 9.9999999999999996e290

   1. Initial program 95.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 64.6

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

   5. Applied rewrites 64.6%

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

   if 9.9999999999999996e290 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 69.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{x}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{y}{z}\right)} + \frac{y}{z}}{x} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{z}} + \frac{y}{z}}{x} \]

      3. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{y}{z}}}{x} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{z}}{x} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{\frac{y}{z}}{x}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{x \cdot z}} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x \cdot z} \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x \cdot z} \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x \cdot z} \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x \cdot z} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x \cdot z} \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{\color{blue}{z \cdot x}} \]

      15. lower-*.f64 47.5

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

   5. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \frac{y}{z \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 10^{+291}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x} \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, 0.5, \frac{1}{x\_m}\right) \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{x\_m}}{z\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+260)
      (/ (* (fma x_m 0.5 (/ 1.0 x_m)) y_m) z_m)
      (* (/ (/ (fma (* x_m x_m) 0.5 1.0) x_m) z_m) y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+260) {
		tmp = (fma(x_m, 0.5, (1.0 / x_m)) * y_m) / z_m;
	} else {
		tmp = ((fma((x_m * x_m), 0.5, 1.0) / x_m) / z_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+260)
		tmp = Float64(Float64(fma(x_m, 0.5, Float64(1.0 / x_m)) * y_m) / z_m);
	else
		tmp = Float64(Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / x_m) / z_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+260], N[(N[(N[(x$95$m * 0.5 + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000013e260

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot y} + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{1 \cdot y + \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x}}{z} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{x}}{z} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)}}{z} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \frac{y}{x} \cdot 1}}{z} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{\frac{y}{x}}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{x}} + \frac{y}{x}}{z} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x}} + \frac{y}{x}}{z} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{\color{blue}{y \cdot 1}}{x}}{z} \]

      11. associate-/l* N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \color{blue}{y \cdot \frac{1}{x}}}{z} \]

      12. distribute-lft-out N/A

          \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right)}}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

   5. Applied rewrites 84.2%

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

   if 2.00000000000000013e260 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 66.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\frac{\color{blue}{1 + \frac{1}{2} \cdot {x}^{2}}}{x}}{z} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\frac{1}{2} \cdot {x}^{2} + 1}}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)}}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right)}{x}}{z} \]

      5. lower-*.f64 71.2

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

   7. Applied rewrites 71.2%

      \[\leadsto y \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}}{x}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 0.5, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{x}}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 1.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, 0.5, \frac{1}{x\_m}\right) \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)}{z\_m \cdot x\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (* (/ y_m x_m) (cosh x_m)) 1.5e+181)
      (/ (* (fma x_m 0.5 (/ 1.0 x_m)) y_m) z_m)
      (* (/ (fma 0.5 (* x_m x_m) 1.0) (* z_m x_m)) y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 1.5e+181) {
		tmp = (fma(x_m, 0.5, (1.0 / x_m)) * y_m) / z_m;
	} else {
		tmp = (fma(0.5, (x_m * x_m), 1.0) / (z_m * x_m)) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 1.5e+181)
		tmp = Float64(Float64(fma(x_m, 0.5, Float64(1.0 / x_m)) * y_m) / z_m);
	else
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) / Float64(z_m * x_m)) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 1.5e+181], N[(N[(N[(x$95$m * 0.5 + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.50000000000000006e181

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot y} + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{1 \cdot y + \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x}}{z} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{x}}{z} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)}}{z} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \frac{y}{x} \cdot 1}}{z} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{\frac{y}{x}}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{x}} + \frac{y}{x}}{z} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x}} + \frac{y}{x}}{z} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{\color{blue}{y \cdot 1}}{x}}{z} \]

      11. associate-/l* N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \color{blue}{y \cdot \frac{1}{x}}}{z} \]

      12. distribute-lft-out N/A

          \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right)}}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

   5. Applied rewrites 83.2%

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

   if 1.50000000000000006e181 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 69.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{y \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{z}}{x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)}}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right)}{z}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)}}{z}}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right)}{z}}{x} \]

      5. lower-*.f64 73.7

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

   7. Applied rewrites 73.7%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}}{x} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z \cdot x}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}}{z \cdot x} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{\color{blue}{z \cdot x}} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z \cdot x}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z \cdot x}} \]

      8. lower-/.f64 53.8

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

   9. Applied rewrites 53.8%

      \[\leadsto \color{blue}{y \cdot \frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 1.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 0.5, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z \cdot x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.6 \cdot 10^{+60}:\\ \;\;\;\;\frac{y\_m \cdot \cosh x\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot 0.001388888888888889\right) \cdot y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 1.6e+60)
      (/ (* y_m (cosh x_m)) (* z_m x_m))
      (/
       (* (* (* (* (* (* x_m x_m) x_m) x_m) x_m) 0.001388888888888889) y_m)
       z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.6e+60) {
		tmp = (y_m * cosh(x_m)) / (z_m * x_m);
	} else {
		tmp = ((((((x_m * x_m) * x_m) * x_m) * x_m) * 0.001388888888888889) * y_m) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
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, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 1.6d+60) then
        tmp = (y_m * cosh(x_m)) / (z_m * x_m)
    else
        tmp = ((((((x_m * x_m) * x_m) * x_m) * x_m) * 0.001388888888888889d0) * y_m) / z_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.6e+60) {
		tmp = (y_m * Math.cosh(x_m)) / (z_m * x_m);
	} else {
		tmp = ((((((x_m * x_m) * x_m) * x_m) * x_m) * 0.001388888888888889) * y_m) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
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, z_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 1.6e+60:
		tmp = (y_m * math.cosh(x_m)) / (z_m * x_m)
	else:
		tmp = ((((((x_m * x_m) * x_m) * x_m) * x_m) * 0.001388888888888889) * y_m) / z_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 1.6e+60)
		tmp = Float64(Float64(y_m * cosh(x_m)) / Float64(z_m * x_m));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(x_m * x_m) * x_m) * x_m) * x_m) * 0.001388888888888889) * y_m) / z_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
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, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 1.6e+60)
		tmp = (y_m * cosh(x_m)) / (z_m * x_m);
	else
		tmp = ((((((x_m * x_m) * x_m) * x_m) * x_m) * 0.001388888888888889) * y_m) / z_m;
	end
	tmp_2 = x_s * (y_s * (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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 1.6e+60], N[(N[(y$95$m * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.59999999999999995e60

   1. Initial program 88.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 95.0

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

   4. Applied rewrites 95.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 85.1

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 85.1

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

   6. Applied rewrites 85.1%

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

   if 1.59999999999999995e60 < x

   1. Initial program 75.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}}}{z} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x} \cdot y}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{1}{720} \cdot {x}^{5}\right) \cdot y}{z} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \frac{\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.001388888888888889\right) \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+60}:\\ \;\;\;\;\frac{y \cdot \cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.001388888888888889\right) \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.6 \cdot 10^{+60}:\\ \;\;\;\;\frac{\cosh x\_m}{z\_m \cdot x\_m} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot 0.001388888888888889\right) \cdot y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 1.6e+60)
      (* (/ (cosh x_m) (* z_m x_m)) y_m)
      (/
       (* (* (* (* (* (* x_m x_m) x_m) x_m) x_m) 0.001388888888888889) y_m)
       z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.6e+60) {
		tmp = (cosh(x_m) / (z_m * x_m)) * y_m;
	} else {
		tmp = ((((((x_m * x_m) * x_m) * x_m) * x_m) * 0.001388888888888889) * y_m) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
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, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 1.6d+60) then
        tmp = (cosh(x_m) / (z_m * x_m)) * y_m
    else
        tmp = ((((((x_m * x_m) * x_m) * x_m) * x_m) * 0.001388888888888889d0) * y_m) / z_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.6e+60) {
		tmp = (Math.cosh(x_m) / (z_m * x_m)) * y_m;
	} else {
		tmp = ((((((x_m * x_m) * x_m) * x_m) * x_m) * 0.001388888888888889) * y_m) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
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, z_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 1.6e+60:
		tmp = (math.cosh(x_m) / (z_m * x_m)) * y_m
	else:
		tmp = ((((((x_m * x_m) * x_m) * x_m) * x_m) * 0.001388888888888889) * y_m) / z_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 1.6e+60)
		tmp = Float64(Float64(cosh(x_m) / Float64(z_m * x_m)) * y_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(x_m * x_m) * x_m) * x_m) * x_m) * 0.001388888888888889) * y_m) / z_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
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, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 1.6e+60)
		tmp = (cosh(x_m) / (z_m * x_m)) * y_m;
	else
		tmp = ((((((x_m * x_m) * x_m) * x_m) * x_m) * 0.001388888888888889) * y_m) / z_m;
	end
	tmp_2 = x_s * (y_s * (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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 1.6e+60], N[(N[(N[Cosh[x$95$m], $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.59999999999999995e60

   1. Initial program 88.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 84.3

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

   4. Applied rewrites 84.3%

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

   if 1.59999999999999995e60 < x

   1. Initial program 75.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}}}{z} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x} \cdot y}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{1}{720} \cdot {x}^{5}\right) \cdot y}{z} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \frac{\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.001388888888888889\right) \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+60}:\\ \;\;\;\;\frac{\cosh x}{z \cdot x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.001388888888888889\right) \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 94.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 0.1:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, \mathsf{fma}\left(\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right) \cdot x\_m\right) \cdot x\_m, x\_m, 0.5 \cdot x\_m\right), 1\right)}{\frac{z\_m}{y\_m}} \cdot \frac{1}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= y_m 0.1)
      (/
       (*
        (/
         (fma
          (fma
           (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
           (* x_m x_m)
           0.5)
          (* x_m x_m)
          1.0)
         x_m)
        y_m)
       z_m)
      (*
       (/
        (fma
         x_m
         (fma
          (*
           (* (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664) x_m)
           x_m)
          x_m
          (* 0.5 x_m))
         1.0)
        (/ z_m y_m))
       (/ 1.0 x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 0.1) {
		tmp = ((fma(fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y_m) / z_m;
	} else {
		tmp = (fma(x_m, fma(((fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664) * x_m) * x_m), x_m, (0.5 * x_m)), 1.0) / (z_m / y_m)) * (1.0 / x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 0.1)
		tmp = Float64(Float64(Float64(fma(fma(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) * y_m) / z_m);
	else
		tmp = Float64(Float64(fma(x_m, fma(Float64(Float64(fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664) * x_m) * x_m), x_m, Float64(0.5 * x_m)), 1.0) / Float64(z_m / y_m)) * Float64(1.0 / x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[y$95$m, 0.1], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(x$95$m * N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + N[(0.5 * x$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 0.10000000000000001

   1. Initial program 82.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}}}{z} \]
   4. Step-by-step derivation

   5. Applied rewrites 89.8%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}}{z} \]

   if 0.10000000000000001 < y

   1. Initial program 94.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{y}{z} + {x}^{2} \cdot \left(\frac{1}{720} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{24} \cdot \frac{y}{z}\right)\right) + \frac{y}{z}}{x}} \]

   5. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \frac{y}{z \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x, \left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.5, x \cdot x, 1\right)\right) \cdot \frac{\color{blue}{y}}{z \cdot x} \]
   8. Step-by-step derivation

   9. Applied rewrites 95.5%

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x\right) \cdot x, x, 0.5 \cdot x\right), 1\right)}{\frac{z}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.1:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x\right) \cdot x, x, 0.5 \cdot x\right), 1\right)}{\frac{z}{y}} \cdot \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 94.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0, x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0 \cdot x\_m, x\_m, 1\right)}{x\_m}}{z\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0
         (fma
          (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
          (* x_m x_m)
          0.5)))
   (*
    x_s
    (*
     y_s
     (*
      z_s
      (if (<= z_m 8e+80)
        (/ (* (/ (fma t_0 (* x_m x_m) 1.0) x_m) y_m) z_m)
        (* (/ (/ (fma (* t_0 x_m) x_m 1.0) x_m) z_m) y_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double t_0 = fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5);
	double tmp;
	if (z_m <= 8e+80) {
		tmp = ((fma(t_0, (x_m * x_m), 1.0) / x_m) * y_m) / z_m;
	} else {
		tmp = ((fma((t_0 * x_m), x_m, 1.0) / x_m) / z_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	t_0 = fma(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664), Float64(x_m * x_m), 0.5)
	tmp = 0.0
	if (z_m <= 8e+80)
		tmp = Float64(Float64(Float64(fma(t_0, Float64(x_m * x_m), 1.0) / x_m) * y_m) / z_m);
	else
		tmp = Float64(Float64(Float64(fma(Float64(t_0 * x_m), x_m, 1.0) / x_m) / z_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 8e+80], N[(N[(N[(N[(t$95$0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[(N[(t$95$0 * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8e80

   1. Initial program 88.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}}}{z} \]
   4. Step-by-step derivation

   5. Applied rewrites 90.7%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}}{z} \]

   if 8e80 < z

   1. Initial program 75.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\frac{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}}{x}}{z} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)}}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right)}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)}{x}}{z} \]

      9. lower-*.f64 87.6

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)}{x}}{z} \]

   7. Applied rewrites 87.6%

      \[\leadsto y \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}}{x}}{z} \]

   8. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\frac{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}}{x}}{z} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1}}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} + 1}{x}}{z} \]

      4. associate-*r* N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right) \cdot x} + 1}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x, x, 1\right)}}{x}}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x}, x, 1\right)}{x}}{z} \]

      7. +-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}\right)} \cdot x, x, 1\right)}{x}}{z} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}\right) \cdot x, x, 1\right)}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)} \cdot x, x, 1\right)}{x}}{z} \]

      10. +-commutative N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right) \cdot x, x, 1\right)}{x}}{z} \]

      11. lower-fma.f64 N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right) \cdot x, x, 1\right)}{x}}{z} \]

      12. unpow2 N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right) \cdot x, x, 1\right)}{x}}{z} \]

      13. lower-*.f64 N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right) \cdot x, x, 1\right)}{x}}{z} \]

      14. unpow2 N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right) \cdot x, x, 1\right)}{x}}{z} \]

      15. lower-*.f64 93.7

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), \color{blue}{x \cdot x}, 0.5\right) \cdot x, x, 1\right)}{x}}{z} \]

   10. Applied rewrites 93.7%

       \[\leadsto y \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right)}}{x}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right)}{x}}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 94.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right) \cdot x\_m, x\_m, 1\right)}{x\_m}}{z\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 8e+80)
      (/
       (*
        (/
         (fma
          (fma (* 0.001388888888888889 (* x_m x_m)) (* x_m x_m) 0.5)
          (* x_m x_m)
          1.0)
         x_m)
        y_m)
       z_m)
      (*
       (/
        (/
         (fma
          (*
           (fma
            (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
            (* x_m x_m)
            0.5)
           x_m)
          x_m
          1.0)
         x_m)
        z_m)
       y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 8e+80) {
		tmp = ((fma(fma((0.001388888888888889 * (x_m * x_m)), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y_m) / z_m;
	} else {
		tmp = ((fma((fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5) * x_m), x_m, 1.0) / x_m) / z_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 8e+80)
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.001388888888888889 * Float64(x_m * x_m)), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) * y_m) / z_m);
	else
		tmp = Float64(Float64(Float64(fma(Float64(fma(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664), Float64(x_m * x_m), 0.5) * x_m), x_m, 1.0) / x_m) / z_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 8e+80], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8e80

   1. Initial program 88.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}}}{z} \]
   4. Step-by-step derivation

   5. Applied rewrites 90.7%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x} \cdot y}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.7%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z} \]

   if 8e80 < z

   1. Initial program 75.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\frac{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}}{x}}{z} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)}}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right)}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)}{x}}{z} \]

      9. lower-*.f64 87.6

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)}{x}}{z} \]

   7. Applied rewrites 87.6%

      \[\leadsto y \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}}{x}}{z} \]

   8. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\frac{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}}{x}}{z} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1}}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} + 1}{x}}{z} \]

      4. associate-*r* N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right) \cdot x} + 1}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x, x, 1\right)}}{x}}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x}, x, 1\right)}{x}}{z} \]

      7. +-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}\right)} \cdot x, x, 1\right)}{x}}{z} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}\right) \cdot x, x, 1\right)}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)} \cdot x, x, 1\right)}{x}}{z} \]

      10. +-commutative N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right) \cdot x, x, 1\right)}{x}}{z} \]

      11. lower-fma.f64 N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right) \cdot x, x, 1\right)}{x}}{z} \]

      12. unpow2 N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right) \cdot x, x, 1\right)}{x}}{z} \]

      13. lower-*.f64 N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right) \cdot x, x, 1\right)}{x}}{z} \]

      14. unpow2 N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right) \cdot x, x, 1\right)}{x}}{z} \]

      15. lower-*.f64 93.7

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), \color{blue}{x \cdot x}, 0.5\right) \cdot x, x, 1\right)}{x}}{z} \]

   10. Applied rewrites 93.7%

       \[\leadsto y \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right)}}{x}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right)}{x}}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 94.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= y_m 2e+47)
      (/
       (*
        (/
         (fma
          (fma (* 0.001388888888888889 (* x_m x_m)) (* x_m x_m) 0.5)
          (* x_m x_m)
          1.0)
         x_m)
        y_m)
       z_m)
      (/
       (/
        (*
         (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)
         y_m)
        z_m)
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 2e+47) {
		tmp = ((fma(fma((0.001388888888888889 * (x_m * x_m)), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y_m) / z_m;
	} else {
		tmp = ((fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) * y_m) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 2e+47)
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.001388888888888889 * Float64(x_m * x_m)), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) * y_m) / z_m);
	else
		tmp = Float64(Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * y_m) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[y$95$m, 2e+47], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.0000000000000001e47

   1. Initial program 83.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}}}{z} \]
   4. Step-by-step derivation

   5. Applied rewrites 90.3%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x} \cdot y}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.3%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z} \]

   if 2.0000000000000001e47 < y

   1. Initial program 93.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{y \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}}{z}}{x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)}}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right)}{z}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)}}{z}}{x} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right)}{z}}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)}{z}}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)}{z}}{x} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)}{z}}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)}{z}}{x} \]

      9. lower-*.f64 93.0

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)}{z}}{x} \]

   7. Applied rewrites 93.0%

      \[\leadsto \frac{\frac{y \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}}{z}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 87.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot y\_m}{z\_m}}{x\_m}\\ \mathbf{elif}\;x\_m \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{y\_m}{z\_m} \cdot 0.001388888888888889\right) \cdot \left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{x\_m}}{z\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (fma (* x_m x_m) 0.5 1.0)))
   (*
    x_s
    (*
     y_s
     (*
      z_s
      (if (<= x_m 4e+60)
        (/ (/ (* t_0 y_m) z_m) x_m)
        (if (<= x_m 1.4e+154)
          (*
           (* (/ y_m z_m) 0.001388888888888889)
           (* (* (* (* x_m x_m) x_m) x_m) x_m))
          (* (/ (/ t_0 x_m) z_m) y_m))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double t_0 = fma((x_m * x_m), 0.5, 1.0);
	double tmp;
	if (x_m <= 4e+60) {
		tmp = ((t_0 * y_m) / z_m) / x_m;
	} else if (x_m <= 1.4e+154) {
		tmp = ((y_m / z_m) * 0.001388888888888889) * ((((x_m * x_m) * x_m) * x_m) * x_m);
	} else {
		tmp = ((t_0 / x_m) / z_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	t_0 = fma(Float64(x_m * x_m), 0.5, 1.0)
	tmp = 0.0
	if (x_m <= 4e+60)
		tmp = Float64(Float64(Float64(t_0 * y_m) / z_m) / x_m);
	elseif (x_m <= 1.4e+154)
		tmp = Float64(Float64(Float64(y_m / z_m) * 0.001388888888888889) * Float64(Float64(Float64(Float64(x_m * x_m) * x_m) * x_m) * x_m));
	else
		tmp = Float64(Float64(Float64(t_0 / x_m) / z_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 4e+60], N[(N[(N[(t$95$0 * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], If[LessEqual[x$95$m, 1.4e+154], N[(N[(N[(y$95$m / z$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x < 3.9999999999999998e60

   1. Initial program 88.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{y \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{z}}{x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)}}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right)}{z}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)}}{z}}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right)}{z}}{x} \]

      5. lower-*.f64 80.1

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

   7. Applied rewrites 80.1%

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

   if 3.9999999999999998e60 < x < 1.4e154

   1. Initial program 89.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{y}{z} + {x}^{2} \cdot \left(\frac{1}{720} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{24} \cdot \frac{y}{z}\right)\right) + \frac{y}{z}}{x}} \]

   5. Applied rewrites 68.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \frac{y}{z \cdot x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{720} \cdot \color{blue}{\frac{{x}^{5} \cdot y}{z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 89.5%

      \[\leadsto \left(\frac{y}{z} \cdot 0.001388888888888889\right) \cdot \color{blue}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)} \]

   if 1.4e154 < x

   1. Initial program 68.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\frac{\color{blue}{1 + \frac{1}{2} \cdot {x}^{2}}}{x}}{z} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\frac{1}{2} \cdot {x}^{2} + 1}}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)}}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right)}{x}}{z} \]

      5. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

      \[\leadsto y \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}}{x}}{z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y}{z}}{x}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{y}{z} \cdot 0.001388888888888889\right) \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{x}}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 89.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000118:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, 0.5, \frac{1}{x\_m}\right) \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot 0.001388888888888889\right) \cdot y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 0.000118)
      (/ (* (fma x_m 0.5 (/ 1.0 x_m)) y_m) z_m)
      (/
       (* (* (* (* (* (* x_m x_m) x_m) x_m) x_m) 0.001388888888888889) y_m)
       z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.000118) {
		tmp = (fma(x_m, 0.5, (1.0 / x_m)) * y_m) / z_m;
	} else {
		tmp = ((((((x_m * x_m) * x_m) * x_m) * x_m) * 0.001388888888888889) * y_m) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.000118)
		tmp = Float64(Float64(fma(x_m, 0.5, Float64(1.0 / x_m)) * y_m) / z_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(x_m * x_m) * x_m) * x_m) * x_m) * 0.001388888888888889) * y_m) / z_m);
	end
	return Float64(x_s * Float64(y_s * Float64(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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 0.000118], N[(N[(N[(x$95$m * 0.5 + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.18e-4

   1. Initial program 87.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot y} + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{1 \cdot y + \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x}}{z} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{x}}{z} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)}}{z} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \frac{y}{x} \cdot 1}}{z} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{\frac{y}{x}}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{x}} + \frac{y}{x}}{z} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x}} + \frac{y}{x}}{z} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{\color{blue}{y \cdot 1}}{x}}{z} \]

      11. associate-/l* N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \color{blue}{y \cdot \frac{1}{x}}}{z} \]

      12. distribute-lft-out N/A

          \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right)}}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

   5. Applied rewrites 73.8%

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

   if 1.18e-4 < x

   1. Initial program 80.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}}}{z} \]
   4. Step-by-step derivation

   5. Applied rewrites 88.7%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x} \cdot y}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 88.7%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{1}{720} \cdot {x}^{5}\right) \cdot y}{z} \]
   10. Step-by-step derivation

   11. Applied rewrites 87.3%

       \[\leadsto \frac{\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.001388888888888889\right) \cdot y}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 84.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 9.8 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)}{z\_m} \cdot y\_m}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{x\_m}}{z\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 9.8e+47)
      (/ (* (/ (fma 0.5 (* x_m x_m) 1.0) z_m) y_m) x_m)
      (* (/ (/ (fma (* x_m x_m) 0.5 1.0) x_m) z_m) y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 9.8e+47) {
		tmp = ((fma(0.5, (x_m * x_m), 1.0) / z_m) * y_m) / x_m;
	} else {
		tmp = ((fma((x_m * x_m), 0.5, 1.0) / x_m) / z_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 9.8e+47)
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) / z_m) * y_m) / x_m);
	else
		tmp = Float64(Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / x_m) / z_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 9.8e+47], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 9.8000000000000006e47

   1. Initial program 88.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{y \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{z}}{x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)}}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right)}{z}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)}}{z}}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right)}{z}}{x} \]

      5. lower-*.f64 82.2

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

   7. Applied rewrites 82.2%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}}{z}}{x} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z} \cdot y}}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z} \cdot y}}{x} \]

      6. lower-/.f64 84.1

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

   9. Applied rewrites 84.1%

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

   if 9.8000000000000006e47 < z

   1. Initial program 77.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\frac{\color{blue}{1 + \frac{1}{2} \cdot {x}^{2}}}{x}}{z} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\frac{1}{2} \cdot {x}^{2} + 1}}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)}}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right)}{x}}{z} \]

      5. lower-*.f64 75.9

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

   7. Applied rewrites 75.9%

      \[\leadsto y \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}}{x}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.8 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z} \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{x}}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 65.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000118:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\_m\right) \cdot y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 0.000118) (/ (/ y_m x_m) z_m) (/ (* (* 0.5 x_m) y_m) z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.000118) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = ((0.5 * x_m) * y_m) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
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, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.000118d0) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = ((0.5d0 * x_m) * y_m) / z_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.000118) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = ((0.5 * x_m) * y_m) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
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, z_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.000118:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = ((0.5 * x_m) * y_m) / z_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.000118)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(Float64(Float64(0.5 * x_m) * y_m) / z_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
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, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.000118)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = ((0.5 * x_m) * y_m) / z_m;
	end
	tmp_2 = x_s * (y_s * (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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 0.000118], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.18e-4

   1. Initial program 87.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 62.1

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

   5. Applied rewrites 62.1%

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

   if 1.18e-4 < x

   1. Initial program 80.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot y} + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{1 \cdot y + \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x}}{z} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{x}}{z} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)}}{z} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \frac{y}{x} \cdot 1}}{z} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{\frac{y}{x}}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{x}} + \frac{y}{x}}{z} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x}} + \frac{y}{x}}{z} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{\color{blue}{y \cdot 1}}{x}}{z} \]

      11. associate-/l* N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \color{blue}{y \cdot \frac{1}{x}}}{z} \]

      12. distribute-lft-out N/A

          \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right)}}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

   5. Applied rewrites 41.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{1}{2} \cdot x\right) \cdot y}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.8%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000118:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\right) \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 65.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000118:\\ \;\;\;\;\frac{y\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\_m\right) \cdot y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
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 z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 0.000118) (/ y_m (* z_m x_m)) (/ (* (* 0.5 x_m) y_m) z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.000118) {
		tmp = y_m / (z_m * x_m);
	} else {
		tmp = ((0.5 * x_m) * y_m) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
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, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.000118d0) then
        tmp = y_m / (z_m * x_m)
    else
        tmp = ((0.5d0 * x_m) * y_m) / z_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.000118) {
		tmp = y_m / (z_m * x_m);
	} else {
		tmp = ((0.5 * x_m) * y_m) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
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, z_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.000118:
		tmp = y_m / (z_m * x_m)
	else:
		tmp = ((0.5 * x_m) * y_m) / z_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.000118)
		tmp = Float64(y_m / Float64(z_m * x_m));
	else
		tmp = Float64(Float64(Float64(0.5 * x_m) * y_m) / z_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
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, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.000118)
		tmp = y_m / (z_m * x_m);
	else
		tmp = ((0.5 * x_m) * y_m) / z_m;
	end
	tmp_2 = x_s * (y_s * (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]
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_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 0.000118], N[(y$95$m / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.18e-4

   1. Initial program 87.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.9

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

   5. Applied rewrites 60.9%

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

   if 1.18e-4 < x

   1. Initial program 80.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot y} + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{1 \cdot y + \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x}}{z} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{x}}{z} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)}}{z} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \frac{y}{x} \cdot 1}}{z} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{\frac{y}{x}}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{x}} + \frac{y}{x}}{z} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x}} + \frac{y}{x}}{z} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{\color{blue}{y \cdot 1}}{x}}{z} \]

      11. associate-/l* N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \color{blue}{y \cdot \frac{1}{x}}}{z} \]

      12. distribute-lft-out N/A

          \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right)}}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

   5. Applied rewrites 41.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{1}{2} \cdot x\right) \cdot y}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.8%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000118:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\right) \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 49.2% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
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 z_s, double x_m, double y_m, double z_m) {
	return x_s * (y_s * (z_s * (y_m / (z_m * x_m))));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
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, z_s, x_m, y_m, z_m):
	return x_s * (y_s * (z_s * (y_m / (z_m * x_m))))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
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, z_s, x_m, y_m, z_m)
	return Float64(x_s * Float64(y_s * Float64(z_s * Float64(y_m / Float64(z_m * x_m)))))
end

MATLAB

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

Derivation

1. Initial program 85.6%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 46.9

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

5. Applied rewrites 46.9%

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

Developer Target 1: 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 2024240 
(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))