Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.7% → 99.7%

Time: 15.2s

Alternatives: 27

Speedup: 2.3×

• 63.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\cosh x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{y\_m \cdot t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{y\_m}{z}\\ \end{array} \end{array} \end{array} \]

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 (/ (cosh x) x)))
   (* y_s (if (<= y_m 5e-29) (/ (* y_m t_0) z) (* t_0 (/ y_m 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 = cosh(x) / x;
	double tmp;
	if (y_m <= 5e-29) {
		tmp = (y_m * t_0) / z;
	} else {
		tmp = t_0 * (y_m / 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) :: t_0
    real(8) :: tmp
    t_0 = cosh(x) / x
    if (y_m <= 5d-29) then
        tmp = (y_m * t_0) / z
    else
        tmp = t_0 * (y_m / 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 t_0 = Math.cosh(x) / x;
	double tmp;
	if (y_m <= 5e-29) {
		tmp = (y_m * t_0) / z;
	} else {
		tmp = t_0 * (y_m / 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):
	t_0 = math.cosh(x) / x
	tmp = 0
	if y_m <= 5e-29:
		tmp = (y_m * t_0) / z
	else:
		tmp = t_0 * (y_m / 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(cosh(x) / x)
	tmp = 0.0
	if (y_m <= 5e-29)
		tmp = Float64(Float64(y_m * t_0) / z);
	else
		tmp = Float64(t_0 * Float64(y_m / 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)
	t_0 = cosh(x) / x;
	tmp = 0.0;
	if (y_m <= 5e-29)
		tmp = (y_m * t_0) / z;
	else
		tmp = t_0 * (y_m / 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_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 5e-29], N[(N[(y$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(t$95$0 * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999986e-29

   1. Initial program 79.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. div-inv N/A

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

      8. /-lowering-/.f64 N/A

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

      9. cosh-lowering-cosh.f64 97.3

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

   4. Applied egg-rr 97.3%

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

   if 4.99999999999999986e-29 < y

   1. Initial program 95.8%

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. div-inv N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. div-inv N/A

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

      11. /-lowering-/.f64 N/A

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

      12. cosh-lowering-cosh.f64 99.8

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y\_m}{x}}{z}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \left(x \cdot x\right) \cdot 0.001388888888888889, 1\right)}{z}}{x}\\ \end{array} \end{array} \end{array} \]

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)) z)))
   (*
    y_s
    (if (<= t_0 2e+66)
      t_0
      (if (<= t_0 INFINITY)
        (* (/ (cosh x) x) (/ y_m z))
        (/
         (*
          y_m
          (/ (fma (* x (* x (* x x))) (* (* x x) 0.001388888888888889) 1.0) 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 t_0 = (cosh(x) * (y_m / x)) / z;
	double tmp;
	if (t_0 <= 2e+66) {
		tmp = t_0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (cosh(x) / x) * (y_m / z);
	} else {
		tmp = (y_m * (fma((x * (x * (x * x))), ((x * x) * 0.001388888888888889), 1.0) / 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)
	t_0 = Float64(Float64(cosh(x) * Float64(y_m / x)) / z)
	tmp = 0.0
	if (t_0 <= 2e+66)
		tmp = t_0;
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(cosh(x) / x) * Float64(y_m / z));
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * Float64(x * Float64(x * x))), Float64(Float64(x * x) * 0.001388888888888889), 1.0) / 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_] := Block[{t$95$0 = N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 2e+66], t$95$0, If[LessEqual[t$95$0, Infinity], N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.99999999999999989e66

   1. Initial program 97.9%

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

   if 1.99999999999999989e66 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < +inf.0

   1. Initial program 94.0%

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. div-inv N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. div-inv N/A

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

      11. /-lowering-/.f64 N/A

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

      12. cosh-lowering-cosh.f64 98.6

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

   4. Applied egg-rr 98.6%

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

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

   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 0.0

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

   5. Simplified 0.0%

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

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 97.1%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 97.1

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

   10. Simplified 97.1%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. /-lowering-/.f64 N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. associate-*l* N/A

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

       9. cube-unmult N/A

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

       10. *-lowering-*.f64 N/A

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

       11. cube-unmult N/A

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

       12. *-lowering-*.f64 N/A

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

       13. *-lowering-*.f64 N/A

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

       14. *-lowering-*.f64 N/A

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

       15. *-lowering-*.f64 100.0

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{+66}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq \infty:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \left(x \cdot x\right) \cdot 0.001388888888888889, 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 2 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{y\_m}{x} \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right), 1\right)}{z}}{x}\\ \end{array} \end{array} \]

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 (<= (/ (* (cosh x) (/ y_m x)) z) 2e+33)
    (/
     (*
      (/ y_m x)
      (fma
       x
       (*
        x
        (fma
         (* x x)
         (fma (* x x) 0.001388888888888889 0.041666666666666664)
         0.5))
       1.0))
     z)
    (/
     (/
      (* y_m (fma x (* x (* (* x x) (* (* x x) 0.001388888888888889))) 1.0))
      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 (((cosh(x) * (y_m / x)) / z) <= 2e+33) {
		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 * fma(x, (x * ((x * x) * ((x * x) * 0.001388888888888889))), 1.0)) / 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 (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 2e+33)
		tmp = Float64(Float64(Float64(y_m / x) * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0)) / z);
	else
		tmp = Float64(Float64(Float64(y_m * fma(x, Float64(x * Float64(Float64(x * x) * Float64(Float64(x * x) * 0.001388888888888889))), 1.0)) / 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[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e+33], N[(N[(N[(y$95$m / x), $MachinePrecision] * 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] / z), $MachinePrecision], N[(N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.9999999999999999e33

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 93.3

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

   5. Simplified 93.3%

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

   if 1.9999999999999999e33 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 67.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 60.8

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

   5. Simplified 60.8%

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

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 93.3%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 93.3

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

   10. Simplified 93.3%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(y\_m \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), y\_m\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right), 1\right)}{z}}{x}\\ \end{array} \end{array} \]

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 (<= (/ (* (cosh x) (/ y_m x)) z) 5e+57)
    (/
     (fma
      x
      (*
       x
       (*
        y_m
        (fma
         (* x x)
         (fma (* x x) 0.001388888888888889 0.041666666666666664)
         0.5)))
      y_m)
     (* x z))
    (/
     (/
      (* y_m (fma x (* x (* (* x x) (* (* x x) 0.001388888888888889))) 1.0))
      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 (((cosh(x) * (y_m / x)) / z) <= 5e+57) {
		tmp = fma(x, (x * (y_m * fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5))), y_m) / (x * z);
	} else {
		tmp = ((y_m * fma(x, (x * ((x * x) * ((x * x) * 0.001388888888888889))), 1.0)) / 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 (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 5e+57)
		tmp = Float64(fma(x, Float64(x * Float64(y_m * fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5))), y_m) / Float64(x * z));
	else
		tmp = Float64(Float64(Float64(y_m * fma(x, Float64(x * Float64(Float64(x * x) * Float64(Float64(x * x) * 0.001388888888888889))), 1.0)) / 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[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e+57], N[(N[(x * N[(x * N[(y$95$m * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4.99999999999999972e57

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 85.8%

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

   if 4.99999999999999972e57 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 67.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 60.4

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

   5. Simplified 60.4%

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

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 93.2%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 93.2

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

   10. Simplified 93.2%

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

Alternative 5: 87.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(y\_m \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), y\_m\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \left(x \cdot x\right) \cdot 0.001388888888888889, 1\right)}{z}}{x}\\ \end{array} \end{array} \]

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 (<= (/ (* (cosh x) (/ y_m x)) z) 2e+60)
    (/
     (fma
      x
      (*
       x
       (*
        y_m
        (fma
         (* x x)
         (fma (* x x) 0.001388888888888889 0.041666666666666664)
         0.5)))
      y_m)
     (* x z))
    (/
     (*
      y_m
      (/ (fma (* x (* x (* x x))) (* (* x x) 0.001388888888888889) 1.0) 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 (((cosh(x) * (y_m / x)) / z) <= 2e+60) {
		tmp = fma(x, (x * (y_m * fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5))), y_m) / (x * z);
	} else {
		tmp = (y_m * (fma((x * (x * (x * x))), ((x * x) * 0.001388888888888889), 1.0) / 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 (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 2e+60)
		tmp = Float64(fma(x, Float64(x * Float64(y_m * fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5))), y_m) / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * Float64(x * Float64(x * x))), Float64(Float64(x * x) * 0.001388888888888889), 1.0) / 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[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e+60], N[(N[(x * N[(x * N[(y$95$m * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 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 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.9999999999999999e60

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 85.9%

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

   if 1.9999999999999999e60 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 66.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 60.1

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

   5. Simplified 60.1%

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

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 93.1%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 93.1

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

   10. Simplified 93.1%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. /-lowering-/.f64 N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. associate-*l* N/A

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

       9. cube-unmult N/A

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

       10. *-lowering-*.f64 N/A

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

       11. cube-unmult N/A

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

       12. *-lowering-*.f64 N/A

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

       13. *-lowering-*.f64 N/A

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

       14. *-lowering-*.f64 N/A

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

       15. *-lowering-*.f64 92.2

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

   12. Applied egg-rr 92.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \left(x \cdot x\right) \cdot 0.001388888888888889, 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(y\_m \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), y\_m\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{z}}{x}\\ \end{array} \end{array} \]

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 (<= (/ (* (cosh x) (/ y_m x)) z) 5e+57)
    (/
     (fma
      x
      (*
       x
       (*
        y_m
        (fma
         (* x x)
         (fma (* x x) 0.001388888888888889 0.041666666666666664)
         0.5)))
      y_m)
     (* x z))
    (/
     (/ (* y_m (fma x (* x (fma (* x x) 0.041666666666666664 0.5)) 1.0)) 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 (((cosh(x) * (y_m / x)) / z) <= 5e+57) {
		tmp = fma(x, (x * (y_m * fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5))), y_m) / (x * z);
	} else {
		tmp = ((y_m * fma(x, (x * fma((x * x), 0.041666666666666664, 0.5)), 1.0)) / 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 (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 5e+57)
		tmp = Float64(fma(x, Float64(x * Float64(y_m * fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5))), y_m) / Float64(x * z));
	else
		tmp = Float64(Float64(Float64(y_m * fma(x, Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5)), 1.0)) / 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[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e+57], N[(N[(x * N[(x * N[(y$95$m * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4.99999999999999972e57

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 85.8%

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

   if 4.99999999999999972e57 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 67.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 57.9

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

   5. Simplified 57.9%

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

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 89.8%

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

Alternative 7: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\frac{y\_m \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{z}}{x}\\ \end{array} \end{array} \]

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 (<= (/ (* (cosh x) (/ y_m x)) z) 5e+57)
    (/
     (*
      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)) 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 (((cosh(x) * (y_m / x)) / z) <= 5e+57) {
		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)) / 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 (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 5e+57)
		tmp = Float64(Float64(y_m * fma(x, Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664), 0.5)), 1.0)) / Float64(x * z));
	else
		tmp = Float64(Float64(Float64(y_m * fma(x, Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5)), 1.0)) / 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[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e+57], N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4.99999999999999972e57

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 93.4

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

   5. Simplified 93.4%

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

      1. associate-/l* N/A

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

      2. associate-/r* N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 87.2%

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

   if 4.99999999999999972e57 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 67.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 57.9

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

   5. Simplified 57.9%

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

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 89.8%

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

Alternative 8: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \frac{y\_m}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{z}}{x}\\ \end{array} \end{array} \]

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 (<= (/ (* (cosh x) (/ y_m x)) z) 5e+57)
    (*
     (fma
      x
      (*
       x
       (fma
        (* x x)
        (fma x (* x 0.001388888888888889) 0.041666666666666664)
        0.5))
      1.0)
     (/ y_m (* x z)))
    (/
     (/ (* y_m (fma x (* x (fma (* x x) 0.041666666666666664 0.5)) 1.0)) 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 (((cosh(x) * (y_m / x)) / z) <= 5e+57) {
		tmp = fma(x, (x * fma((x * x), fma(x, (x * 0.001388888888888889), 0.041666666666666664), 0.5)), 1.0) * (y_m / (x * z));
	} else {
		tmp = ((y_m * fma(x, (x * fma((x * x), 0.041666666666666664, 0.5)), 1.0)) / 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 (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 5e+57)
		tmp = Float64(fma(x, Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664), 0.5)), 1.0) * Float64(y_m / Float64(x * z)));
	else
		tmp = Float64(Float64(Float64(y_m * fma(x, Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5)), 1.0)) / 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[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e+57], N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4.99999999999999972e57

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 93.4

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

   5. Simplified 93.4%

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

      1. associate-/l* N/A

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

      2. associate-/r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 85.8

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

   7. Applied egg-rr 85.8%

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

   if 4.99999999999999972e57 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 67.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 57.9

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

   5. Simplified 57.9%

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

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 89.8%

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

Alternative 9: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 2 \cdot 10^{+33}:\\ \;\;\;\;y\_m \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{z}}{x}\\ \end{array} \end{array} \]

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 (<= (/ (* (cosh x) (/ y_m x)) z) 2e+33)
    (*
     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)) 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 (((cosh(x) * (y_m / x)) / z) <= 2e+33) {
		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)) / 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 (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 2e+33)
		tmp = Float64(y_m * Float64(fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0) / Float64(x * z)));
	else
		tmp = Float64(Float64(Float64(y_m * fma(x, Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5)), 1.0)) / 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[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e+33], N[(y$95$m * N[(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[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.9999999999999999e33

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 93.3

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

   5. Simplified 93.3%

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

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 88.4%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

   9. Applied egg-rr 85.7%

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

   if 1.9999999999999999e33 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 67.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 58.3

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

   5. Simplified 58.3%

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

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 89.9%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{+33}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 2 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{y\_m}{x} \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{z}}{x}\\ \end{array} \end{array} \]

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 (<= (/ (* (cosh x) (/ y_m x)) z) 2e+33)
    (/
     (* (/ y_m x) (fma x (* x (fma x (* x 0.041666666666666664) 0.5)) 1.0))
     z)
    (/
     (/ (* y_m (fma x (* x (fma (* x x) 0.041666666666666664 0.5)) 1.0)) 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 (((cosh(x) * (y_m / x)) / z) <= 2e+33) {
		tmp = ((y_m / x) * fma(x, (x * fma(x, (x * 0.041666666666666664), 0.5)), 1.0)) / z;
	} else {
		tmp = ((y_m * fma(x, (x * fma((x * x), 0.041666666666666664, 0.5)), 1.0)) / 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 (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 2e+33)
		tmp = Float64(Float64(Float64(y_m / x) * fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), 0.5)), 1.0)) / z);
	else
		tmp = Float64(Float64(Float64(y_m * fma(x, Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5)), 1.0)) / 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[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e+33], N[(N[(N[(y$95$m / x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.9999999999999999e33

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 93.3

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

   5. Simplified 93.3%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 88.5

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

   8. Simplified 88.5%

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

   if 1.9999999999999999e33 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 67.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 58.3

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

   5. Simplified 58.3%

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

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 89.9%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(y\_m \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\_m\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y\_m \cdot \left(x \cdot x\right), \left(x \cdot x\right) \cdot 0.041666666666666664, y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

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 (<= (/ (* (cosh x) (/ y_m x)) z) 5e+57)
    (/
     (fma x (* x (* y_m (fma x (* x 0.041666666666666664) 0.5))) y_m)
     (* x z))
    (/ (/ (fma (* y_m (* x x)) (* (* x x) 0.041666666666666664) 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 (((cosh(x) * (y_m / x)) / z) <= 5e+57) {
		tmp = fma(x, (x * (y_m * fma(x, (x * 0.041666666666666664), 0.5))), y_m) / (x * z);
	} else {
		tmp = (fma((y_m * (x * x)), ((x * x) * 0.041666666666666664), 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 (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 5e+57)
		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(fma(Float64(y_m * Float64(x * x)), Float64(Float64(x * x) * 0.041666666666666664), 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[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e+57], 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[(N[(N[(y$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4.99999999999999972e57

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 83.0%

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

   if 4.99999999999999972e57 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 67.2%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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 81.6%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 81.6

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

   8. Simplified 81.6%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 86.5

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

   10. Applied egg-rr 86.5%

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

Alternative 12: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-145}:\\ \;\;\;\;-\frac{-1}{z \cdot \frac{x}{y\_m}}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;y\_m \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \left(x \cdot x\right) \cdot 0.001388888888888889, 1\right)}{z}}{x}\\ \end{array} \end{array} \]

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 (<= x 1.35e-145)
    (- (/ -1.0 (* z (/ x y_m))))
    (if (<= x 7.2e+51)
      (* y_m (/ (cosh x) (* x z)))
      (/
       (*
        y_m
        (/ (fma (* x (* x (* x x))) (* (* x x) 0.001388888888888889) 1.0) 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 (x <= 1.35e-145) {
		tmp = -(-1.0 / (z * (x / y_m)));
	} else if (x <= 7.2e+51) {
		tmp = y_m * (cosh(x) / (x * z));
	} else {
		tmp = (y_m * (fma((x * (x * (x * x))), ((x * x) * 0.001388888888888889), 1.0) / 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 (x <= 1.35e-145)
		tmp = Float64(-Float64(-1.0 / Float64(z * Float64(x / y_m))));
	elseif (x <= 7.2e+51)
		tmp = Float64(y_m * Float64(cosh(x) / Float64(x * z)));
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * Float64(x * Float64(x * x))), Float64(Float64(x * x) * 0.001388888888888889), 1.0) / 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[x, 1.35e-145], (-N[(-1.0 / N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[x, 7.2e+51], N[(y$95$m * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.35e-145

   1. Initial program 84.8%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 50.8

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

   5. Simplified 50.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 50.7

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

   7. Applied egg-rr 50.7%

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 53.7

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

   9. Applied egg-rr 53.7%

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

       1. frac-2neg N/A

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

       2. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{-1}}{\mathsf{neg}\left(z\right)}}{\frac{x}{y}} \]

       3. associate-/l/ N/A

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

       4. /-lowering-/.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. /-lowering-/.f64 N/A

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

       7. neg-lowering-neg.f64 53.7

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

   11. Applied egg-rr 53.7%

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

   if 1.35e-145 < x < 7.20000000000000022e51

   1. Initial program 93.7%

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. cosh-lowering-cosh.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 97.7

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

   4. Applied egg-rr 97.7%

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

   if 7.20000000000000022e51 < x

   1. Initial program 73.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 73.1

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

   5. Simplified 73.1%

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

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 100.0

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

   10. Simplified 100.0%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. /-lowering-/.f64 N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. associate-*l* N/A

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

       9. cube-unmult N/A

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

       10. *-lowering-*.f64 N/A

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

       11. cube-unmult N/A

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

       12. *-lowering-*.f64 N/A

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

       13. *-lowering-*.f64 N/A

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

       14. *-lowering-*.f64 N/A

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

       15. *-lowering-*.f64 100.0

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-145}:\\ \;\;\;\;-\frac{-1}{z \cdot \frac{x}{y}}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \left(x \cdot x\right) \cdot 0.001388888888888889, 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.3 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{x}}{\frac{1}{\mathsf{fma}\left(y\_m, \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right), y\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y\_m}{z}\\ \end{array} \end{array} \]

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 (<= y_m 3.3e-125)
    (*
     (/ 1.0 z)
     (/
      (/ 1.0 x)
      (/
       1.0
       (fma
        y_m
        (* (* x (* x (* x x))) (* (* x x) 0.001388888888888889))
        y_m))))
    (* (/ (cosh x) x) (/ y_m 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 (y_m <= 3.3e-125) {
		tmp = (1.0 / z) * ((1.0 / x) / (1.0 / fma(y_m, ((x * (x * (x * x))) * ((x * x) * 0.001388888888888889)), y_m)));
	} else {
		tmp = (cosh(x) / x) * (y_m / 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 (y_m <= 3.3e-125)
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(1.0 / x) / Float64(1.0 / fma(y_m, Float64(Float64(x * Float64(x * Float64(x * x))) * Float64(Float64(x * x) * 0.001388888888888889)), y_m))));
	else
		tmp = Float64(Float64(cosh(x) / x) * Float64(y_m / 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[y$95$m, 3.3e-125], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / N[(1.0 / N[(y$95$m * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.3000000000000001e-125

   1. Initial program 78.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 73.0

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

   5. Simplified 73.0%

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

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 87.8%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 87.6

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

   10. Simplified 87.6%

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

       1. div-inv N/A

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

       2. clear-num N/A

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

       3. associate-*l/ N/A

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

       4. div-inv N/A

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

       5. times-frac N/A

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

       6. *-lowering-*.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. /-lowering-/.f64 N/A

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

       9. /-lowering-/.f64 N/A

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

       10. /-lowering-/.f64 N/A

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

       11. distribute-lft-in N/A

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

       12. *-rgt-identity N/A

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

   12. Applied egg-rr 90.5%

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

   if 3.3000000000000001e-125 < y

   1. Initial program 94.4%

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. div-inv N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. div-inv N/A

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

      11. /-lowering-/.f64 N/A

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

      12. cosh-lowering-cosh.f64 99.8

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{x}}{\frac{1}{\mathsf{fma}\left(y, \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right), y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 10^{+52}:\\ \;\;\;\;\frac{y\_m \cdot \cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \left(x \cdot x\right) \cdot 0.001388888888888889, 1\right)}{z}}{x}\\ \end{array} \end{array} \]

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 (<= x 1e+52)
    (/ (* y_m (cosh x)) (* x z))
    (/
     (*
      y_m
      (/ (fma (* x (* x (* x x))) (* (* x x) 0.001388888888888889) 1.0) 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 (x <= 1e+52) {
		tmp = (y_m * cosh(x)) / (x * z);
	} else {
		tmp = (y_m * (fma((x * (x * (x * x))), ((x * x) * 0.001388888888888889), 1.0) / 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 (x <= 1e+52)
		tmp = Float64(Float64(y_m * cosh(x)) / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * Float64(x * Float64(x * x))), Float64(Float64(x * x) * 0.001388888888888889), 1.0) / 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[x, 1e+52], N[(N[(y$95$m * N[Cosh[x], $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999999e51

   1. Initial program 86.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. div-inv N/A

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

      8. /-lowering-/.f64 N/A

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

      9. cosh-lowering-cosh.f64 96.1

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

   4. Applied egg-rr 96.1%

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

      1. associate-/l* N/A

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

      2. frac-times N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cosh-lowering-cosh.f64 N/A

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

      6. *-lowering-*.f64 87.8

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

   6. Applied egg-rr 87.8%

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

   if 9.9999999999999999e51 < x

   1. Initial program 73.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 73.1

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

   5. Simplified 73.1%

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

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 100.0

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

   10. Simplified 100.0%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. /-lowering-/.f64 N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. associate-*l* N/A

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

       9. cube-unmult N/A

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

       10. *-lowering-*.f64 N/A

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

       11. cube-unmult N/A

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

       12. *-lowering-*.f64 N/A

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

       13. *-lowering-*.f64 N/A

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

       14. *-lowering-*.f64 N/A

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

       15. *-lowering-*.f64 100.0

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+52}:\\ \;\;\;\;\frac{y \cdot \cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \left(x \cdot x\right) \cdot 0.001388888888888889, 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 91.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 17000:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y\_m \cdot \left(x \cdot x\right), \left(x \cdot x\right) \cdot 0.041666666666666664, y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

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 17000.0)
    (/
     (* y_m (/ (fma (* x x) (fma x (* x 0.041666666666666664) 0.5) 1.0) x))
     z)
    (/ (/ (fma (* y_m (* x x)) (* (* x x) 0.041666666666666664) 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 <= 17000.0) {
		tmp = (y_m * (fma((x * x), fma(x, (x * 0.041666666666666664), 0.5), 1.0) / x)) / z;
	} else {
		tmp = (fma((y_m * (x * x)), ((x * x) * 0.041666666666666664), 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 <= 17000.0)
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), fma(x, Float64(x * 0.041666666666666664), 0.5), 1.0) / x)) / z);
	else
		tmp = Float64(Float64(fma(Float64(y_m * Float64(x * x)), Float64(Float64(x * x) * 0.041666666666666664), 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, 17000.0], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(y$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 17000

   1. Initial program 80.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. div-inv N/A

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

      8. /-lowering-/.f64 N/A

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

      9. cosh-lowering-cosh.f64 97.4

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

   4. Applied egg-rr 97.4%

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

   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. /-lowering-/.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. accelerator-lowering-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. *-lowering-*.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. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 86.9

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

   7. Simplified 86.9%

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

   if 17000 < y

   1. Initial program 95.3%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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 89.8%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 89.5

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

   8. Simplified 89.5%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 95.5

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

   10. Applied egg-rr 95.5%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 17000:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot \left(x \cdot x\right), \left(x \cdot x\right) \cdot 0.041666666666666664, y\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 78.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 3.7:\\ \;\;\;\;\frac{\frac{y\_m}{x} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(0.041666666666666664 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}{x}}{z}\\ \end{array} \end{array} \]

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 3.7)
    (/ (* (/ y_m x) (fma 0.5 (* x x) 1.0)) z)
    (/ (/ (* y_m (* 0.041666666666666664 (* (* x x) (* 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 <= 3.7) {
		tmp = ((y_m / x) * fma(0.5, (x * x), 1.0)) / z;
	} else {
		tmp = ((y_m * (0.041666666666666664 * ((x * x) * (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 <= 3.7)
		tmp = Float64(Float64(Float64(y_m / x) * fma(0.5, Float64(x * x), 1.0)) / z);
	else
		tmp = Float64(Float64(Float64(y_m * Float64(0.041666666666666664 * Float64(Float64(x * x) * Float64(x * x)))) / 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, 3.7], N[(N[(N[(y$95$m / x), $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y$95$m * N[(0.041666666666666664 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.7000000000000002

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 71.8

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

   5. Simplified 71.8%

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

   if 3.7000000000000002 < x

   1. Initial program 76.9%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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 79.5%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 79.5

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

   8. Simplified 79.5%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot y}}{x}}{z} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{4}\right)}}{x}}{z} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{4}\right)}}{x}}{z} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{y \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right)}}{x}}{z} \]

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 82.5

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

   11. Simplified 82.5%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(0.041666666666666664 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 78.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 3.7:\\ \;\;\;\;\frac{\frac{y\_m}{x} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(0.041666666666666664 \cdot \left(y\_m \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{x}}{z}\\ \end{array} \end{array} \]

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 3.7)
    (/ (* (/ y_m x) (fma 0.5 (* x x) 1.0)) z)
    (/ (/ (* x (* 0.041666666666666664 (* y_m (* x (* 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 <= 3.7) {
		tmp = ((y_m / x) * fma(0.5, (x * x), 1.0)) / z;
	} else {
		tmp = ((x * (0.041666666666666664 * (y_m * (x * (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 <= 3.7)
		tmp = Float64(Float64(Float64(y_m / x) * fma(0.5, Float64(x * x), 1.0)) / z);
	else
		tmp = Float64(Float64(Float64(x * Float64(0.041666666666666664 * Float64(y_m * Float64(x * Float64(x * x))))) / 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, 3.7], N[(N[(N[(y$95$m / x), $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x * N[(0.041666666666666664 * N[(y$95$m * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.7000000000000002

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 71.8

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

   5. Simplified 71.8%

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

   if 3.7000000000000002 < x

   1. Initial program 76.9%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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 79.5%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot y}}{x}}{z} \]

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. unpow3 N/A

          \[\leadsto \frac{\frac{x \cdot \left(\frac{1}{24} \cdot \left(y \cdot \color{blue}{{x}^{3}}\right)\right)}{x}}{z} \]

      16. associate-*r* N/A

          \[\leadsto \frac{\frac{x \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot y\right) \cdot {x}^{3}\right)}}{x}}{z} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(\left(\frac{1}{24} \cdot y\right) \cdot {x}^{3}\right)}}{x}}{z} \]

      18. associate-*r* N/A

          \[\leadsto \frac{\frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(y \cdot {x}^{3}\right)\right)}}{x}}{z} \]

      19. *-commutative N/A

          \[\leadsto \frac{\frac{x \cdot \left(\frac{1}{24} \cdot \color{blue}{\left({x}^{3} \cdot y\right)}\right)}{x}}{z} \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)\right)}}{x}}{z} \]

      21. *-commutative N/A

          \[\leadsto \frac{\frac{x \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(y \cdot {x}^{3}\right)}\right)}{x}}{z} \]

      22. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(y \cdot {x}^{3}\right)}\right)}{x}}{z} \]

      23. cube-mult N/A

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

      24. unpow2 N/A

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

      25. *-lowering-*.f64 N/A

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

   8. Simplified 82.5%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(0.041666666666666664 \cdot \left(y \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 80.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(y\_m \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\_m\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)}{z}\\ \end{array} \end{array} \]

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 1.4e+58)
    (/
     (fma x (* x (* y_m (fma x (* x 0.041666666666666664) 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 <= 1.4e+58) {
		tmp = fma(x, (x * (y_m * fma(x, (x * 0.041666666666666664), 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 <= 1.4e+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 * 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, 1.4e+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 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999e58

   1. Initial program 86.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 75.4%

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

   if 1.3999999999999999e58 < x

   1. Initial program 72.5%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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 94.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(\frac{1}{24} \cdot y + \frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}}{z} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(\frac{1}{24} \cdot y\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}}{z} \]

      2. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{24} \cdot y\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{24} \cdot y\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} \cdot y\right)\right)} + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{x \cdot \left({x}^{2} \cdot \color{blue}{\left(y \cdot \frac{1}{24}\right)}\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      6. associate-*r* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left({x}^{2} \cdot y\right) \cdot \frac{1}{24}\right)} + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)} + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. associate-*l/ N/A

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

      15. associate-/l* N/A

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

      16. associate-/l* N/A

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

      17. *-inverses N/A

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

      18. metadata-eval N/A

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

      19. *-commutative N/A

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

   8. Simplified 98.1%

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

Alternative 19: 81.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+58}:\\ \;\;\;\;\frac{y\_m \cdot \mathsf{fma}\left(x, t\_0, 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot t\_0}{z}\\ \end{array} \end{array} \end{array} \]

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 1.12e+58)
      (/ (* 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 <= 1.12e+58) {
		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 <= 1.12e+58)
		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, 1.12e+58], 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 < 1.12e58

   1. Initial program 86.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 75.8

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

   5. Simplified 75.8%

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

      1. associate-/l* N/A

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

      2. associate-/r* N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 77.2

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

   7. Applied egg-rr 77.2%

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

   if 1.12e58 < x

   1. Initial program 72.5%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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 94.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(\frac{1}{24} \cdot y + \frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}}{z} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(\frac{1}{24} \cdot y\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}}{z} \]

      2. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{24} \cdot y\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{24} \cdot y\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} \cdot y\right)\right)} + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{x \cdot \left({x}^{2} \cdot \color{blue}{\left(y \cdot \frac{1}{24}\right)}\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      6. associate-*r* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left({x}^{2} \cdot y\right) \cdot \frac{1}{24}\right)} + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)} + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. associate-*l/ N/A

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

      15. associate-/l* N/A

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

      16. associate-/l* N/A

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

      17. *-inverses N/A

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

      18. metadata-eval N/A

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

      19. *-commutative N/A

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

   8. Simplified 98.1%

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

Alternative 20: 77.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{\frac{y\_m}{x} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)}{z}\\ \end{array} \end{array} \]

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 x) (fma 0.5 (* x x) 1.0)) 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 / x) * fma(0.5, (x * x), 1.0)) / 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(Float64(y_m / x) * fma(0.5, Float64(x * x), 1.0)) / 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[(N[(y$95$m / x), $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $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 86.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 71.8

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

   5. Simplified 71.8%

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

   if 2.25 < x

   1. Initial program 76.9%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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 79.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(\frac{1}{24} \cdot y + \frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}}{z} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(\frac{1}{24} \cdot y\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}}{z} \]

      2. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{24} \cdot y\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{24} \cdot y\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} \cdot y\right)\right)} + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{x \cdot \left({x}^{2} \cdot \color{blue}{\left(y \cdot \frac{1}{24}\right)}\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      6. associate-*r* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left({x}^{2} \cdot y\right) \cdot \frac{1}{24}\right)} + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)} + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. associate-*l/ N/A

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

      15. associate-/l* N/A

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

      16. associate-/l* N/A

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

      17. *-inverses N/A

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

      18. metadata-eval N/A

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

      19. *-commutative N/A

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

   8. Simplified 81.1%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 76.1% 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 86.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   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. accelerator-lowering-fma.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. /-lowering-/.f64 70.6

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

   5. Simplified 70.6%

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

   if 2.25 < x

   1. Initial program 76.9%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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 79.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(\frac{1}{24} \cdot y + \frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}}{z} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(\frac{1}{24} \cdot y\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}}{z} \]

      2. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{24} \cdot y\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{24} \cdot y\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} \cdot y\right)\right)} + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{x \cdot \left({x}^{2} \cdot \color{blue}{\left(y \cdot \frac{1}{24}\right)}\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      6. associate-*r* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left({x}^{2} \cdot y\right) \cdot \frac{1}{24}\right)} + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)} + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. associate-*l/ N/A

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

      15. associate-/l* N/A

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

      16. associate-/l* N/A

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

      17. *-inverses N/A

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

      18. metadata-eval N/A

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

      19. *-commutative N/A

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

   8. Simplified 81.1%

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

Alternative 22: 68.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.3:\\ \;\;\;\;-\frac{-1}{z \cdot \frac{x}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)}{z}\\ \end{array} \end{array} \]

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 1.3)
    (- (/ -1.0 (* z (/ x y_m))))
    (/ (* 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 <= 1.3) {
		tmp = -(-1.0 / (z * (x / y_m)));
	} 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 <= 1.3)
		tmp = Float64(-Float64(-1.0 / Float64(z * Float64(x / y_m))));
	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, 1.3], (-N[(-1.0 / N[(z * N[(x / y$95$m), $MachinePrecision]), $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 < 1.30000000000000004

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 58.5

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

   5. Simplified 58.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 58.1

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

   7. Applied egg-rr 58.1%

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 60.1

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

   9. Applied egg-rr 60.1%

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

       1. frac-2neg N/A

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

       2. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{-1}}{\mathsf{neg}\left(z\right)}}{\frac{x}{y}} \]

       3. associate-/l/ N/A

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

       4. /-lowering-/.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. /-lowering-/.f64 N/A

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

       7. neg-lowering-neg.f64 60.1

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

   11. Applied egg-rr 60.1%

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

   if 1.30000000000000004 < x

   1. Initial program 76.9%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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 79.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(\frac{1}{24} \cdot y + \frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}}{z} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(\frac{1}{24} \cdot y\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}}{z} \]

      2. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{24} \cdot y\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{24} \cdot y\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} \cdot y\right)\right)} + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{x \cdot \left({x}^{2} \cdot \color{blue}{\left(y \cdot \frac{1}{24}\right)}\right) + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      6. associate-*r* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left({x}^{2} \cdot y\right) \cdot \frac{1}{24}\right)} + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)} + {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2}}\right)}{z} \]

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. associate-*l/ N/A

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

      15. associate-/l* N/A

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

      16. associate-/l* N/A

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

      17. *-inverses N/A

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

      18. metadata-eval N/A

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

      19. *-commutative N/A

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

   8. Simplified 81.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3:\\ \;\;\;\;-\frac{-1}{z \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 68.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;-\frac{-1}{z \cdot \frac{x}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(y\_m \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \end{array} \]

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)
    (- (/ -1.0 (* z (/ x y_m))))
    (/ (* 0.041666666666666664 (* y_m (* 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 = -(-1.0 / (z * (x / y_m)));
	} else {
		tmp = (0.041666666666666664 * (y_m * (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 = -((-1.0d0) / (z * (x / y_m)))
    else
        tmp = (0.041666666666666664d0 * (y_m * (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 = -(-1.0 / (z * (x / y_m)));
	} else {
		tmp = (0.041666666666666664 * (y_m * (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 = -(-1.0 / (z * (x / y_m)))
	else:
		tmp = (0.041666666666666664 * (y_m * (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(-1.0 / Float64(z * Float64(x / y_m))));
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(y_m * 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 = -(-1.0 / (z * (x / y_m)));
	else
		tmp = (0.041666666666666664 * (y_m * (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[(-1.0 / N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.041666666666666664 * N[(y$95$m * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 58.5

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

   5. Simplified 58.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 58.1

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

   7. Applied egg-rr 58.1%

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 60.1

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

   9. Applied egg-rr 60.1%

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

       1. frac-2neg N/A

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

       2. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{-1}}{\mathsf{neg}\left(z\right)}}{\frac{x}{y}} \]

       3. associate-/l/ N/A

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

       4. /-lowering-/.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. /-lowering-/.f64 N/A

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

       7. neg-lowering-neg.f64 60.1

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

   11. Applied egg-rr 60.1%

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

   if 2.25 < x

   1. Initial program 76.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 59.6

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

   5. Simplified 59.6%

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

      1. associate-/l* N/A

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

      2. associate-/r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 55.0

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

   7. Applied egg-rr 55.0%

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

   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. *-commutative N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(y \cdot {x}^{3}\right)}}{z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot y\right) \cdot {x}^{3}}}{z} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{24} \cdot y\right) \cdot {x}^{3}}{z}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left(y \cdot {x}^{3}\right)}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left({x}^{3} \cdot y\right)}}{z} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(y \cdot {x}^{3}\right)}}{z} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(y \cdot {x}^{3}\right)}}{z} \]

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 81.1

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

   10. Simplified 81.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;-\frac{-1}{z \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(y \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 57.7% accurate, 3.6× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 58.5

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

   5. Simplified 58.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 58.1

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

   7. Applied egg-rr 58.1%

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 60.1

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

   9. Applied egg-rr 60.1%

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

       1. frac-2neg N/A

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

       2. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{-1}}{\mathsf{neg}\left(z\right)}}{\frac{x}{y}} \]

       3. associate-/l/ N/A

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

       4. /-lowering-/.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. /-lowering-/.f64 N/A

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

       7. neg-lowering-neg.f64 60.1

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

   11. Applied egg-rr 60.1%

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

   if 1.3999999999999999 < x

   1. Initial program 76.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-fma.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. /-lowering-/.f64 34.0

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

   5. Simplified 34.0%

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

   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. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{2} \cdot y\right)}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot \frac{1}{2}\right)}}{z} \]

      8. *-lowering-*.f64 34.0

         \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot 0.5\right)}}{z} \]

   8. Simplified 34.0%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot 0.5\right)}{z}} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \frac{1}{2}}}{z} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{\frac{1}{2}}{z}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{\frac{1}{2}}{z}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{\frac{1}{2}}{z} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{\frac{1}{2}}{z} \]

      6. /-lowering-/.f64 34.0

         \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\frac{0.5}{z}} \]

   10. Applied egg-rr 34.0%

       \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{0.5}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;-\frac{-1}{z \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{0.5}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 57.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot x\right) \cdot \frac{0.5}{z}\\ \end{array} \end{array} \]

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 1.4) (/ (/ 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 <= 1.4) {
		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 <= 1.4d0) 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 <= 1.4) {
		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 <= 1.4:
		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 <= 1.4)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(Float64(y_m * 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 <= 1.4)
		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, 1.4], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m * x), $MachinePrecision] * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 86.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 59.5

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   5. Simplified 59.5%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   if 1.3999999999999999 < x

   1. Initial program 76.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x \cdot \frac{1}{2}, \frac{y}{x}\right)}}{z} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{x \cdot \frac{1}{2}}, \frac{y}{x}\right)}{z} \]

      18. /-lowering-/.f64 34.0

          \[\leadsto \frac{\mathsf{fma}\left(y, x \cdot 0.5, \color{blue}{\frac{y}{x}}\right)}{z} \]

   5. Simplified 34.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x \cdot 0.5, \frac{y}{x}\right)}}{z} \]

   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. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{z}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{z}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \frac{1}{2}}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \frac{1}{2}\right)}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{2} \cdot y\right)}}{z} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{2} \cdot y\right)}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot \frac{1}{2}\right)}}{z} \]

      8. *-lowering-*.f64 34.0

         \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot 0.5\right)}}{z} \]

   8. Simplified 34.0%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot 0.5\right)}{z}} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \frac{1}{2}}}{z} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{\frac{1}{2}}{z}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{\frac{1}{2}}{z}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{\frac{1}{2}}{z} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{\frac{1}{2}}{z} \]

      6. /-lowering-/.f64 34.0

         \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\frac{0.5}{z}} \]

   10. Applied egg-rr 34.0%

       \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{0.5}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 58.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{y\_m}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot x\right) \cdot \frac{0.5}{z}\\ \end{array} \end{array} \]

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 1.4) (/ 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 <= 1.4) {
		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 <= 1.4d0) 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 <= 1.4) {
		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 <= 1.4:
		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 <= 1.4)
		tmp = Float64(y_m / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * 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 <= 1.4)
		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, 1.4], N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * x), $MachinePrecision] * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 86.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. *-lowering-*.f64 58.5

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

   5. Simplified 58.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 1.3999999999999999 < x

   1. Initial program 76.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x \cdot \frac{1}{2}, \frac{y}{x}\right)}}{z} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{x \cdot \frac{1}{2}}, \frac{y}{x}\right)}{z} \]

      18. /-lowering-/.f64 34.0

          \[\leadsto \frac{\mathsf{fma}\left(y, x \cdot 0.5, \color{blue}{\frac{y}{x}}\right)}{z} \]

   5. Simplified 34.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x \cdot 0.5, \frac{y}{x}\right)}}{z} \]

   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. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{z}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{z}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \frac{1}{2}}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \frac{1}{2}\right)}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{2} \cdot y\right)}}{z} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{2} \cdot y\right)}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot \frac{1}{2}\right)}}{z} \]

      8. *-lowering-*.f64 34.0

         \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot 0.5\right)}}{z} \]

   8. Simplified 34.0%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot 0.5\right)}{z}} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \frac{1}{2}}}{z} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{\frac{1}{2}}{z}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{\frac{1}{2}}{z}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{\frac{1}{2}}{z} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{\frac{1}{2}}{z} \]

      6. /-lowering-/.f64 34.0

         \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\frac{0.5}{z}} \]

   10. Applied egg-rr 34.0%

       \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{0.5}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 49.9% 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 84.0%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   2. *-lowering-*.f64 45.6

      \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

5. Simplified 45.6%

   \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
6. Add Preprocessing

Developer Target 1: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y z) x) (cosh x))))
   (if (< y -4.618902267687042e-52)
     t_0
     (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y / z) / x) * cosh(x)
    if (y < (-4.618902267687042d-52)) then
        tmp = t_0
    else if (y < 1.038530535935153d-39) then
        tmp = ((cosh(x) * y) / x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * Math.cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((Math.cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / z) / x) * math.cosh(x)
	tmp = 0
	if y < -4.618902267687042e-52:
		tmp = t_0
	elif y < 1.038530535935153e-39:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) / x) * cosh(x))
	tmp = 0.0
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / z) / x) * cosh(x);
	tmp = 0.0;
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -4.618902267687042e-52], t$95$0, If[Less[y, 1.038530535935153e-39], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024204 
(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))