Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.8% → 99.3%

Time: 13.4s

Alternatives: 26

Speedup: 2.3×

• 64.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.8% 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: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), y\_m\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\cosh x\_m}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 2e+292)
      (/
       (/
        (fma
         x_m
         (*
          x_m
          (*
           y_m
           (fma
            (* x_m x_m)
            (fma x_m (* x_m 0.001388888888888889) 0.041666666666666664)
            0.5)))
         y_m)
        x_m)
       z_m)
      (* y_m (/ (/ (cosh x_m) x_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 (((cosh(x_m) * (y_m / x_m)) / z_m) <= 2e+292) {
		tmp = (fma(x_m, (x_m * (y_m * fma((x_m * x_m), fma(x_m, (x_m * 0.001388888888888889), 0.041666666666666664), 0.5))), y_m) / x_m) / z_m;
	} else {
		tmp = y_m * ((cosh(x_m) / x_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 (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 2e+292)
		tmp = Float64(Float64(fma(x_m, Float64(x_m * Float64(y_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.001388888888888889), 0.041666666666666664), 0.5))), y_m) / x_m) / z_m);
	else
		tmp = Float64(y_m * Float64(Float64(cosh(x_m) / x_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[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 2e+292], N[(N[(N[(x$95$m * N[(x$95$m * N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * N[(N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 2e292

   1. Initial program 97.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 91.2%

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

   if 2e292 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 71.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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%

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

Alternative 2: 94.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, 0.041666666666666664\right)\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, t\_0, 0.5\right)\right), y\_m\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot t\_0, 0.5\right), 1\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

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 0.001388888888888889) 0.041666666666666664)))
   (*
    x_s
    (*
     y_s
     (*
      z_s
      (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 2e+292)
        (/ (/ (fma x_m (* x_m (* y_m (fma (* x_m x_m) t_0 0.5))) y_m) x_m) z_m)
        (*
         y_m
         (/ (/ (fma x_m (* x_m (fma x_m (* x_m t_0) 0.5)) 1.0) x_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 t_0 = fma(x_m, (x_m * 0.001388888888888889), 0.041666666666666664);
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 2e+292) {
		tmp = (fma(x_m, (x_m * (y_m * fma((x_m * x_m), t_0, 0.5))), y_m) / x_m) / z_m;
	} else {
		tmp = y_m * ((fma(x_m, (x_m * fma(x_m, (x_m * t_0), 0.5)), 1.0) / x_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)
	t_0 = fma(x_m, Float64(x_m * 0.001388888888888889), 0.041666666666666664)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 2e+292)
		tmp = Float64(Float64(fma(x_m, Float64(x_m * Float64(y_m * fma(Float64(x_m * x_m), t_0, 0.5))), y_m) / x_m) / z_m);
	else
		tmp = Float64(y_m * Float64(Float64(fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * t_0), 0.5)), 1.0) / x_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_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 2e+292], N[(N[(N[(x$95$m * N[(x$95$m * N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * N[(N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * t$95$0), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 2e292

   1. Initial program 97.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 91.2%

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

   if 2e292 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 71.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 93.1%

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

Alternative 3: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right)\right), y\_m\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 2e+292)
      (/
       (/
        (fma
         x_m
         (* x_m (* y_m (fma x_m (* x_m 0.041666666666666664) 0.5)))
         y_m)
        x_m)
       z_m)
      (*
       y_m
       (/
        (/
         (fma
          x_m
          (*
           x_m
           (fma
            x_m
            (* x_m (fma x_m (* x_m 0.001388888888888889) 0.041666666666666664))
            0.5))
          1.0)
         x_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 (((cosh(x_m) * (y_m / x_m)) / z_m) <= 2e+292) {
		tmp = (fma(x_m, (x_m * (y_m * fma(x_m, (x_m * 0.041666666666666664), 0.5))), y_m) / x_m) / z_m;
	} else {
		tmp = y_m * ((fma(x_m, (x_m * fma(x_m, (x_m * fma(x_m, (x_m * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0) / x_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 (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 2e+292)
		tmp = Float64(Float64(fma(x_m, Float64(x_m * Float64(y_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5))), y_m) / x_m) / z_m);
	else
		tmp = Float64(y_m * Float64(Float64(fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0) / x_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[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 2e+292], N[(N[(N[(x$95$m * N[(x$95$m * N[(y$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * N[(N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 2e292

   1. Initial program 97.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 89.1%

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

   if 2e292 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 71.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 93.1%

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

Alternative 4: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right)\right), y\_m\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889\right), 0.5\right), 1\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 2e+292)
      (/
       (/
        (fma
         x_m
         (* x_m (* y_m (fma x_m (* x_m 0.041666666666666664) 0.5)))
         y_m)
        x_m)
       z_m)
      (*
       y_m
       (/
        (/
         (fma
          x_m
          (* x_m (fma x_m (* x_m (* (* x_m x_m) 0.001388888888888889)) 0.5))
          1.0)
         x_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 (((cosh(x_m) * (y_m / x_m)) / z_m) <= 2e+292) {
		tmp = (fma(x_m, (x_m * (y_m * fma(x_m, (x_m * 0.041666666666666664), 0.5))), y_m) / x_m) / z_m;
	} else {
		tmp = y_m * ((fma(x_m, (x_m * fma(x_m, (x_m * ((x_m * x_m) * 0.001388888888888889)), 0.5)), 1.0) / x_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 (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 2e+292)
		tmp = Float64(Float64(fma(x_m, Float64(x_m * Float64(y_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5))), y_m) / x_m) / z_m);
	else
		tmp = Float64(y_m * Float64(Float64(fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * Float64(Float64(x_m * x_m) * 0.001388888888888889)), 0.5)), 1.0) / x_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[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 2e+292], N[(N[(N[(x$95$m * N[(x$95$m * N[(y$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * N[(N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 2e292

   1. Initial program 97.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 89.1%

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

   if 2e292 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 71.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \frac{1}{720} \cdot \color{blue}{{x}^{3}}, \frac{1}{2}\right), 1\right)}{x}}{z} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.1%

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

Alternative 5: 91.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right)\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot t\_0\right), y\_m\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot t\_0, 1\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

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 0.041666666666666664) 0.5)))
   (*
    x_s
    (*
     y_s
     (*
      z_s
      (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 2e+292)
        (/ (/ (fma x_m (* x_m (* y_m t_0)) y_m) x_m) z_m)
        (* y_m (/ (/ (fma x_m (* x_m t_0) 1.0) x_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 t_0 = fma(x_m, (x_m * 0.041666666666666664), 0.5);
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 2e+292) {
		tmp = (fma(x_m, (x_m * (y_m * t_0)), y_m) / x_m) / z_m;
	} else {
		tmp = y_m * ((fma(x_m, (x_m * t_0), 1.0) / x_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)
	t_0 = fma(x_m, Float64(x_m * 0.041666666666666664), 0.5)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 2e+292)
		tmp = Float64(Float64(fma(x_m, Float64(x_m * Float64(y_m * t_0)), y_m) / x_m) / z_m);
	else
		tmp = Float64(y_m * Float64(Float64(fma(x_m, Float64(x_m * t_0), 1.0) / x_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_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 2e+292], N[(N[(N[(x$95$m * N[(x$95$m * N[(y$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * N[(N[(N[(x$95$m * N[(x$95$m * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 2e292

   1. Initial program 97.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 89.1%

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

   if 2e292 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 71.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 87.1%

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

Alternative 6: 91.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, x\_m \cdot 0.5, \frac{y\_m}{x\_m}\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= (* (cosh x_m) (/ y_m x_m)) 2e+144)
      (/ (fma y_m (* x_m 0.5) (/ y_m x_m)) z_m)
      (*
       y_m
       (/
        (/
         (fma x_m (* x_m (fma x_m (* x_m 0.041666666666666664) 0.5)) 1.0)
         x_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 ((cosh(x_m) * (y_m / x_m)) <= 2e+144) {
		tmp = fma(y_m, (x_m * 0.5), (y_m / x_m)) / z_m;
	} else {
		tmp = y_m * ((fma(x_m, (x_m * fma(x_m, (x_m * 0.041666666666666664), 0.5)), 1.0) / x_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 (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 2e+144)
		tmp = Float64(fma(y_m, Float64(x_m * 0.5), Float64(y_m / x_m)) / z_m);
	else
		tmp = Float64(y_m * Float64(Float64(fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5)), 1.0) / x_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[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 2e+144], N[(N[(y$95$m * N[(x$95$m * 0.5), $MachinePrecision] + N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * N[(N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000005e144

   1. Initial program 95.9%

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

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. *-commutative N/A

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

      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. associate-/l* N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 73.3

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

   5. Applied rewrites 73.3%

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

   if 2.00000000000000005e144 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 73.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 90.3%

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

Alternative 7: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 10^{+307}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, x\_m \cdot 0.5, \frac{y\_m}{x\_m}\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x\_m, x\_m \cdot 0.5, 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= (* (cosh x_m) (/ y_m x_m)) 1e+307)
      (/ (fma y_m (* x_m 0.5) (/ y_m x_m)) z_m)
      (/ (* y_m (/ (fma x_m (* x_m 0.5) 1.0) 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 ((cosh(x_m) * (y_m / x_m)) <= 1e+307) {
		tmp = fma(y_m, (x_m * 0.5), (y_m / x_m)) / z_m;
	} else {
		tmp = (y_m * (fma(x_m, (x_m * 0.5), 1.0) / 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(cosh(x_m) * Float64(y_m / x_m)) <= 1e+307)
		tmp = Float64(fma(y_m, Float64(x_m * 0.5), Float64(y_m / x_m)) / z_m);
	else
		tmp = Float64(Float64(y_m * Float64(fma(x_m, Float64(x_m * 0.5), 1.0) / 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[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 1e+307], N[(N[(y$95$m * N[(x$95$m * 0.5), $MachinePrecision] + N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y$95$m * N[(N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 9.99999999999999986e306

   1. Initial program 96.3%

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

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. *-commutative N/A

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

      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. associate-/l* N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if 9.99999999999999986e306 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 69.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 47.4

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

   5. Applied rewrites 47.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 51.2

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

   7. Applied rewrites 51.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 77.0

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

   9. Applied rewrites 77.0%

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

Alternative 8: 57.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 4e+133)
      (/ (/ y_m x_m) z_m)
      (/ (/ 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 (((cosh(x_m) * (y_m / x_m)) / z_m) <= 4e+133) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (y_m / z_m) / x_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 (((cosh(x_m) * (y_m / x_m)) / z_m) <= 4d+133) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = (y_m / z_m) / x_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 (((Math.cosh(x_m) * (y_m / x_m)) / z_m) <= 4e+133) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (y_m / z_m) / x_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 ((math.cosh(x_m) * (y_m / x_m)) / z_m) <= 4e+133:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = (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(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 4e+133)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(Float64(y_m / z_m) / x_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 (((cosh(x_m) * (y_m / x_m)) / z_m) <= 4e+133)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = (y_m / z_m) / x_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[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 4e+133], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y$95$m / 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) < 4.0000000000000001e133

   1. Initial program 97.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   if 4.0000000000000001e133 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 73.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lower-*.f64 31.2

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

   5. Applied rewrites 31.2%

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

   7. Applied rewrites 35.9%

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

Alternative 9: 57.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 10^{+15}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 1e+15)
      (/ y_m (* x_m z_m))
      (/ (/ 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 (((cosh(x_m) * (y_m / x_m)) / z_m) <= 1e+15) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (y_m / z_m) / x_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 (((cosh(x_m) * (y_m / x_m)) / z_m) <= 1d+15) then
        tmp = y_m / (x_m * z_m)
    else
        tmp = (y_m / z_m) / x_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 (((Math.cosh(x_m) * (y_m / x_m)) / z_m) <= 1e+15) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (y_m / z_m) / x_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 ((math.cosh(x_m) * (y_m / x_m)) / z_m) <= 1e+15:
		tmp = y_m / (x_m * z_m)
	else:
		tmp = (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(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 1e+15)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(y_m / z_m) / x_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 (((cosh(x_m) * (y_m / x_m)) / z_m) <= 1e+15)
		tmp = y_m / (x_m * z_m);
	else
		tmp = (y_m / z_m) / x_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[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 1e+15], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / 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) < 1e15

   1. Initial program 96.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

   if 1e15 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 74.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lower-*.f64 33.8

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

   5. Applied rewrites 33.8%

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

   7. Applied rewrites 39.7%

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

Alternative 10: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 10^{+139}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)}{x\_m \cdot z\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= (* (cosh x_m) (/ y_m x_m)) 1e+139)
      (/ (/ y_m x_m) z_m)
      (/ (* y_m (fma 0.5 (* x_m x_m) 1.0)) (* x_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 ((cosh(x_m) * (y_m / x_m)) <= 1e+139) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (y_m * fma(0.5, (x_m * x_m), 1.0)) / (x_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 (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 1e+139)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(Float64(y_m * fma(0.5, Float64(x_m * x_m), 1.0)) / Float64(x_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[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 1e+139], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y$95$m * N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.00000000000000003e139

   1. Initial program 95.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 59.0

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

   5. Applied rewrites 59.0%

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

   if 1.00000000000000003e139 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 73.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 54.1

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

   5. Applied rewrites 54.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

Alternative 11: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 10^{+139}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, \left(x\_m \cdot x\_m\right) \cdot 0.5, y\_m\right)}{x\_m \cdot z\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= (* (cosh x_m) (/ y_m x_m)) 1e+139)
      (/ (/ y_m x_m) z_m)
      (/ (fma y_m (* (* x_m x_m) 0.5) y_m) (* x_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 ((cosh(x_m) * (y_m / x_m)) <= 1e+139) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = fma(y_m, ((x_m * x_m) * 0.5), y_m) / (x_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 (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 1e+139)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(fma(y_m, Float64(Float64(x_m * x_m) * 0.5), y_m) / Float64(x_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[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 1e+139], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.00000000000000003e139

   1. Initial program 95.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 59.0

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

   5. Applied rewrites 59.0%

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

   if 1.00000000000000003e139 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 73.3%

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

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft1-in N/A

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

      6. +-commutative N/A

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

      7. associate-*l/ N/A

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

      8. times-frac N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt1-in N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 57.4%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 10^{+139}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \left(x \cdot x\right) \cdot 0.5, y\right)}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4.2 \cdot 10^{+48}:\\ \;\;\;\;y\_m \cdot \frac{\cosh x\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889\right), 0.5\right), 1\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 4.2e+48)
      (* y_m (/ (cosh x_m) (* x_m z_m)))
      (*
       y_m
       (/
        (/
         (fma
          x_m
          (* x_m (fma x_m (* x_m (* (* x_m x_m) 0.001388888888888889)) 0.5))
          1.0)
         x_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 <= 4.2e+48) {
		tmp = y_m * (cosh(x_m) / (x_m * z_m));
	} else {
		tmp = y_m * ((fma(x_m, (x_m * fma(x_m, (x_m * ((x_m * x_m) * 0.001388888888888889)), 0.5)), 1.0) / x_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 <= 4.2e+48)
		tmp = Float64(y_m * Float64(cosh(x_m) / Float64(x_m * z_m)));
	else
		tmp = Float64(y_m * Float64(Float64(fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * Float64(Float64(x_m * x_m) * 0.001388888888888889)), 0.5)), 1.0) / x_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, 4.2e+48], N[(y$95$m * N[(N[Cosh[x$95$m], $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.1999999999999997e48

   1. Initial program 87.8%

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

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

      11. lower-*.f64 84.2

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

   4. Applied rewrites 84.2%

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

   if 4.1999999999999997e48 < x

   1. Initial program 79.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \frac{1}{720} \cdot \color{blue}{{x}^{3}}, \frac{1}{2}\right), 1\right)}{x}}{z} \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

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

Alternative 13: 90.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 7.8 \cdot 10^{+174}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, x\_m \cdot \left(x\_m \cdot \left(0.001388888888888889 \cdot \left(x\_m \cdot \left(x\_m \cdot y\_m\right)\right)\right)\right), y\_m\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= y_m 7.8e+174)
      (/
       (/
        (fma
         (* x_m x_m)
         (* x_m (* x_m (* 0.001388888888888889 (* x_m (* x_m y_m)))))
         y_m)
        x_m)
       z_m)
      (*
       y_m
       (/
        (/
         (fma x_m (* x_m (fma x_m (* x_m 0.041666666666666664) 0.5)) 1.0)
         x_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 (y_m <= 7.8e+174) {
		tmp = (fma((x_m * x_m), (x_m * (x_m * (0.001388888888888889 * (x_m * (x_m * y_m))))), y_m) / x_m) / z_m;
	} else {
		tmp = y_m * ((fma(x_m, (x_m * fma(x_m, (x_m * 0.041666666666666664), 0.5)), 1.0) / x_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 (y_m <= 7.8e+174)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), Float64(x_m * Float64(x_m * Float64(0.001388888888888889 * Float64(x_m * Float64(x_m * y_m))))), y_m) / x_m) / z_m);
	else
		tmp = Float64(y_m * Float64(Float64(fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5)), 1.0) / x_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[y$95$m, 7.8e+174], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(0.001388888888888889 * N[(x$95$m * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * N[(N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 7.79999999999999962e174

   1. Initial program 85.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 74.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{720} \cdot \left({x}^{4} \cdot y\right)\right), y\right)}{x \cdot z} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.9%

      \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot \left(0.001388888888888889 \cdot \left(x \cdot \left(x \cdot y\right)\right)\right)\right), y\right)}{x \cdot z} \]
   9. Step-by-step derivation

   10. Applied rewrites 89.9%

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

   if 7.79999999999999962e174 < y

   1. Initial program 87.5%

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

      1. lift-/.f64 N/A

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

      2. 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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 92.4%

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

Alternative 14: 89.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.2 \cdot 10^{+66}:\\ \;\;\;\;y\_m \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), 1\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 3.2e+66)
      (*
       y_m
       (/
        (fma
         (fma
          (fma x_m (* x_m 0.001388888888888889) 0.041666666666666664)
          (* x_m x_m)
          0.5)
         (* x_m x_m)
         1.0)
        (* x_m z_m)))
      (/
       (/
        (*
         y_m
         (fma x_m (* x_m (fma (* x_m x_m) 0.041666666666666664 0.5)) 1.0))
        x_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 <= 3.2e+66) {
		tmp = y_m * (fma(fma(fma(x_m, (x_m * 0.001388888888888889), 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / (x_m * z_m));
	} else {
		tmp = ((y_m * fma(x_m, (x_m * fma((x_m * x_m), 0.041666666666666664, 0.5)), 1.0)) / x_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 <= 3.2e+66)
		tmp = Float64(y_m * Float64(fma(fma(fma(x_m, Float64(x_m * 0.001388888888888889), 0.041666666666666664), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / Float64(x_m * z_m)));
	else
		tmp = Float64(Float64(Float64(y_m * fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), 0.041666666666666664, 0.5)), 1.0)) / x_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, 3.2e+66], N[(y$95$m * N[(N[(N[(N[(x$95$m * N[(x$95$m * 0.001388888888888889), $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] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.2e66

   1. Initial program 88.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 97.1

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 90.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.1

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

   9. Applied rewrites 78.1%

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

   if 3.2e66 < x

   1. Initial program 77.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.1

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

   5. Applied rewrites 53.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 75.8

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 98.0

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

   10. Applied rewrites 98.0%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+66}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 89.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{+174}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), 1\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= y_m 4e+174)
      (/
       (/
        (*
         y_m
         (fma x_m (* x_m (fma (* x_m x_m) 0.041666666666666664 0.5)) 1.0))
        x_m)
       z_m)
      (*
       y_m
       (/
        (/
         (fma x_m (* x_m (fma x_m (* x_m 0.041666666666666664) 0.5)) 1.0)
         x_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 (y_m <= 4e+174) {
		tmp = ((y_m * fma(x_m, (x_m * fma((x_m * x_m), 0.041666666666666664, 0.5)), 1.0)) / x_m) / z_m;
	} else {
		tmp = y_m * ((fma(x_m, (x_m * fma(x_m, (x_m * 0.041666666666666664), 0.5)), 1.0) / x_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 (y_m <= 4e+174)
		tmp = Float64(Float64(Float64(y_m * fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), 0.041666666666666664, 0.5)), 1.0)) / x_m) / z_m);
	else
		tmp = Float64(y_m * Float64(Float64(fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5)), 1.0) / x_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[y$95$m, 4e+174], N[(N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * N[(N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.00000000000000028e174

   1. Initial program 85.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 64.3

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

   5. Applied rewrites 64.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 77.5

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

   7. Applied rewrites 77.5%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 89.7

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

   10. Applied rewrites 89.7%

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

   if 4.00000000000000028e174 < y

   1. Initial program 87.5%

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

      1. lift-/.f64 N/A

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

      2. 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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 92.4%

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

Alternative 16: 84.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4 \cdot 10^{+124}:\\ \;\;\;\;\frac{y\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), 1\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.5\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 1.4e+124)
      (/
       (* y_m (fma x_m (* x_m (fma (* x_m x_m) 0.041666666666666664 0.5)) 1.0))
       (* x_m z_m))
      (/ (/ (* y_m (* (* x_m x_m) 0.5)) x_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 <= 1.4e+124) {
		tmp = (y_m * fma(x_m, (x_m * fma((x_m * x_m), 0.041666666666666664, 0.5)), 1.0)) / (x_m * z_m);
	} else {
		tmp = ((y_m * ((x_m * x_m) * 0.5)) / x_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 <= 1.4e+124)
		tmp = Float64(Float64(y_m * fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), 0.041666666666666664, 0.5)), 1.0)) / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(Float64(x_m * x_m) * 0.5)) / x_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, 1.4e+124], N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e124

   1. Initial program 88.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 66.9

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

   5. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 67.6

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

   7. Applied rewrites 67.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 76.7

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

   10. Applied rewrites 76.7%

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

   if 1.4e124 < x

   1. Initial program 67.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.8

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 96.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+124}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 84.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4 \cdot 10^{+124}:\\ \;\;\;\;y\_m \cdot \frac{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.5\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 1.4e+124)
      (*
       y_m
       (/
        (fma x_m (* x_m (fma x_m (* x_m 0.041666666666666664) 0.5)) 1.0)
        (* x_m z_m)))
      (/ (/ (* y_m (* (* x_m x_m) 0.5)) x_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 <= 1.4e+124) {
		tmp = y_m * (fma(x_m, (x_m * fma(x_m, (x_m * 0.041666666666666664), 0.5)), 1.0) / (x_m * z_m));
	} else {
		tmp = ((y_m * ((x_m * x_m) * 0.5)) / x_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 <= 1.4e+124)
		tmp = Float64(y_m * Float64(fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5)), 1.0) / Float64(x_m * z_m)));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(Float64(x_m * x_m) * 0.5)) / x_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, 1.4e+124], N[(y$95$m * N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e124

   1. Initial program 88.5%

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

      1. lift-/.f64 N/A

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

      2. 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 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 90.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.9

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

   9. Applied rewrites 78.9%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 75.0

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

   12. Applied rewrites 75.0%

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

   if 1.4e124 < x

   1. Initial program 67.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.8

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 96.8%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+124}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 82.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 8.8 \cdot 10^{+123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, y\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), y\_m\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.5\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 8.8e+123)
      (/
       (fma (* x_m x_m) (* y_m (fma (* x_m x_m) 0.041666666666666664 0.5)) y_m)
       (* x_m z_m))
      (/ (/ (* y_m (* (* x_m x_m) 0.5)) x_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 <= 8.8e+123) {
		tmp = fma((x_m * x_m), (y_m * fma((x_m * x_m), 0.041666666666666664, 0.5)), y_m) / (x_m * z_m);
	} else {
		tmp = ((y_m * ((x_m * x_m) * 0.5)) / x_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 <= 8.8e+123)
		tmp = Float64(fma(Float64(x_m * x_m), Float64(y_m * fma(Float64(x_m * x_m), 0.041666666666666664, 0.5)), y_m) / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(Float64(x_m * x_m) * 0.5)) / x_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, 8.8e+123], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 8.79999999999999969e123

   1. Initial program 88.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 77.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.6%

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

   if 8.79999999999999969e123 < x

   1. Initial program 67.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.8

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 96.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{+123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, y \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 82.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m, x\_m \cdot 0.5, 1\right)\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.55 \cdot 10^{+29}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{y\_m}{z\_m}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot \frac{z\_m}{t\_0}}\\ \end{array}\right)\right) \end{array} \end{array} \]

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 (<= z_m 1.55e+29)
        (/ (* t_0 (/ y_m z_m)) x_m)
        (/ y_m (* x_m (/ z_m t_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 t_0 = fma(x_m, (x_m * 0.5), 1.0);
	double tmp;
	if (z_m <= 1.55e+29) {
		tmp = (t_0 * (y_m / z_m)) / x_m;
	} else {
		tmp = y_m / (x_m * (z_m / t_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)
	t_0 = fma(x_m, Float64(x_m * 0.5), 1.0)
	tmp = 0.0
	if (z_m <= 1.55e+29)
		tmp = Float64(Float64(t_0 * Float64(y_m / z_m)) / x_m);
	else
		tmp = Float64(y_m / Float64(x_m * Float64(z_m / t_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_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.55e+29], N[(N[(t$95$0 * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision], N[(y$95$m / N[(x$95$m * N[(z$95$m / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.5499999999999999e29

   1. Initial program 86.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lower-*.f64 45.3

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

   8. Applied rewrites 77.1%

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

   if 1.5499999999999999e29 < z

   1. Initial program 83.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 58.7

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

   5. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 56.4

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

   7. Applied rewrites 56.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

          \[\leadsto \left(y \cdot \frac{\mathsf{fma}\left(x, \mathsf{Rewrite=>}\left(lower-*.f64, \left(x \cdot \frac{1}{2}\right)\right), 1\right)}{z}\right) \cdot \color{blue}{\frac{1}{x}} \]

   9. Applied rewrites 63.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-/.f64 N/A

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

       5. clear-num N/A

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

       6. lift-/.f64 N/A

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

       7. frac-times N/A

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

       8. metadata-eval N/A

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

       9. un-div-inv N/A

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

       10. lower-/.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 70.1

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

   11. Applied rewrites 70.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{+29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot 0.5, 1\right) \cdot \frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \frac{z}{\mathsf{fma}\left(x, x \cdot 0.5, 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 79.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.8 \cdot 10^{+66}:\\ \;\;\;\;\left(y\_m \cdot \mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)\right) \cdot \frac{1}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.5\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 1.8e+66)
      (* (* y_m (fma 0.5 (* x_m x_m) 1.0)) (/ 1.0 (* x_m z_m)))
      (/ (/ (* y_m (* (* x_m x_m) 0.5)) x_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 <= 1.8e+66) {
		tmp = (y_m * fma(0.5, (x_m * x_m), 1.0)) * (1.0 / (x_m * z_m));
	} else {
		tmp = ((y_m * ((x_m * x_m) * 0.5)) / x_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 <= 1.8e+66)
		tmp = Float64(Float64(y_m * fma(0.5, Float64(x_m * x_m), 1.0)) * Float64(1.0 / Float64(x_m * z_m)));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(Float64(x_m * x_m) * 0.5)) / x_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, 1.8e+66], N[(N[(y$95$m * N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.8e66

   1. Initial program 88.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 69.8

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

   5. Applied rewrites 69.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. div-inv N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 70.0

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

   7. Applied rewrites 70.0%

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

   if 1.8e66 < x

   1. Initial program 77.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.1

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

   5. Applied rewrites 53.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 75.8

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 75.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{+66}:\\ \;\;\;\;\left(y \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\right) \cdot \frac{1}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 69.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.7 \cdot 10^{+107}:\\ \;\;\;\;\left(y\_m \cdot \mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)\right) \cdot \frac{1}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, x\_m \cdot 0.5, \frac{y\_m}{x\_m}\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= z_m 2.7e+107)
      (* (* y_m (fma 0.5 (* x_m x_m) 1.0)) (/ 1.0 (* x_m z_m)))
      (/ (fma y_m (* x_m 0.5) (/ y_m x_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 (z_m <= 2.7e+107) {
		tmp = (y_m * fma(0.5, (x_m * x_m), 1.0)) * (1.0 / (x_m * z_m));
	} else {
		tmp = fma(y_m, (x_m * 0.5), (y_m / x_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 (z_m <= 2.7e+107)
		tmp = Float64(Float64(y_m * fma(0.5, Float64(x_m * x_m), 1.0)) * Float64(1.0 / Float64(x_m * z_m)));
	else
		tmp = Float64(fma(y_m, Float64(x_m * 0.5), Float64(y_m / x_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[z$95$m, 2.7e+107], N[(N[(y$95$m * N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m * 0.5), $MachinePrecision] + N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.7000000000000001e107

   1. Initial program 86.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. div-inv N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 69.7

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

   7. Applied rewrites 69.7%

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

   if 2.7000000000000001e107 < z

   1. Initial program 82.1%

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

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. *-commutative N/A

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

      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. associate-/l* N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 51.3

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

   5. Applied rewrites 51.3%

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

Alternative 22: 69.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.7 \cdot 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, \left(x\_m \cdot x\_m\right) \cdot 0.5, y\_m\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, x\_m \cdot 0.5, \frac{y\_m}{x\_m}\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= z_m 2.7e+107)
      (/ (fma y_m (* (* x_m x_m) 0.5) y_m) (* x_m z_m))
      (/ (fma y_m (* x_m 0.5) (/ y_m x_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 (z_m <= 2.7e+107) {
		tmp = fma(y_m, ((x_m * x_m) * 0.5), y_m) / (x_m * z_m);
	} else {
		tmp = fma(y_m, (x_m * 0.5), (y_m / x_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 (z_m <= 2.7e+107)
		tmp = Float64(fma(y_m, Float64(Float64(x_m * x_m) * 0.5), y_m) / Float64(x_m * z_m));
	else
		tmp = Float64(fma(y_m, Float64(x_m * 0.5), Float64(y_m / x_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[z$95$m, 2.7e+107], N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m * 0.5), $MachinePrecision] + N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.7000000000000001e107

   1. Initial program 86.8%

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

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft1-in N/A

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

      6. +-commutative N/A

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

      7. associate-*l/ N/A

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

      8. times-frac N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt1-in N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 69.7%

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

   if 2.7000000000000001e107 < z

   1. Initial program 82.1%

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

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. *-commutative N/A

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

      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. associate-/l* N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 51.3

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

   5. Applied rewrites 51.3%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \left(x \cdot x\right) \cdot 0.5, y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x \cdot 0.5, \frac{y}{x}\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 69.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.55 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot 0.5\right), y\_m\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.5\right)}{x\_m \cdot z\_m}\\ \end{array}\right)\right) \end{array} \]

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 2.55e+154)
      (/ (fma x_m (* x_m (* y_m 0.5)) y_m) (* x_m z_m))
      (/ (* y_m (* (* x_m x_m) 0.5)) (* x_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 <= 2.55e+154) {
		tmp = fma(x_m, (x_m * (y_m * 0.5)), y_m) / (x_m * z_m);
	} else {
		tmp = (y_m * ((x_m * x_m) * 0.5)) / (x_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 <= 2.55e+154)
		tmp = Float64(fma(x_m, Float64(x_m * Float64(y_m * 0.5)), y_m) / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(y_m * Float64(Float64(x_m * x_m) * 0.5)) / Float64(x_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, 2.55e+154], N[(N[(x$95$m * N[(x$95$m * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.55e154

   1. Initial program 88.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 77.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 64.7%

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

   if 2.55e154 < x

   1. Initial program 60.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 60.0

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

   5. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 56.0

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

   7. Applied rewrites 56.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 56.0%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.55 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot 0.5\right), y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 69.2% accurate, 3.4× speedup

Math

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

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 1.4)
      (* y_m (/ 1.0 (* x_m z_m)))
      (/ (* y_m (* (* x_m x_m) 0.5)) (* x_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 <= 1.4) {
		tmp = y_m * (1.0 / (x_m * z_m));
	} else {
		tmp = (y_m * ((x_m * x_m) * 0.5)) / (x_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 <= 1.4d0) then
        tmp = y_m * (1.0d0 / (x_m * z_m))
    else
        tmp = (y_m * ((x_m * x_m) * 0.5d0)) / (x_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 <= 1.4) {
		tmp = y_m * (1.0 / (x_m * z_m));
	} else {
		tmp = (y_m * ((x_m * x_m) * 0.5)) / (x_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 <= 1.4:
		tmp = y_m * (1.0 / (x_m * z_m))
	else:
		tmp = (y_m * ((x_m * x_m) * 0.5)) / (x_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 <= 1.4)
		tmp = Float64(y_m * Float64(1.0 / Float64(x_m * z_m)));
	else
		tmp = Float64(Float64(y_m * Float64(Float64(x_m * x_m) * 0.5)) / Float64(x_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 <= 1.4)
		tmp = y_m * (1.0 / (x_m * z_m));
	else
		tmp = (y_m * ((x_m * x_m) * 0.5)) / (x_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, 1.4], N[(y$95$m * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 87.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 96.9

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 57.6

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

   7. Applied rewrites 57.6%

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

   if 1.3999999999999999 < x

   1. Initial program 82.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 44.0

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

   5. Applied rewrites 44.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.2

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

   7. Applied rewrites 42.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 42.2%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;y \cdot \frac{1}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 49.1% accurate, 5.8× speedup

Math

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

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 (/ 1.0 (* x_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) {
	return x_s * (y_s * (z_s * (y_m * (1.0 / (x_m * z_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 * (1.0d0 / (x_m * z_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 * (1.0 / (x_m * z_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 * (1.0 / (x_m * z_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(1.0 / Float64(x_m * z_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 * (1.0 / (x_m * z_m)))));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.0%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
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. 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 97.6

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

4. Applied rewrites 97.6%

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

5. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 45.6

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

7. Applied rewrites 45.6%

   \[\leadsto y \cdot \color{blue}{\frac{1}{x \cdot z}} \]
8. Add Preprocessing

Alternative 26: 49.5% accurate, 7.5× speedup

Math

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

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 (* x_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) {
	return x_s * (y_s * (z_s * (y_m / (x_m * z_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 / (x_m * z_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 / (x_m * z_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 / (x_m * z_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(x_m * z_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 / (x_m * z_m))));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.0%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
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. lower-*.f64 45.6

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

5. Applied rewrites 45.6%

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

Developer Target 1: 97.2% 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 2024232 
(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))