Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 85.4% → 97.7%

Time: 16.0s

Alternatives: 33

Speedup: 2.3×

• 63.7% 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: 85.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.6 \cdot 10^{+182}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 3.6e+182)
    (/ (/ (* y_m (cosh x)) x) z)
    (/
     y_m
     (*
      z
      (/
       x
       (fma
        x
        (*
         x
         (fma
          (* x x)
          (fma (* x x) 0.001388888888888889 0.041666666666666664)
          0.5))
        1.0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.6e+182) {
		tmp = ((y_m * cosh(x)) / x) / z;
	} else {
		tmp = y_m / (z * (x / fma(x, (x * fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0)));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 3.6e+182)
		tmp = Float64(Float64(Float64(y_m * cosh(x)) / x) / z);
	else
		tmp = Float64(y_m / Float64(z * Float64(x / fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 3.6e+182], N[(N[(N[(y$95$m * N[Cosh[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(z * N[(x / N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.6e182

   1. Initial program 85.0%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

      9. lower-/.f64 96.9

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

   4. Applied egg-rr 96.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 96.9

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 97.0

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

   6. Applied egg-rr 97.0%

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

   if 3.6e182 < y

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 96.9

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

   5. Simplified 96.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 97.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

   9. Applied egg-rr 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+182}:\\ \;\;\;\;\frac{\frac{y \cdot \cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.7% accurate, 0.5× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y_m x))))
   (*
    y_s
    (if (<= t_0 INFINITY)
      (/ t_0 z)
      (/
       y_m
       (*
        x
        (/
         z
         (fma
          (* x x)
          (fma x (* 0.001388888888888889 (* x (* x x))) 0.5)
          1.0))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = cosh(x) * (y_m / x);
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 / z;
	} else {
		tmp = y_m / (x * (z / fma((x * x), fma(x, (0.001388888888888889 * (x * (x * x))), 0.5), 1.0)));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(cosh(x) * Float64(y_m / x))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(y_m / Float64(x * Float64(z / fma(Float64(x * x), fma(x, Float64(0.001388888888888889 * Float64(x * Float64(x * x))), 0.5), 1.0))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, Infinity], N[(t$95$0 / z), $MachinePrecision], N[(y$95$m / N[(x * N[(z / N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.001388888888888889 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < +inf.0

   1. Initial program 96.6%

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

   if +inf.0 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 0.1

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

   5. Simplified 0.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{y}{\frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{720} \cdot {x}^{3}}, \frac{1}{2}\right), 1\right)} \cdot x} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 100.0

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

   12. Simplified 100.0%

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

4. Final simplification 97.0%

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

Alternative 3: 95.7% accurate, 0.5× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* (cosh x) (/ y_m x)) INFINITY)
    (* (/ y_m x) (/ (cosh x) z))
    (/
     y_m
     (*
      x
      (/
       z
       (fma
        (* x x)
        (fma x (* 0.001388888888888889 (* x (* x x))) 0.5)
        1.0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((cosh(x) * (y_m / x)) <= ((double) INFINITY)) {
		tmp = (y_m / x) * (cosh(x) / z);
	} else {
		tmp = y_m / (x * (z / fma((x * x), fma(x, (0.001388888888888889 * (x * (x * x))), 0.5), 1.0)));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y_m / x)) <= Inf)
		tmp = Float64(Float64(y_m / x) * Float64(cosh(x) / z));
	else
		tmp = Float64(y_m / Float64(x * Float64(z / fma(Float64(x * x), fma(x, Float64(0.001388888888888889 * Float64(x * Float64(x * x))), 0.5), 1.0))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(y$95$m / x), $MachinePrecision] * N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(x * N[(z / N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.001388888888888889 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < +inf.0

   1. Initial program 96.6%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 96.6

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

   4. Applied egg-rr 96.6%

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

   if +inf.0 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 0.1

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

   5. Simplified 0.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{y}{\frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{720} \cdot {x}^{3}}, \frac{1}{2}\right), 1\right)} \cdot x} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 100.0

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

   12. Simplified 100.0%

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

4. Final simplification 96.9%

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

Alternative 4: 92.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y\_m}{x} \leq 10^{+270}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, y\_m \cdot \mathsf{fma}\left(x \cdot x, t\_0, 0.5\right), y\_m\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x \cdot \frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot t\_0, 0.5\right), 1\right)}}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (fma (* x x) 0.001388888888888889 0.041666666666666664)))
   (*
    y_s
    (if (<= (* (cosh x) (/ y_m x)) 1e+270)
      (/ (/ (fma (* x x) (* y_m (fma (* x x) t_0 0.5)) y_m) x) z)
      (/ y_m (* x (/ z (fma (* x x) (fma x (* x t_0) 0.5) 1.0))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = fma((x * x), 0.001388888888888889, 0.041666666666666664);
	double tmp;
	if ((cosh(x) * (y_m / x)) <= 1e+270) {
		tmp = (fma((x * x), (y_m * fma((x * x), t_0, 0.5)), y_m) / x) / z;
	} else {
		tmp = y_m / (x * (z / fma((x * x), fma(x, (x * t_0), 0.5), 1.0)));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y_m / x)) <= 1e+270)
		tmp = Float64(Float64(fma(Float64(x * x), Float64(y_m * fma(Float64(x * x), t_0, 0.5)), y_m) / x) / z);
	else
		tmp = Float64(y_m / Float64(x * Float64(z / fma(Float64(x * x), fma(x, Float64(x * t_0), 0.5), 1.0))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]}, N[(y$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], 1e+270], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(y$95$m * N[(N[(x * x), $MachinePrecision] * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(x * N[(z / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * t$95$0), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1e270

   1. Initial program 96.1%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

      9. lower-/.f64 96.0

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 91.6%

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

   if 1e270 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 73.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 64.3

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

   5. Simplified 64.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

   9. Applied egg-rr 91.1%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 10^{+270}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), y\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y\_m}{x} \leq 10^{+270}:\\ \;\;\;\;\frac{\frac{y\_m \cdot t\_0}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x \cdot \frac{z}{t\_0}}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0
         (fma
          (* x x)
          (fma
           x
           (* x (fma (* x x) 0.001388888888888889 0.041666666666666664))
           0.5)
          1.0)))
   (*
    y_s
    (if (<= (* (cosh x) (/ y_m x)) 1e+270)
      (/ (/ (* y_m t_0) x) z)
      (/ y_m (* x (/ z t_0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = fma((x * x), fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0);
	double tmp;
	if ((cosh(x) * (y_m / x)) <= 1e+270) {
		tmp = ((y_m * t_0) / x) / z;
	} else {
		tmp = y_m / (x * (z / t_0));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y_m / x)) <= 1e+270)
		tmp = Float64(Float64(Float64(y_m * t_0) / x) / z);
	else
		tmp = Float64(y_m / Float64(x * Float64(z / t_0)));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(y$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], 1e+270], N[(N[(N[(y$95$m * t$95$0), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(x * N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1e270

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 91.6

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

   5. Simplified 91.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

   9. Applied egg-rr 92.2%

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

   if 1e270 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 73.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 64.3

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

   5. Simplified 64.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

   9. Applied egg-rr 91.1%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 10^{+270}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y\_m}{x} \leq 10^{+270}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, t\_0, 0.5\right), 1\right) \cdot \frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x \cdot \frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot t\_0, 0.5\right), 1\right)}}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (fma (* x x) 0.001388888888888889 0.041666666666666664)))
   (*
    y_s
    (if (<= (* (cosh x) (/ y_m x)) 1e+270)
      (/ (* (fma x (* x (fma (* x x) t_0 0.5)) 1.0) (/ y_m x)) z)
      (/ y_m (* x (/ z (fma (* x x) (fma x (* x t_0) 0.5) 1.0))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = fma((x * x), 0.001388888888888889, 0.041666666666666664);
	double tmp;
	if ((cosh(x) * (y_m / x)) <= 1e+270) {
		tmp = (fma(x, (x * fma((x * x), t_0, 0.5)), 1.0) * (y_m / x)) / z;
	} else {
		tmp = y_m / (x * (z / fma((x * x), fma(x, (x * t_0), 0.5), 1.0)));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y_m / x)) <= 1e+270)
		tmp = Float64(Float64(fma(x, Float64(x * fma(Float64(x * x), t_0, 0.5)), 1.0) * Float64(y_m / x)) / z);
	else
		tmp = Float64(y_m / Float64(x * Float64(z / fma(Float64(x * x), fma(x, Float64(x * t_0), 0.5), 1.0))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]}, N[(y$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], 1e+270], N[(N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(x * N[(z / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * t$95$0), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1e270

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 91.6

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

   5. Simplified 91.6%

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

   if 1e270 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 73.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 64.3

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

   5. Simplified 64.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

   9. Applied egg-rr 91.1%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 10^{+270}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y\_m}{x} \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{y\_m}{x} \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x \cdot \frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* (cosh x) (/ y_m x)) 2e+258)
    (*
     (/ y_m x)
     (/
      (fma
       x
       (*
        x
        (fma
         x
         (* x (fma x (* x 0.001388888888888889) 0.041666666666666664))
         0.5))
       1.0)
      z))
    (/
     y_m
     (*
      x
      (/
       z
       (fma
        (* x x)
        (fma
         x
         (* x (fma (* x x) 0.001388888888888889 0.041666666666666664))
         0.5)
        1.0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((cosh(x) * (y_m / x)) <= 2e+258) {
		tmp = (y_m / x) * (fma(x, (x * fma(x, (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0) / z);
	} else {
		tmp = y_m / (x * (z / fma((x * x), fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0)));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y_m / x)) <= 2e+258)
		tmp = Float64(Float64(y_m / x) * Float64(fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0) / z));
	else
		tmp = Float64(y_m / Float64(x * Float64(z / fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], 2e+258], N[(N[(y$95$m / x), $MachinePrecision] * N[(N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(x * N[(z / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000011e258

   1. Initial program 96.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 91.4

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

   5. Simplified 91.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 91.4%

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

   8. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

   10. Simplified 91.4%

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

   if 2.00000000000000011e258 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 74.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 65.3

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

   5. Simplified 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 66.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

   9. Applied egg-rr 91.3%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y\_m}{x} \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{y\_m}{x} \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x \cdot \frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.001388888888888889 \cdot \left(x \cdot \left(x \cdot x\right)\right), 0.5\right), 1\right)}}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* (cosh x) (/ y_m x)) 2e+258)
    (*
     (/ y_m x)
     (/
      (fma
       x
       (*
        x
        (fma
         x
         (* x (fma x (* x 0.001388888888888889) 0.041666666666666664))
         0.5))
       1.0)
      z))
    (/
     y_m
     (*
      x
      (/
       z
       (fma
        (* x x)
        (fma x (* 0.001388888888888889 (* x (* x x))) 0.5)
        1.0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((cosh(x) * (y_m / x)) <= 2e+258) {
		tmp = (y_m / x) * (fma(x, (x * fma(x, (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0) / z);
	} else {
		tmp = y_m / (x * (z / fma((x * x), fma(x, (0.001388888888888889 * (x * (x * x))), 0.5), 1.0)));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y_m / x)) <= 2e+258)
		tmp = Float64(Float64(y_m / x) * Float64(fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0) / z));
	else
		tmp = Float64(y_m / Float64(x * Float64(z / fma(Float64(x * x), fma(x, Float64(0.001388888888888889 * Float64(x * Float64(x * x))), 0.5), 1.0))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], 2e+258], N[(N[(y$95$m / x), $MachinePrecision] * N[(N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(x * N[(z / N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.001388888888888889 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000011e258

   1. Initial program 96.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 91.4

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

   5. Simplified 91.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 91.4%

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

   8. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

   10. Simplified 91.4%

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

   if 2.00000000000000011e258 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 74.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 65.3

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

   5. Simplified 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 66.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

   9. Applied egg-rr 91.3%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{y}{\frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{720} \cdot {x}^{3}}, \frac{1}{2}\right), 1\right)} \cdot x} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 91.3

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

   12. Simplified 91.3%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.001388888888888889 \cdot \left(x \cdot \left(x \cdot x\right)\right), 0.5\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y\_m}{x} \leq 10^{+270}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, x \cdot \left(y\_m \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\_m\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x \cdot \frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.001388888888888889 \cdot \left(x \cdot \left(x \cdot x\right)\right), 0.5\right), 1\right)}}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* (cosh x) (/ y_m x)) 1e+270)
    (/
     (/ (fma x (* x (* y_m (fma x (* x 0.041666666666666664) 0.5))) y_m) x)
     z)
    (/
     y_m
     (*
      x
      (/
       z
       (fma
        (* x x)
        (fma x (* 0.001388888888888889 (* x (* x x))) 0.5)
        1.0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((cosh(x) * (y_m / x)) <= 1e+270) {
		tmp = (fma(x, (x * (y_m * fma(x, (x * 0.041666666666666664), 0.5))), y_m) / x) / z;
	} else {
		tmp = y_m / (x * (z / fma((x * x), fma(x, (0.001388888888888889 * (x * (x * x))), 0.5), 1.0)));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y_m / x)) <= 1e+270)
		tmp = Float64(Float64(fma(x, Float64(x * Float64(y_m * fma(x, Float64(x * 0.041666666666666664), 0.5))), y_m) / x) / z);
	else
		tmp = Float64(y_m / Float64(x * Float64(z / fma(Float64(x * x), fma(x, Float64(0.001388888888888889 * Float64(x * Float64(x * x))), 0.5), 1.0))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], 1e+270], N[(N[(N[(x * N[(x * N[(y$95$m * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(x * N[(z / N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.001388888888888889 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1e270

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}}}{z} \]
   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. Simplified 89.4%

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

   if 1e270 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 73.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 64.3

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

   5. Simplified 64.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

   9. Applied egg-rr 91.1%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{y}{\frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{720} \cdot {x}^{3}}, \frac{1}{2}\right), 1\right)} \cdot x} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 91.1

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

   12. Simplified 91.1%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 10^{+270}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.001388888888888889 \cdot \left(x \cdot \left(x \cdot x\right)\right), 0.5\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.6 \cdot 10^{+182}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 3.6e+182)
    (/ (* y_m (/ (cosh x) x)) z)
    (/
     y_m
     (*
      z
      (/
       x
       (fma
        x
        (*
         x
         (fma
          (* x x)
          (fma (* x x) 0.001388888888888889 0.041666666666666664)
          0.5))
        1.0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.6e+182) {
		tmp = (y_m * (cosh(x) / x)) / z;
	} else {
		tmp = y_m / (z * (x / fma(x, (x * fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0)));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 3.6e+182)
		tmp = Float64(Float64(y_m * Float64(cosh(x) / x)) / z);
	else
		tmp = Float64(y_m / Float64(z * Float64(x / fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 3.6e+182], N[(N[(y$95$m * N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(z * N[(x / N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.6e182

   1. Initial program 85.0%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

      9. lower-/.f64 96.9

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

   4. Applied egg-rr 96.9%

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

   if 3.6e182 < y

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 96.9

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

   5. Simplified 96.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 97.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

   9. Applied egg-rr 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+182}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\frac{y\_m \cdot \cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x \cdot \frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.001388888888888889 \cdot \left(x \cdot \left(x \cdot x\right)\right), 0.5\right), 1\right)}}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 7e+51)
    (/ (* y_m (cosh x)) (* x z))
    (/
     y_m
     (*
      x
      (/
       z
       (fma
        (* x x)
        (fma x (* 0.001388888888888889 (* x (* x x))) 0.5)
        1.0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 7e+51) {
		tmp = (y_m * cosh(x)) / (x * z);
	} else {
		tmp = y_m / (x * (z / fma((x * x), fma(x, (0.001388888888888889 * (x * (x * x))), 0.5), 1.0)));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 7e+51)
		tmp = Float64(Float64(y_m * cosh(x)) / Float64(x * z));
	else
		tmp = Float64(y_m / Float64(x * Float64(z / fma(Float64(x * x), fma(x, Float64(0.001388888888888889 * Float64(x * Float64(x * x))), 0.5), 1.0))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 7e+51], N[(N[(y$95$m * N[Cosh[x], $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(x * N[(z / N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.001388888888888889 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7e51

   1. Initial program 90.2%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

      9. lower-/.f64 96.1

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

   4. Applied egg-rr 96.1%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 84.7

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

   6. Applied egg-rr 84.7%

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

   if 7e51 < x

   1. Initial program 72.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 72.2

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

   5. Simplified 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{y}{\frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{720} \cdot {x}^{3}}, \frac{1}{2}\right), 1\right)} \cdot x} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 100.0

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

   12. Simplified 100.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\frac{y \cdot \cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.001388888888888889 \cdot \left(x \cdot \left(x \cdot x\right)\right), 0.5\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+51}:\\ \;\;\;\;y\_m \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x \cdot \frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.001388888888888889 \cdot \left(x \cdot \left(x \cdot x\right)\right), 0.5\right), 1\right)}}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 7e+51)
    (* y_m (/ (cosh x) (* x z)))
    (/
     y_m
     (*
      x
      (/
       z
       (fma
        (* x x)
        (fma x (* 0.001388888888888889 (* x (* x x))) 0.5)
        1.0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 7e+51) {
		tmp = y_m * (cosh(x) / (x * z));
	} else {
		tmp = y_m / (x * (z / fma((x * x), fma(x, (0.001388888888888889 * (x * (x * x))), 0.5), 1.0)));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 7e+51)
		tmp = Float64(y_m * Float64(cosh(x) / Float64(x * z)));
	else
		tmp = Float64(y_m / Float64(x * Float64(z / fma(Float64(x * x), fma(x, Float64(0.001388888888888889 * Float64(x * Float64(x * x))), 0.5), 1.0))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 7e+51], N[(y$95$m * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(x * N[(z / N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.001388888888888889 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7e51

   1. Initial program 90.2%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 84.2

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

   4. Applied egg-rr 84.2%

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

   if 7e51 < x

   1. Initial program 72.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 72.2

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

   5. Simplified 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{y}{\frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{720} \cdot {x}^{3}}, \frac{1}{2}\right), 1\right)} \cdot x} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 100.0

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

   12. Simplified 100.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+51}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \frac{z}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.001388888888888889 \cdot \left(x \cdot \left(x \cdot x\right)\right), 0.5\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+64}:\\ \;\;\;\;\frac{y\_m \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 2.8e+64)
    (/
     (*
      y_m
      (fma
       (* x x)
       (fma
        (* x x)
        (fma x (* x 0.001388888888888889) 0.041666666666666664)
        0.5)
       1.0))
     (* x z))
    (/
     (* y_m (/ (fma (* x x) (fma (* x x) 0.041666666666666664 0.5) 1.0) x))
     z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.8e+64) {
		tmp = (y_m * fma((x * x), fma((x * x), fma(x, (x * 0.001388888888888889), 0.041666666666666664), 0.5), 1.0)) / (x * z);
	} else {
		tmp = (y_m * (fma((x * x), fma((x * x), 0.041666666666666664, 0.5), 1.0) / x)) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.8e+64)
		tmp = Float64(Float64(y_m * fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664), 0.5), 1.0)) / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0) / x)) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.8e+64], N[(N[(y$95$m * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.80000000000000024e64

   1. Initial program 90.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 82.6

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

   5. Simplified 82.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-*.f64 N/A

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

   7. Applied egg-rr 78.6%

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

   if 2.80000000000000024e64 < x

   1. Initial program 69.4%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

      9. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \cdot y}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot y}{z} \]

      10. lower-*.f64 98.0

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot y}{z} \]

   7. Simplified 98.0%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}} \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+64}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 83.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+64}:\\ \;\;\;\;y\_m \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 2.8e+64)
    (*
     y_m
     (/
      (fma
       x
       (*
        x
        (fma
         (* x x)
         (fma (* x x) 0.001388888888888889 0.041666666666666664)
         0.5))
       1.0)
      (* x z)))
    (/
     (* y_m (/ (fma (* x x) (fma (* x x) 0.041666666666666664 0.5) 1.0) x))
     z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.8e+64) {
		tmp = y_m * (fma(x, (x * fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0) / (x * z));
	} else {
		tmp = (y_m * (fma((x * x), fma((x * x), 0.041666666666666664, 0.5), 1.0) / x)) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.8e+64)
		tmp = Float64(y_m * Float64(fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0) / Float64(x * z)));
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0) / x)) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.8e+64], N[(y$95$m * N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.80000000000000024e64

   1. Initial program 90.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      15. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      16. lower-*.f64 82.6

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 82.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right)} + \frac{1}{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right)}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right)\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \cdot \frac{y}{x}}{z} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}}{z} \]

      9. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{z}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}}} \]

      10. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{z} \cdot \left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}\right)} \]

   7. Applied egg-rr 88.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{z}}{\frac{x}{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}}} \]

   9. Applied egg-rr 78.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z \cdot x} \cdot y} \]

   if 2.80000000000000024e64 < x

   1. Initial program 69.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{1}{\frac{x}{y}}}}{z} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot y\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right) \cdot y}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right) \cdot y}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \cdot y}{z} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

      9. lower-/.f64 100.0

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x} \cdot y}}{z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \cdot y}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \cdot y}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot y}{z} \]

      10. lower-*.f64 98.0

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot y}{z} \]

   7. Simplified 98.0%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}} \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+64}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 91.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \frac{y\_m}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 5e-82)
    (/
     (* y_m (/ (fma (* x x) (fma (* x x) 0.041666666666666664 0.5) 1.0) x))
     z)
    (/
     (* (fma (* x x) (fma x (* x 0.041666666666666664) 0.5) 1.0) (/ y_m z))
     x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5e-82) {
		tmp = (y_m * (fma((x * x), fma((x * x), 0.041666666666666664, 0.5), 1.0) / x)) / z;
	} else {
		tmp = (fma((x * x), fma(x, (x * 0.041666666666666664), 0.5), 1.0) * (y_m / z)) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 5e-82)
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0) / x)) / z);
	else
		tmp = Float64(Float64(fma(Float64(x * x), fma(x, Float64(x * 0.041666666666666664), 0.5), 1.0) * Float64(y_m / z)) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 5e-82], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999998e-82

   1. Initial program 80.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{1}{\frac{x}{y}}}}{z} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot y\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right) \cdot y}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right) \cdot y}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \cdot y}{z} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

      9. lower-/.f64 96.1

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

   4. Applied egg-rr 96.1%

      \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x} \cdot y}}{z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \cdot y}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \cdot y}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot y}{z} \]

      10. lower-*.f64 82.9

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot y}{z} \]

   7. Simplified 82.9%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}} \cdot y}{z} \]

   if 4.9999999999999998e-82 < y

   1. Initial program 98.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      15. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      16. lower-*.f64 95.1

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 95.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right)} + \frac{1}{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right)}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right)\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \cdot \frac{y}{x}}{z} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot y}{x}}}{z} \]

   7. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z} \cdot \frac{y}{x}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}}{z} \cdot \frac{y}{x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{z} \cdot \frac{y}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{z} \cdot \frac{y}{x} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{z} \cdot \frac{y}{x} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{z} \cdot \frac{y}{x} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{z} \cdot \frac{y}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{z} \cdot \frac{y}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{z} \cdot \frac{y}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{z} \cdot \frac{y}{x} \]

      9. lower-*.f64 94.6

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{z} \cdot \frac{y}{x} \]

   10. Simplified 94.6%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}}{z} \cdot \frac{y}{x} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}\right) + 1}{z} \cdot \frac{y}{x} \]

       2. lift-fma.f64 N/A

          \[\leadsto \frac{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right)} + 1}{z} \cdot \frac{y}{x} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right) + 1}{z} \cdot \frac{y}{x} \]

       4. lift-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), 1\right)}}{z} \cdot \frac{y}{x} \]

       5. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{z}} \cdot \frac{y}{x} \]

       6. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{z} \cdot y}{x}} \]

       7. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{z} \cdot y}{x}} \]

   12. Applied egg-rr 94.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \frac{y}{z}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \frac{y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 90.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.2e+99)
    (/
     (* y_m (/ (fma (* x x) (fma (* x x) 0.041666666666666664 0.5) 1.0) x))
     z)
    (*
     (/ y_m z)
     (/ (fma (* x x) (fma x (* x 0.041666666666666664) 0.5) 1.0) x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.2e+99) {
		tmp = (y_m * (fma((x * x), fma((x * x), 0.041666666666666664, 0.5), 1.0) / x)) / z;
	} else {
		tmp = (y_m / z) * (fma((x * x), fma(x, (x * 0.041666666666666664), 0.5), 1.0) / x);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.2e+99)
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0) / x)) / z);
	else
		tmp = Float64(Float64(y_m / z) * Float64(fma(Float64(x * x), fma(x, Float64(x * 0.041666666666666664), 0.5), 1.0) / x));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.2e+99], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.2000000000000001e99

   1. Initial program 84.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{1}{\frac{x}{y}}}}{z} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot y\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right) \cdot y}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right) \cdot y}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \cdot y}{z} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

      9. lower-/.f64 96.7

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

   4. Applied egg-rr 96.7%

      \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x} \cdot y}}{z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \cdot y}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \cdot y}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot y}{z} \]

      10. lower-*.f64 84.9

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot y}{z} \]

   7. Simplified 84.9%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}} \cdot y}{z} \]

   if 1.2000000000000001e99 < y

   1. Initial program 97.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 95.7

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 95.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)} \cdot \frac{y}{x}}{z} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{x}}}{z} \]

      8. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{z \cdot x}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{\color{blue}{x \cdot z}} \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot \frac{y}{z}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot \frac{y}{z}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x}} \cdot \frac{y}{z} \]

      13. lower-/.f64 97.8

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot \color{blue}{\frac{y}{z}} \]

   7. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot \frac{y}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 83.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 7.4 \cdot 10^{+63}:\\ \;\;\;\;\frac{y\_m}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}{x}}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 7.4e+63)
    (*
     (/ y_m x)
     (/ (fma (* x x) (fma x (* x 0.041666666666666664) 0.5) 1.0) z))
    (/ (* y_m (/ (* (* x x) (* (* x x) 0.041666666666666664)) x)) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 7.4e+63) {
		tmp = (y_m / x) * (fma((x * x), fma(x, (x * 0.041666666666666664), 0.5), 1.0) / z);
	} else {
		tmp = (y_m * (((x * x) * ((x * x) * 0.041666666666666664)) / x)) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 7.4e+63)
		tmp = Float64(Float64(y_m / x) * Float64(fma(Float64(x * x), fma(x, Float64(x * 0.041666666666666664), 0.5), 1.0) / z));
	else
		tmp = Float64(Float64(y_m * Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * 0.041666666666666664)) / x)) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 7.4e+63], N[(N[(y$95$m / x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7.39999999999999937e63

   1. Initial program 90.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 78.7

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 78.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)} \cdot \frac{y}{x}}{z} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{x}}}{z} \]

      8. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{z \cdot x}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{z} \cdot \frac{y}{x}} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{z} \cdot \color{blue}{\frac{y}{x}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{z} \cdot \frac{y}{x}} \]

      12. lower-/.f64 79.6

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}} \cdot \frac{y}{x} \]

   7. Applied egg-rr 79.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z} \cdot \frac{y}{x}} \]

   if 7.39999999999999937e63 < x

   1. Initial program 69.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{1}{\frac{x}{y}}}}{z} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot y\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right) \cdot y}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right) \cdot y}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \cdot y}{z} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

      9. lower-/.f64 100.0

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x} \cdot y}}{z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \cdot y}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \cdot y}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot y}{z} \]

      10. lower-*.f64 98.0

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot y}{z} \]

   7. Simplified 98.0%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}} \cdot y}{z} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{24} \cdot {x}^{4}}}{x} \cdot y}{z} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{1}{24} \cdot {x}^{\color{blue}{\left(2 \cdot 2\right)}}}{x} \cdot y}{z} \]

      2. pow-sqr N/A

         \[\leadsto \frac{\frac{\frac{1}{24} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}}{x} \cdot y}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}}}{x} \cdot y}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)}}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)}}{x} \cdot y}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)}{x} \cdot y}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)}{x} \cdot y}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}}{x} \cdot y}{z} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}}{x} \cdot y}{z} \]

      10. unpow2 N/A

          \[\leadsto \frac{\frac{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24}\right)}{x} \cdot y}{z} \]

      11. lower-*.f64 98.0

          \[\leadsto \frac{\frac{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.041666666666666664\right)}{x} \cdot y}{z} \]

   10. Simplified 98.0%

       \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}}{x} \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.4 \cdot 10^{+63}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 81.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(y\_m \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\_m\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}{x}}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 2.1e+58)
    (/
     (fma x (* x (* y_m (fma x (* x 0.041666666666666664) 0.5))) y_m)
     (* x z))
    (/ (* y_m (/ (* (* x x) (* (* x x) 0.041666666666666664)) x)) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.1e+58) {
		tmp = fma(x, (x * (y_m * fma(x, (x * 0.041666666666666664), 0.5))), y_m) / (x * z);
	} else {
		tmp = (y_m * (((x * x) * ((x * x) * 0.041666666666666664)) / x)) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.1e+58)
		tmp = Float64(fma(x, Float64(x * Float64(y_m * fma(x, Float64(x * 0.041666666666666664), 0.5))), y_m) / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * 0.041666666666666664)) / x)) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.1e+58], N[(N[(x * N[(x * N[(y$95$m * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000012e58

   1. Initial program 90.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}}{x}} \]

   5. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\right)}{x \cdot z}} \]

   if 2.10000000000000012e58 < x

   1. Initial program 71.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{1}{\frac{x}{y}}}}{z} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot y\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right) \cdot y}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right) \cdot y}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \cdot y}{z} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

      9. lower-/.f64 100.0

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x} \cdot y}}{z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \cdot y}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \cdot y}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot y}{z} \]

      10. lower-*.f64 96.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot y}{z} \]

   7. Simplified 96.3%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}} \cdot y}{z} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{24} \cdot {x}^{4}}}{x} \cdot y}{z} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{1}{24} \cdot {x}^{\color{blue}{\left(2 \cdot 2\right)}}}{x} \cdot y}{z} \]

      2. pow-sqr N/A

         \[\leadsto \frac{\frac{\frac{1}{24} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}}{x} \cdot y}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}}}{x} \cdot y}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)}}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)}}{x} \cdot y}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)}{x} \cdot y}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)}{x} \cdot y}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}}{x} \cdot y}{z} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}}{x} \cdot y}{z} \]

      10. unpow2 N/A

          \[\leadsto \frac{\frac{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24}\right)}{x} \cdot y}{z} \]

      11. lower-*.f64 96.3

          \[\leadsto \frac{\frac{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.041666666666666664\right)}{x} \cdot y}{z} \]

   10. Simplified 96.3%

       \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}}{x} \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 80.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(y\_m \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\_m\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 2.1e+58)
    (/
     (fma x (* x (* y_m (fma x (* x 0.041666666666666664) 0.5))) y_m)
     (* x z))
    (/ (* y_m (* x (* (* x x) 0.041666666666666664))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.1e+58) {
		tmp = fma(x, (x * (y_m * fma(x, (x * 0.041666666666666664), 0.5))), y_m) / (x * z);
	} else {
		tmp = (y_m * (x * ((x * x) * 0.041666666666666664))) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.1e+58)
		tmp = Float64(fma(x, Float64(x * Float64(y_m * fma(x, Float64(x * 0.041666666666666664), 0.5))), y_m) / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * Float64(x * Float64(Float64(x * x) * 0.041666666666666664))) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.1e+58], N[(N[(x * N[(x * N[(y$95$m * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000012e58

   1. Initial program 90.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}}{x}} \]

   5. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\right)}{x \cdot z}} \]

   if 2.10000000000000012e58 < x

   1. Initial program 71.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{1}{\frac{x}{y}}}}{z} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot y\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right) \cdot y}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right) \cdot y}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \cdot y}{z} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

      9. lower-/.f64 100.0

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x} \cdot y}}{z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \cdot y}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \cdot y}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot y}{z} \]

      10. lower-*.f64 96.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot y}{z} \]

   7. Simplified 96.3%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}} \cdot y}{z} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)} \cdot y}{z} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot \frac{1}{24}\right)} \cdot y}{z} \]

      2. cube-mult N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \frac{1}{24}\right) \cdot y}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{1}{24}\right) \cdot y}{z} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left({x}^{2} \cdot \frac{1}{24}\right)\right)} \cdot y}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right) \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}\right) \cdot y}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}\right) \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24}\right)\right) \cdot y}{z} \]

      10. lower-*.f64 94.5

          \[\leadsto \frac{\left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.041666666666666664\right)\right) \cdot y}{z} \]

   10. Simplified 94.5%

       \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 81.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{y\_m \cdot \mathsf{fma}\left(x, t\_0, 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot t\_0}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* x (fma (* x x) 0.041666666666666664 0.5))))
   (*
    y_s
    (if (<= x 3.5e+60)
      (/ (* y_m (fma x t_0 1.0)) (* x z))
      (/ (* y_m t_0) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = x * fma((x * x), 0.041666666666666664, 0.5);
	double tmp;
	if (x <= 3.5e+60) {
		tmp = (y_m * fma(x, t_0, 1.0)) / (x * z);
	} else {
		tmp = (y_m * t_0) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5))
	tmp = 0.0
	if (x <= 3.5e+60)
		tmp = Float64(Float64(y_m * fma(x, t_0, 1.0)) / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * t_0) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[x, 3.5e+60], N[(N[(y$95$m * N[(x * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5000000000000002e60

   1. Initial program 90.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{1}{\frac{x}{y}}}}{z} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot y\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right) \cdot y}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right) \cdot y}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \cdot y}{z} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

      9. lower-/.f64 96.2

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

   4. Applied egg-rr 96.2%

      \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x} \cdot y}}{z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \cdot y}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \cdot y}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot y}{z} \]

      10. lower-*.f64 84.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot y}{z} \]

   7. Simplified 84.3%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}} \cdot y}{z} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24} + \frac{1}{2}\right) + 1}{x} \cdot y}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}\right) + 1}{x} \cdot y}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right)} + 1}{x} \cdot y}{z} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), 1\right)}}{x} \cdot y}{z} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x}} \cdot y}{z} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot \frac{y}{z}} \]

      7. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x}} \cdot \frac{y}{z} \]

      8. frac-times N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{x \cdot z}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{\color{blue}{x \cdot z}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{x \cdot z}} \]

   9. Applied egg-rr 74.9%

      \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z}} \]

   if 3.5000000000000002e60 < x

   1. Initial program 70.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{1}{\frac{x}{y}}}}{z} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot y\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right) \cdot y}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right) \cdot y}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \cdot y}{z} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

      9. lower-/.f64 100.0

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x} \cdot y}}{z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \cdot y}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \cdot y}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot y}{z} \]

      10. lower-*.f64 96.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot y}{z} \]

   7. Simplified 96.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}} \cdot y}{z} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot y}{z} \]
   9. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot y}{z} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot y}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \cdot y}{z} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right)}\right) \cdot y}{z} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right) \cdot x\right)} \cdot y}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)\right)} \cdot x\right) \cdot y}{z} \]

      7. lft-mult-inverse N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \left(\frac{1}{2} \cdot \color{blue}{1}\right) \cdot x\right) \cdot y}{z} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \color{blue}{\frac{1}{2}} \cdot x\right) \cdot y}{z} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)\right)} \cdot y}{z} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right) \cdot y}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot y}{z} \]

      12. +-commutative N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)}\right) \cdot y}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}\right)\right) \cdot y}{z} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}\right) \cdot y}{z} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right)\right) \cdot y}{z} \]

      16. lower-*.f64 94.3

          \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right)\right) \cdot y}{z} \]

   10. Simplified 94.3%

       \[\leadsto \frac{\color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)} \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 78.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{y\_m}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 5.2e+49)
    (* (/ y_m x) (/ (fma (* x x) 0.5 1.0) z))
    (/ (* y_m (* x (* (* x x) 0.041666666666666664))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 5.2e+49) {
		tmp = (y_m / x) * (fma((x * x), 0.5, 1.0) / z);
	} else {
		tmp = (y_m * (x * ((x * x) * 0.041666666666666664))) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 5.2e+49)
		tmp = Float64(Float64(y_m / x) * Float64(fma(Float64(x * x), 0.5, 1.0) / z));
	else
		tmp = Float64(Float64(y_m * Float64(x * Float64(Float64(x * x) * 0.041666666666666664))) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 5.2e+49], N[(N[(y$95$m / x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.19999999999999977e49

   1. Initial program 90.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      15. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      16. lower-*.f64 82.2

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 82.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right)} + \frac{1}{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right)}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right)\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \cdot \frac{y}{x}}{z} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot y}{x}}}{z} \]

   7. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z} \cdot \frac{y}{x}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}}{z} \cdot \frac{y}{x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{z} \cdot \frac{y}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{z} \cdot \frac{y}{x} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{z} \cdot \frac{y}{x} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{z} \cdot \frac{y}{x} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{z} \cdot \frac{y}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{z} \cdot \frac{y}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{z} \cdot \frac{y}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{z} \cdot \frac{y}{x} \]

      9. lower-*.f64 80.0

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{z} \cdot \frac{y}{x} \]

   10. Simplified 80.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}}{z} \cdot \frac{y}{x} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\color{blue}{1 + \frac{1}{2} \cdot {x}^{2}}}{z} \cdot \frac{y}{x} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot {x}^{2} + 1}}{z} \cdot \frac{y}{x} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1}{z} \cdot \frac{y}{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)}}{z} \cdot \frac{y}{x} \]

       4. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right)}{z} \cdot \frac{y}{x} \]

       5. lower-*.f64 76.4

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right)}{z} \cdot \frac{y}{x} \]

   13. Simplified 76.4%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}}{z} \cdot \frac{y}{x} \]

   if 5.19999999999999977e49 < x

   1. Initial program 72.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{1}{\frac{x}{y}}}}{z} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot y\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right) \cdot y}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right) \cdot y}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \cdot y}{z} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

      9. lower-/.f64 100.0

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x} \cdot y}}{z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \cdot y}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \cdot y}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot y}{z} \]

      10. lower-*.f64 92.9

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot y}{z} \]

   7. Simplified 92.9%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}} \cdot y}{z} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)} \cdot y}{z} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot \frac{1}{24}\right)} \cdot y}{z} \]

      2. cube-mult N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \frac{1}{24}\right) \cdot y}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{1}{24}\right) \cdot y}{z} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left({x}^{2} \cdot \frac{1}{24}\right)\right)} \cdot y}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right) \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}\right) \cdot y}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}\right) \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24}\right)\right) \cdot y}{z} \]

      10. lower-*.f64 91.1

          \[\leadsto \frac{\left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.041666666666666664\right)\right) \cdot y}{z} \]

   10. Simplified 91.1%

       \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 76.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, x \cdot 0.5, \frac{y\_m}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 2.25)
    (/ (fma y_m (* x 0.5) (/ y_m x)) z)
    (/ (* y_m (* x (fma (* x x) 0.041666666666666664 0.5))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.25) {
		tmp = fma(y_m, (x * 0.5), (y_m / x)) / z;
	} else {
		tmp = (y_m * (x * fma((x * x), 0.041666666666666664, 0.5))) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.25)
		tmp = Float64(fma(y_m, Float64(x * 0.5), Float64(y_m / x)) / z);
	else
		tmp = Float64(Float64(y_m * Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5))) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.25], N[(N[(y$95$m * N[(x * 0.5), $MachinePrecision] + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   1. Initial program 89.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   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 74.0

          \[\leadsto \frac{\mathsf{fma}\left(y, x \cdot 0.5, \color{blue}{\frac{y}{x}}\right)}{z} \]

   5. Simplified 74.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x \cdot 0.5, \frac{y}{x}\right)}}{z} \]

   if 2.25 < x

   1. Initial program 77.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{1}{\frac{x}{y}}}}{z} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot y\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right) \cdot y}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right) \cdot y}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \cdot y}{z} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

      9. lower-/.f64 100.0

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x} \cdot y}}{z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \cdot y}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \cdot y}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot y}{z} \]

      10. lower-*.f64 84.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot y}{z} \]

   7. Simplified 84.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}} \cdot y}{z} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot y}{z} \]
   9. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot y}{z} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot y}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \cdot y}{z} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right)}\right) \cdot y}{z} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right) \cdot x\right)} \cdot y}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)\right)} \cdot x\right) \cdot y}{z} \]

      7. lft-mult-inverse N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \left(\frac{1}{2} \cdot \color{blue}{1}\right) \cdot x\right) \cdot y}{z} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \color{blue}{\frac{1}{2}} \cdot x\right) \cdot y}{z} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)\right)} \cdot y}{z} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right) \cdot y}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot y}{z} \]

      12. +-commutative N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)}\right) \cdot y}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}\right)\right) \cdot y}{z} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}\right) \cdot y}{z} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right)\right) \cdot y}{z} \]

      16. lower-*.f64 82.8

          \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right)\right) \cdot y}{z} \]

   10. Simplified 82.8%

       \[\leadsto \frac{\color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)} \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x \cdot 0.5, \frac{y}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 76.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{y\_m \cdot \mathsf{fma}\left(x, 0.5, \frac{1}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 2.25)
    (/ (* y_m (fma x 0.5 (/ 1.0 x))) z)
    (/ (* y_m (* x (fma (* x x) 0.041666666666666664 0.5))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.25) {
		tmp = (y_m * fma(x, 0.5, (1.0 / x))) / z;
	} else {
		tmp = (y_m * (x * fma((x * x), 0.041666666666666664, 0.5))) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.25)
		tmp = Float64(Float64(y_m * fma(x, 0.5, Float64(1.0 / x))) / z);
	else
		tmp = Float64(Float64(y_m * Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5))) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.25], N[(N[(y$95$m * N[(x * 0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   1. Initial program 89.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   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 74.0

          \[\leadsto \frac{\mathsf{fma}\left(y, x \cdot 0.5, \color{blue}{\frac{y}{x}}\right)}{z} \]

   5. Simplified 74.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x \cdot 0.5, \frac{y}{x}\right)}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(x \cdot \frac{1}{2}\right)} + \frac{y}{x}}{z} \]

      2. div-inv N/A

         \[\leadsto \frac{y \cdot \left(x \cdot \frac{1}{2}\right) + \color{blue}{y \cdot \frac{1}{x}}}{z} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot \frac{1}{2} + \frac{1}{x}\right)}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot \frac{1}{2} + \frac{1}{x}\right)}}{z} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{x \cdot \frac{1}{2}} + \frac{1}{x}\right)}{z} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \frac{1}{x}\right)}}{z} \]

      7. lower-/.f64 74.0

         \[\leadsto \frac{y \cdot \mathsf{fma}\left(x, 0.5, \color{blue}{\frac{1}{x}}\right)}{z} \]

   7. Applied egg-rr 74.0%

      \[\leadsto \frac{\color{blue}{y \cdot \mathsf{fma}\left(x, 0.5, \frac{1}{x}\right)}}{z} \]

   if 2.25 < x

   1. Initial program 77.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{1}{\frac{x}{y}}}}{z} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot y\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right) \cdot y}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right) \cdot y}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \cdot y}{z} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

      9. lower-/.f64 100.0

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x} \cdot y}}{z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \cdot y}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \cdot y}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot y}{z} \]

      10. lower-*.f64 84.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot y}{z} \]

   7. Simplified 84.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}} \cdot y}{z} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot y}{z} \]
   9. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot y}{z} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot y}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \cdot y}{z} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right)}\right) \cdot y}{z} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right) \cdot x\right)} \cdot y}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)\right)} \cdot x\right) \cdot y}{z} \]

      7. lft-mult-inverse N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \left(\frac{1}{2} \cdot \color{blue}{1}\right) \cdot x\right) \cdot y}{z} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \color{blue}{\frac{1}{2}} \cdot x\right) \cdot y}{z} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)\right)} \cdot y}{z} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right) \cdot y}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot y}{z} \]

      12. +-commutative N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)}\right) \cdot y}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}\right)\right) \cdot y}{z} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}\right) \cdot y}{z} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right)\right) \cdot y}{z} \]

      16. lower-*.f64 82.8

          \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right)\right) \cdot y}{z} \]

   10. Simplified 82.8%

       \[\leadsto \frac{\color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)} \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(x, 0.5, \frac{1}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 78.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, \left(x \cdot x\right) \cdot 0.5, y\_m\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 2.25)
    (/ (fma y_m (* (* x x) 0.5) y_m) (* x z))
    (/ (* y_m (* x (fma (* x x) 0.041666666666666664 0.5))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.25) {
		tmp = fma(y_m, ((x * x) * 0.5), y_m) / (x * z);
	} else {
		tmp = (y_m * (x * fma((x * x), 0.041666666666666664, 0.5))) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.25)
		tmp = Float64(fma(y_m, Float64(Float64(x * x) * 0.5), y_m) / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5))) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.25], N[(N[(y$95$m * N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   1. Initial program 89.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. Simplified 73.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 0.5 \cdot \left(x \cdot x\right), y\right)}{x \cdot z}} \]

   if 2.25 < x

   1. Initial program 77.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{1}{\frac{x}{y}}}}{z} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot y\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right) \cdot y}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right) \cdot y}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \cdot y}{z} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

      9. lower-/.f64 100.0

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x} \cdot y}}{z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \cdot y}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \cdot y}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot y}{z} \]

      10. lower-*.f64 84.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot y}{z} \]

   7. Simplified 84.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}} \cdot y}{z} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot y}{z} \]
   9. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot y}{z} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot y}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \cdot y}{z} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right)}\right) \cdot y}{z} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right) \cdot x\right)} \cdot y}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)\right)} \cdot x\right) \cdot y}{z} \]

      7. lft-mult-inverse N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \left(\frac{1}{2} \cdot \color{blue}{1}\right) \cdot x\right) \cdot y}{z} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \color{blue}{\frac{1}{2}} \cdot x\right) \cdot y}{z} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)\right)} \cdot y}{z} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right) \cdot y}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot y}{z} \]

      12. +-commutative N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)}\right) \cdot y}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}\right)\right) \cdot y}{z} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}\right) \cdot y}{z} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right)\right) \cdot y}{z} \]

      16. lower-*.f64 82.8

          \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right)\right) \cdot y}{z} \]

   10. Simplified 82.8%

       \[\leadsto \frac{\color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)} \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \left(x \cdot x\right) \cdot 0.5, y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 67.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 0.88)
    (/ (/ y_m x) z)
    (/ (* y_m (* x (fma (* x x) 0.041666666666666664 0.5))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.88) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (y_m * (x * fma((x * x), 0.041666666666666664, 0.5))) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(Float64(y_m * Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5))) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 0.88], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 89.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 65.4

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   5. Simplified 65.4%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   if 0.880000000000000004 < x

   1. Initial program 77.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{1}{\frac{x}{y}}}}{z} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot y\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right) \cdot y}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right) \cdot y}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \cdot y}{z} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

      9. lower-/.f64 100.0

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x} \cdot y}}{z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \cdot y}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \cdot y}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot y}{z} \]

      10. lower-*.f64 84.4

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot y}{z} \]

   7. Simplified 84.4%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}} \cdot y}{z} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot y}{z} \]
   9. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot y}{z} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot y}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \cdot y}{z} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right)}\right) \cdot y}{z} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right) \cdot x\right)} \cdot y}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)\right)} \cdot x\right) \cdot y}{z} \]

      7. lft-mult-inverse N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \left(\frac{1}{2} \cdot \color{blue}{1}\right) \cdot x\right) \cdot y}{z} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x + \color{blue}{\frac{1}{2}} \cdot x\right) \cdot y}{z} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)\right)} \cdot y}{z} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right) \cdot y}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot y}{z} \]

      12. +-commutative N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)}\right) \cdot y}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}\right)\right) \cdot y}{z} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}\right) \cdot y}{z} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right)\right) \cdot y}{z} \]

      16. lower-*.f64 83.0

          \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right)\right) \cdot y}{z} \]

   10. Simplified 83.0%

       \[\leadsto \frac{\color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)} \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 67.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 2.25)
    (/ (/ y_m x) z)
    (/ (* y_m (* x (* (* x x) 0.041666666666666664))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.25) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (y_m * (x * ((x * x) * 0.041666666666666664))) / z;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.25d0) then
        tmp = (y_m / x) / z
    else
        tmp = (y_m * (x * ((x * x) * 0.041666666666666664d0))) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.25) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (y_m * (x * ((x * x) * 0.041666666666666664))) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 2.25:
		tmp = (y_m / x) / z
	else:
		tmp = (y_m * (x * ((x * x) * 0.041666666666666664))) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.25)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(Float64(y_m * Float64(x * Float64(Float64(x * x) * 0.041666666666666664))) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 2.25)
		tmp = (y_m / x) / z;
	else
		tmp = (y_m * (x * ((x * x) * 0.041666666666666664))) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.25], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   1. Initial program 89.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
   4. Step-by-step derivation

      1. lower-/.f64 65.6

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   5. Simplified 65.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   if 2.25 < x

   1. Initial program 77.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{1}{\frac{x}{y}}}}{z} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot y\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right) \cdot y}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right) \cdot y}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \cdot y}{z} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

      9. lower-/.f64 100.0

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x} \cdot y}}{z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \cdot y}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \cdot y}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot y}{z} \]

      10. lower-*.f64 84.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot y}{z} \]

   7. Simplified 84.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}} \cdot y}{z} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)} \cdot y}{z} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot \frac{1}{24}\right)} \cdot y}{z} \]

      2. cube-mult N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \frac{1}{24}\right) \cdot y}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{1}{24}\right) \cdot y}{z} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left({x}^{2} \cdot \frac{1}{24}\right)\right)} \cdot y}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right) \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}\right) \cdot y}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}\right) \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24}\right)\right) \cdot y}{z} \]

      10. lower-*.f64 82.8

          \[\leadsto \frac{\left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.041666666666666664\right)\right) \cdot y}{z} \]

   10. Simplified 82.8%

       \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 67.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 2.25)
    (/ (/ y_m x) z)
    (* y_m (/ (* 0.041666666666666664 (* x (* x x))) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.25) {
		tmp = (y_m / x) / z;
	} else {
		tmp = y_m * ((0.041666666666666664 * (x * (x * x))) / z);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.25d0) then
        tmp = (y_m / x) / z
    else
        tmp = y_m * ((0.041666666666666664d0 * (x * (x * x))) / z)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.25) {
		tmp = (y_m / x) / z;
	} else {
		tmp = y_m * ((0.041666666666666664 * (x * (x * x))) / z);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 2.25:
		tmp = (y_m / x) / z
	else:
		tmp = y_m * ((0.041666666666666664 * (x * (x * x))) / z)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.25)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(y_m * Float64(Float64(0.041666666666666664 * Float64(x * Float64(x * x))) / z));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 2.25)
		tmp = (y_m / x) / z;
	else
		tmp = y_m * ((0.041666666666666664 * (x * (x * x))) / z);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.25], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(0.041666666666666664 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   1. Initial program 89.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
   4. Step-by-step derivation

      1. lower-/.f64 65.6

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   5. Simplified 65.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   if 2.25 < x

   1. Initial program 77.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 63.3

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 63.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot y\right) \cdot \frac{1}{24}}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(y \cdot \frac{1}{24}\right)}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{{x}^{3} \cdot \color{blue}{\left(\frac{1}{24} \cdot y\right)}}{z} \]

      4. unpow3 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(\frac{1}{24} \cdot y\right)}{z} \]

      5. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(\frac{1}{24} \cdot y\right)}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{1}{24} \cdot y\right)\right)}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{1}{24} \cdot y\right)\right)}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(\frac{1}{24} \cdot y\right)\right)}{z} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(\frac{1}{24} \cdot y\right)\right)}{z} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot y\right)\right)}}{z} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(y \cdot \frac{1}{24}\right)}\right)}{z} \]

      12. lower-*.f64 82.8

          \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(y \cdot 0.041666666666666664\right)}\right)}{z} \]

   8. Simplified 82.8%

      \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(y \cdot 0.041666666666666664\right)\right)}}{z} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(y \cdot \frac{1}{24}\right)\right)}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(y \cdot \frac{1}{24}\right)}\right)}{z} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(y \cdot \frac{1}{24}\right)\right)}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(y \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(y \cdot \frac{1}{24}\right)\right)}}{z} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(y \cdot \frac{1}{24}\right)\right)}}{z} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(y \cdot \frac{1}{24}\right)}}{z} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \cdot \left(y \cdot \frac{1}{24}\right)}{z} \]

      9. pow3 N/A

         \[\leadsto \frac{\color{blue}{{x}^{3}} \cdot \left(y \cdot \frac{1}{24}\right)}{z} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{x}^{3} \cdot \color{blue}{\left(y \cdot \frac{1}{24}\right)}}{z} \]

      11. *-commutative N/A

          \[\leadsto \frac{{x}^{3} \cdot \color{blue}{\left(\frac{1}{24} \cdot y\right)}}{z} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot \frac{1}{24}\right) \cdot y}}{z} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\left({x}^{3} \cdot \frac{1}{24}\right) \cdot y}{\color{blue}{z \cdot 1}} \]

      14. times-frac N/A

          \[\leadsto \color{blue}{\frac{{x}^{3} \cdot \frac{1}{24}}{z} \cdot \frac{y}{1}} \]

      15. /-rgt-identity N/A

          \[\leadsto \frac{{x}^{3} \cdot \frac{1}{24}}{z} \cdot \color{blue}{y} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{{x}^{3} \cdot \frac{1}{24}}{z} \cdot y} \]

      17. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{x}^{3} \cdot \frac{1}{24}}{z}} \cdot y \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \frac{1}{24}}}{z} \cdot y \]

      19. cube-unmult N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \frac{1}{24}}{z} \cdot y \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \frac{1}{24}}{z} \cdot y \]

      21. lower-*.f64 81.4

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot 0.041666666666666664}{z} \cdot y \]

   10. Applied egg-rr 81.4%

       \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664}{z} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 66.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x \cdot \frac{\left(x \cdot x\right) \cdot 0.041666666666666664}{z}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 2.25)
    (/ (/ y_m x) z)
    (* y_m (* x (/ (* (* x x) 0.041666666666666664) z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.25) {
		tmp = (y_m / x) / z;
	} else {
		tmp = y_m * (x * (((x * x) * 0.041666666666666664) / z));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.25d0) then
        tmp = (y_m / x) / z
    else
        tmp = y_m * (x * (((x * x) * 0.041666666666666664d0) / z))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.25) {
		tmp = (y_m / x) / z;
	} else {
		tmp = y_m * (x * (((x * x) * 0.041666666666666664) / z));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 2.25:
		tmp = (y_m / x) / z
	else:
		tmp = y_m * (x * (((x * x) * 0.041666666666666664) / z))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.25)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(y_m * Float64(x * Float64(Float64(Float64(x * x) * 0.041666666666666664) / z)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 2.25)
		tmp = (y_m / x) / z;
	else
		tmp = y_m * (x * (((x * x) * 0.041666666666666664) / z));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.25], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(x * N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   1. Initial program 89.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
   4. Step-by-step derivation

      1. lower-/.f64 65.6

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   5. Simplified 65.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   if 2.25 < x

   1. Initial program 77.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{1}{\frac{x}{y}}}}{z} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot y\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right) \cdot y}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \cdot y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} \cdot \cosh x\right) \cdot y}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \cdot y}{z} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

      9. lower-/.f64 100.0

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x}} \cdot y}{z} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x} \cdot y}}{z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \cdot y}{z} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \cdot y}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \cdot y}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \cdot y}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \cdot y}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot y}{z} \]

      10. lower-*.f64 84.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot y}{z} \]

   7. Simplified 84.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}} \cdot y}{z} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}{z}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot y}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{24} \cdot {x}^{3}}{z}} \]

      5. unpow3 N/A

         \[\leadsto y \cdot \frac{\frac{1}{24} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{z} \]

      6. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{1}{24} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right)}{z} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\frac{1}{24} \cdot {x}^{2}}{z} \cdot x\right)} \]

      9. associate-*r/ N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot \frac{{x}^{2}}{z}\right)} \cdot x\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(\left(\frac{1}{24} \cdot \frac{{x}^{2}}{z}\right) \cdot x\right)} \]

      11. associate-*r/ N/A

          \[\leadsto y \cdot \left(\color{blue}{\frac{\frac{1}{24} \cdot {x}^{2}}{z}} \cdot x\right) \]

      12. associate-*l/ N/A

          \[\leadsto y \cdot \color{blue}{\frac{\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x}{z}} \]

      13. associate-*r* N/A

          \[\leadsto y \cdot \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{2} \cdot x\right)}}{z} \]

      14. unpow2 N/A

          \[\leadsto y \cdot \frac{\frac{1}{24} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right)}{z} \]

      15. unpow3 N/A

          \[\leadsto y \cdot \frac{\frac{1}{24} \cdot \color{blue}{{x}^{3}}}{z} \]

      16. *-commutative N/A

          \[\leadsto y \cdot \frac{\color{blue}{{x}^{3} \cdot \frac{1}{24}}}{z} \]

      17. cube-mult N/A

          \[\leadsto y \cdot \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \frac{1}{24}}{z} \]

      18. unpow2 N/A

          \[\leadsto y \cdot \frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{1}{24}}{z} \]

      19. associate-*l* N/A

          \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left({x}^{2} \cdot \frac{1}{24}\right)}}{z} \]

      20. *-commutative N/A

          \[\leadsto y \cdot \frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}}{z} \]

      21. associate-*r/ N/A

          \[\leadsto y \cdot \color{blue}{\left(x \cdot \frac{\frac{1}{24} \cdot {x}^{2}}{z}\right)} \]

   10. Simplified 81.4%

       \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{\left(x \cdot x\right) \cdot 0.041666666666666664}{z}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 29: 65.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \frac{y\_m \cdot \left(x \cdot x\right)}{z}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 2.25)
    (/ (/ y_m x) z)
    (* x (* 0.041666666666666664 (/ (* y_m (* x x)) z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.25) {
		tmp = (y_m / x) / z;
	} else {
		tmp = x * (0.041666666666666664 * ((y_m * (x * x)) / z));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.25d0) then
        tmp = (y_m / x) / z
    else
        tmp = x * (0.041666666666666664d0 * ((y_m * (x * x)) / z))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.25) {
		tmp = (y_m / x) / z;
	} else {
		tmp = x * (0.041666666666666664 * ((y_m * (x * x)) / z));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 2.25:
		tmp = (y_m / x) / z
	else:
		tmp = x * (0.041666666666666664 * ((y_m * (x * x)) / z))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.25)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(x * Float64(0.041666666666666664 * Float64(Float64(y_m * Float64(x * x)) / z)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 2.25)
		tmp = (y_m / x) / z;
	else
		tmp = x * (0.041666666666666664 * ((y_m * (x * x)) / z));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.25], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(0.041666666666666664 * N[(N[(y$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   1. Initial program 89.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
   4. Step-by-step derivation

      1. lower-/.f64 65.6

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   5. Simplified 65.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   if 2.25 < x

   1. Initial program 77.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 63.3

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 63.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} \cdot y}{z} \cdot \frac{1}{24}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left({x}^{3} \cdot \frac{y}{z}\right)} \cdot \frac{1}{24} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{y}{z} \cdot \frac{1}{24}\right)} \]

      4. *-commutative N/A

         \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{y}{z}\right)} \]

      5. cube-mult N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{24} \cdot \frac{y}{z}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{24} \cdot \frac{y}{z}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{y}{z}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{y}{z}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{y}{z} \cdot \frac{1}{24}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left({x}^{2} \cdot \frac{y}{z}\right) \cdot \frac{1}{24}\right)} \]

      11. associate-/l* N/A

          \[\leadsto x \cdot \left(\color{blue}{\frac{{x}^{2} \cdot y}{z}} \cdot \frac{1}{24}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{{x}^{2} \cdot y}{z} \cdot \frac{1}{24}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto x \cdot \left(\color{blue}{\frac{{x}^{2} \cdot y}{z}} \cdot \frac{1}{24}\right) \]

      14. *-commutative N/A

          \[\leadsto x \cdot \left(\frac{\color{blue}{y \cdot {x}^{2}}}{z} \cdot \frac{1}{24}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{\color{blue}{y \cdot {x}^{2}}}{z} \cdot \frac{1}{24}\right) \]

      16. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{y \cdot \color{blue}{\left(x \cdot x\right)}}{z} \cdot \frac{1}{24}\right) \]

      17. lower-*.f64 81.4

          \[\leadsto x \cdot \left(\frac{y \cdot \color{blue}{\left(x \cdot x\right)}}{z} \cdot 0.041666666666666664\right) \]

   8. Simplified 81.4%

      \[\leadsto \color{blue}{x \cdot \left(\frac{y \cdot \left(x \cdot x\right)}{z} \cdot 0.041666666666666664\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \frac{y \cdot \left(x \cdot x\right)}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 57.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 0.88) (/ (/ y_m x) z) (* y_m (* x (/ 0.5 z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.88) {
		tmp = (y_m / x) / z;
	} else {
		tmp = y_m * (x * (0.5 / z));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.88d0) then
        tmp = (y_m / x) / z
    else
        tmp = y_m * (x * (0.5d0 / z))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.88) {
		tmp = (y_m / x) / z;
	} else {
		tmp = y_m * (x * (0.5 / z));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 0.88:
		tmp = (y_m / x) / z
	else:
		tmp = y_m * (x * (0.5 / z))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(y_m * Float64(x * Float64(0.5 / z)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 0.88)
		tmp = (y_m / x) / z;
	else
		tmp = y_m * (x * (0.5 / z));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 0.88], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(x * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 89.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 65.4

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   5. Simplified 65.4%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   if 0.880000000000000004 < x

   1. Initial program 77.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 39.6

          \[\leadsto \frac{\mathsf{fma}\left(y, x \cdot 0.5, \color{blue}{\frac{y}{x}}\right)}{z} \]

   5. Simplified 39.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x \cdot 0.5, \frac{y}{x}\right)}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{z} \cdot \frac{1}{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{z}\right)} \cdot \frac{1}{2} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{y}{z} \cdot \frac{1}{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{y}{z}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \frac{y}{z}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{2} \cdot y}{z}} \]

      7. *-commutative N/A

         \[\leadsto x \cdot \frac{\color{blue}{y \cdot \frac{1}{2}}}{z} \]

      8. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{\frac{1}{2}}{z}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{\frac{1}{2}}{z}\right)} \]

      10. lower-/.f64 35.5

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\frac{0.5}{z}}\right) \]

   8. Simplified 35.5%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{z}\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x \cdot \left(y \cdot \color{blue}{\frac{\frac{1}{2}}{z}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{\frac{1}{2}}{z}\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot \frac{\frac{1}{2}}{z}\right) \cdot x} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot \frac{\frac{1}{2}}{z}\right)} \cdot x \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{\frac{1}{2}}{z} \cdot x\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{\frac{1}{2}}{z} \cdot x\right)} \]

      7. lower-*.f64 50.7

         \[\leadsto y \cdot \color{blue}{\left(\frac{0.5}{z} \cdot x\right)} \]

   10. Applied egg-rr 50.7%

       \[\leadsto \color{blue}{y \cdot \left(\frac{0.5}{z} \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 31: 57.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{y\_m}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 0.88) (/ y_m (* x z)) (* y_m (* x (/ 0.5 z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.88) {
		tmp = y_m / (x * z);
	} else {
		tmp = y_m * (x * (0.5 / z));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.88d0) then
        tmp = y_m / (x * z)
    else
        tmp = y_m * (x * (0.5d0 / z))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.88) {
		tmp = y_m / (x * z);
	} else {
		tmp = y_m * (x * (0.5 / z));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 0.88:
		tmp = y_m / (x * z)
	else:
		tmp = y_m * (x * (0.5 / z))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(y_m / Float64(x * z));
	else
		tmp = Float64(y_m * Float64(x * Float64(0.5 / z)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 0.88)
		tmp = y_m / (x * z);
	else
		tmp = y_m * (x * (0.5 / z));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 0.88], N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 89.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. lower-*.f64 65.3

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

   5. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 0.880000000000000004 < x

   1. Initial program 77.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 39.6

          \[\leadsto \frac{\mathsf{fma}\left(y, x \cdot 0.5, \color{blue}{\frac{y}{x}}\right)}{z} \]

   5. Simplified 39.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x \cdot 0.5, \frac{y}{x}\right)}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{z} \cdot \frac{1}{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{z}\right)} \cdot \frac{1}{2} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{y}{z} \cdot \frac{1}{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{y}{z}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \frac{y}{z}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{2} \cdot y}{z}} \]

      7. *-commutative N/A

         \[\leadsto x \cdot \frac{\color{blue}{y \cdot \frac{1}{2}}}{z} \]

      8. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{\frac{1}{2}}{z}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{\frac{1}{2}}{z}\right)} \]

      10. lower-/.f64 35.5

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\frac{0.5}{z}}\right) \]

   8. Simplified 35.5%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{z}\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x \cdot \left(y \cdot \color{blue}{\frac{\frac{1}{2}}{z}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{\frac{1}{2}}{z}\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot \frac{\frac{1}{2}}{z}\right) \cdot x} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot \frac{\frac{1}{2}}{z}\right)} \cdot x \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{\frac{1}{2}}{z} \cdot x\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{\frac{1}{2}}{z} \cdot x\right)} \]

      7. lower-*.f64 50.7

         \[\leadsto y \cdot \color{blue}{\left(\frac{0.5}{z} \cdot x\right)} \]

   10. Applied egg-rr 50.7%

       \[\leadsto \color{blue}{y \cdot \left(\frac{0.5}{z} \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 32: 55.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{y\_m}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y\_m \cdot \frac{0.5}{z}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 0.88) (/ y_m (* x z)) (* x (* y_m (/ 0.5 z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.88) {
		tmp = y_m / (x * z);
	} else {
		tmp = x * (y_m * (0.5 / z));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.88d0) then
        tmp = y_m / (x * z)
    else
        tmp = x * (y_m * (0.5d0 / z))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.88) {
		tmp = y_m / (x * z);
	} else {
		tmp = x * (y_m * (0.5 / z));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 0.88:
		tmp = y_m / (x * z)
	else:
		tmp = x * (y_m * (0.5 / z))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(y_m / Float64(x * z));
	else
		tmp = Float64(x * Float64(y_m * Float64(0.5 / z)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 0.88)
		tmp = y_m / (x * z);
	else
		tmp = x * (y_m * (0.5 / z));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 0.88], N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(x * N[(y$95$m * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 89.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. lower-*.f64 65.3

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

   5. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 0.880000000000000004 < x

   1. Initial program 77.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 39.6

          \[\leadsto \frac{\mathsf{fma}\left(y, x \cdot 0.5, \color{blue}{\frac{y}{x}}\right)}{z} \]

   5. Simplified 39.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x \cdot 0.5, \frac{y}{x}\right)}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{z} \cdot \frac{1}{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{z}\right)} \cdot \frac{1}{2} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{y}{z} \cdot \frac{1}{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{y}{z}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \frac{y}{z}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{2} \cdot y}{z}} \]

      7. *-commutative N/A

         \[\leadsto x \cdot \frac{\color{blue}{y \cdot \frac{1}{2}}}{z} \]

      8. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{\frac{1}{2}}{z}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{\frac{1}{2}}{z}\right)} \]

      10. lower-/.f64 35.5

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\frac{0.5}{z}}\right) \]

   8. Simplified 35.5%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{z}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 33: 49.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \frac{y\_m}{x \cdot z} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (/ y_m (* x z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (y_m / (x * z));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (y_m / (x * z))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (y_m / (x * z));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * (y_m / (x * z))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(y_m / Float64(x * z)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (y_m / (x * z));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.4%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
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 51.7

      \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

5. Simplified 51.7%

   \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
6. Add Preprocessing

Developer Target 1: 97.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y z) x) (cosh x))))
   (if (< y -4.618902267687042e-52)
     t_0
     (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y / z) / x) * cosh(x)
    if (y < (-4.618902267687042d-52)) then
        tmp = t_0
    else if (y < 1.038530535935153d-39) then
        tmp = ((cosh(x) * y) / x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * Math.cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((Math.cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / z) / x) * math.cosh(x)
	tmp = 0
	if y < -4.618902267687042e-52:
		tmp = t_0
	elif y < 1.038530535935153e-39:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) / x) * cosh(x))
	tmp = 0.0
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / z) / x) * cosh(x);
	tmp = 0.0;
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -4.618902267687042e-52], t$95$0, If[Less[y, 1.038530535935153e-39], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024212 
(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))