Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.4% → 99.2%

Time: 14.8s

Alternatives: 22

Speedup: 6.7×

• 64.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 10^{+290}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(0.041666666666666664 + x\_m \cdot \left(x\_m \cdot 0.001388888888888889\right)\right)\right)\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\cosh x\_m}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 1e+290)
      (/
       (/
        (*
         y_m
         (+
          1.0
          (*
           (* x_m x_m)
           (+
            0.5
            (*
             (* x_m x_m)
             (+ 0.041666666666666664 (* x_m (* x_m 0.001388888888888889))))))))
        x_m)
       z_m)
      (* y_m (/ (/ (cosh x_m) x_m) z_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 1e+290) {
		tmp = ((y_m * (1.0 + ((x_m * x_m) * (0.5 + ((x_m * x_m) * (0.041666666666666664 + (x_m * (x_m * 0.001388888888888889)))))))) / x_m) / z_m;
	} else {
		tmp = y_m * ((cosh(x_m) / x_m) / z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 1d+290) then
        tmp = ((y_m * (1.0d0 + ((x_m * x_m) * (0.5d0 + ((x_m * x_m) * (0.041666666666666664d0 + (x_m * (x_m * 0.001388888888888889d0)))))))) / x_m) / z_m
    else
        tmp = y_m * ((cosh(x_m) / x_m) / z_m)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((Math.cosh(x_m) * (y_m / x_m)) / z_m) <= 1e+290) {
		tmp = ((y_m * (1.0 + ((x_m * x_m) * (0.5 + ((x_m * x_m) * (0.041666666666666664 + (x_m * (x_m * 0.001388888888888889)))))))) / x_m) / z_m;
	} else {
		tmp = y_m * ((Math.cosh(x_m) / x_m) / z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if ((math.cosh(x_m) * (y_m / x_m)) / z_m) <= 1e+290:
		tmp = ((y_m * (1.0 + ((x_m * x_m) * (0.5 + ((x_m * x_m) * (0.041666666666666664 + (x_m * (x_m * 0.001388888888888889)))))))) / x_m) / z_m
	else:
		tmp = y_m * ((math.cosh(x_m) / x_m) / z_m)
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 1e+290)
		tmp = Float64(Float64(Float64(y_m * Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(0.041666666666666664 + Float64(x_m * Float64(x_m * 0.001388888888888889)))))))) / x_m) / z_m);
	else
		tmp = Float64(y_m * Float64(Float64(cosh(x_m) / x_m) / z_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 1e+290)
		tmp = ((y_m * (1.0 + ((x_m * x_m) * (0.5 + ((x_m * x_m) * (0.041666666666666664 + (x_m * (x_m * 0.001388888888888889)))))))) / x_m) / z_m;
	else
		tmp = y_m * ((cosh(x_m) / x_m) / z_m);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 1e+290], N[(N[(N[(y$95$m * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.041666666666666664 + N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * N[(N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.00000000000000006e290

   1. Initial program 95.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}\right)}, z\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)\right), x\right), z\right) \]

   5. Simplified 91.7%

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

   if 1.00000000000000006e290 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 63.8%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{x} \cdot \cosh x}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{x} \cdot \cosh x\right), \color{blue}{z}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\cosh x \cdot \frac{1}{x}\right), z\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{\cosh x}{x}\right), z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cosh x, x\right), z\right)\right) \]

      10. cosh-lowering-cosh.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), x\right), z\right)\right) \]

   4. Applied egg-rr 99.9%

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

Alternative 2: 89.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := 1 + x\_m \cdot \left(x\_m \cdot 0.5\right)\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(t\_0 + x\_m \cdot \left(0.041666666666666664 \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)\right)}{x\_m}}{z\_m}\\ \mathbf{elif}\;y\_m \leq 1.12 \cdot 10^{+115}:\\ \;\;\;\;\frac{y\_m}{x\_m} \cdot \frac{1 + x\_m \cdot \left(x\_m \cdot \left(0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(0.041666666666666664 + x\_m \cdot \left(x\_m \cdot 0.001388888888888889\right)\right)\right)\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m} \cdot \left(t\_0 + x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)\right)\right)}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x_m (* x_m 0.5)))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= y_m 3.2e-148)
        (/
         (/
          (* y_m (+ t_0 (* x_m (* 0.041666666666666664 (* x_m (* x_m x_m))))))
          x_m)
         z_m)
        (if (<= y_m 1.12e+115)
          (*
           (/ y_m x_m)
           (/
            (+
             1.0
             (*
              x_m
              (*
               x_m
               (+
                0.5
                (*
                 (* x_m x_m)
                 (+
                  0.041666666666666664
                  (* x_m (* x_m 0.001388888888888889))))))))
            z_m))
          (/
           (*
            (/ y_m z_m)
            (+ t_0 (* x_m (* x_m (* (* x_m x_m) 0.041666666666666664)))))
           x_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = 1.0 + (x_m * (x_m * 0.5));
	double tmp;
	if (y_m <= 3.2e-148) {
		tmp = ((y_m * (t_0 + (x_m * (0.041666666666666664 * (x_m * (x_m * x_m)))))) / x_m) / z_m;
	} else if (y_m <= 1.12e+115) {
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * (0.5 + ((x_m * x_m) * (0.041666666666666664 + (x_m * (x_m * 0.001388888888888889)))))))) / z_m);
	} else {
		tmp = ((y_m / z_m) * (t_0 + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))))) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x_m * (x_m * 0.5d0))
    if (y_m <= 3.2d-148) then
        tmp = ((y_m * (t_0 + (x_m * (0.041666666666666664d0 * (x_m * (x_m * x_m)))))) / x_m) / z_m
    else if (y_m <= 1.12d+115) then
        tmp = (y_m / x_m) * ((1.0d0 + (x_m * (x_m * (0.5d0 + ((x_m * x_m) * (0.041666666666666664d0 + (x_m * (x_m * 0.001388888888888889d0)))))))) / z_m)
    else
        tmp = ((y_m / z_m) * (t_0 + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664d0))))) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = 1.0 + (x_m * (x_m * 0.5));
	double tmp;
	if (y_m <= 3.2e-148) {
		tmp = ((y_m * (t_0 + (x_m * (0.041666666666666664 * (x_m * (x_m * x_m)))))) / x_m) / z_m;
	} else if (y_m <= 1.12e+115) {
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * (0.5 + ((x_m * x_m) * (0.041666666666666664 + (x_m * (x_m * 0.001388888888888889)))))))) / z_m);
	} else {
		tmp = ((y_m / z_m) * (t_0 + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))))) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	t_0 = 1.0 + (x_m * (x_m * 0.5))
	tmp = 0
	if y_m <= 3.2e-148:
		tmp = ((y_m * (t_0 + (x_m * (0.041666666666666664 * (x_m * (x_m * x_m)))))) / x_m) / z_m
	elif y_m <= 1.12e+115:
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * (0.5 + ((x_m * x_m) * (0.041666666666666664 + (x_m * (x_m * 0.001388888888888889)))))))) / z_m)
	else:
		tmp = ((y_m / z_m) * (t_0 + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))))) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(1.0 + Float64(x_m * Float64(x_m * 0.5)))
	tmp = 0.0
	if (y_m <= 3.2e-148)
		tmp = Float64(Float64(Float64(y_m * Float64(t_0 + Float64(x_m * Float64(0.041666666666666664 * Float64(x_m * Float64(x_m * x_m)))))) / x_m) / z_m);
	elseif (y_m <= 1.12e+115)
		tmp = Float64(Float64(y_m / x_m) * Float64(Float64(1.0 + Float64(x_m * Float64(x_m * Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(0.041666666666666664 + Float64(x_m * Float64(x_m * 0.001388888888888889)))))))) / z_m));
	else
		tmp = Float64(Float64(Float64(y_m / z_m) * Float64(t_0 + Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664))))) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	t_0 = 1.0 + (x_m * (x_m * 0.5));
	tmp = 0.0;
	if (y_m <= 3.2e-148)
		tmp = ((y_m * (t_0 + (x_m * (0.041666666666666664 * (x_m * (x_m * x_m)))))) / x_m) / z_m;
	elseif (y_m <= 1.12e+115)
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * (0.5 + ((x_m * x_m) * (0.041666666666666664 + (x_m * (x_m * 0.001388888888888889)))))))) / z_m);
	else
		tmp = ((y_m / z_m) * (t_0 + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))))) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(1.0 + N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 3.2e-148], N[(N[(N[(y$95$m * N[(t$95$0 + N[(x$95$m * N[(0.041666666666666664 * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], If[LessEqual[y$95$m, 1.12e+115], N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(N[(1.0 + N[(x$95$m * N[(x$95$m * N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.041666666666666664 + N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m / z$95$m), $MachinePrecision] * N[(t$95$0 + N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if y < 3.19999999999999993e-148

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}\right)}, z\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)\right), x\right), z\right) \]

   5. Simplified 91.9%

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

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \left(\left(x \cdot x\right) \cdot \frac{1}{2} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right), x\right), z\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right), z\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(\left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right), z\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{720}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{720}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{720}\right)\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

   7. Applied egg-rr 91.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(y + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)\right)}, x\right), z\right) \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)\right), x\right), z\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right) \cdot {x}^{2}\right), x\right), z\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot y + \frac{1}{2} \cdot y\right) \cdot {x}^{2}\right), x\right), z\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + \left(y \cdot \left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)\right) \cdot {x}^{2}\right), x\right), z\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + \left(y \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right), x\right), z\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + y \cdot \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}\right)\right), x\right), z\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + y \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), x\right), z\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), x\right), z\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right) \cdot y\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right), y\right), x\right), z\right) \]

   10. Simplified 87.3%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{24}\right)\right), y\right), x\right), z\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \left(\frac{1}{2} \cdot \left(x \cdot x\right) + \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       3. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right), y\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right)\right), y\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x\right) \cdot x\right)\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} \cdot x\right)\right)\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right)\right), y\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x\right)\right)\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right)\right), y\right), x\right), z\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right), y\right), x\right), z\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right), y\right), x\right), z\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(x \cdot \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot x\right)\right)\right), y\right), x\right), z\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \left(\left(\frac{1}{24} \cdot \left(x \cdot x\right)\right) \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)\right), y\right), x\right), z\right) \]

       17. pow3 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{3}\right)\right)\right), y\right), x\right), z\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \left({x}^{3}\right)\right)\right)\right), y\right), x\right), z\right) \]

       19. cube-unmult N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right), y\right), x\right), z\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right)\right), y\right), x\right), z\right) \]

       21. *-lowering-*.f64 87.3%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), y\right), x\right), z\right) \]

   12. Applied egg-rr 87.3%

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

   if 3.19999999999999993e-148 < y < 1.12e115

   1. Initial program 91.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}\right)}, z\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)\right), x\right), z\right) \]

   5. Simplified 99.8%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)}{z}\right), \color{blue}{\left(\frac{y}{x}\right)}\right) \]

   7. Applied egg-rr 91.7%

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

   if 1.12e115 < y

   1. Initial program 78.6%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   5. Simplified 93.7%

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

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left(1 + \left(\frac{1}{2} \cdot \left(x \cdot x\right) + \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right) + \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right), x\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

      15. *-lowering-*.f64 93.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 93.7%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\left(1 + x \cdot \left(x \cdot 0.5\right)\right) + x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{x}}{z}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+115}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \left(\left(1 + x \cdot \left(x \cdot 0.5\right)\right) + x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.12 \cdot 10^{+115}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(0.041666666666666664 + x\_m \cdot \left(x\_m \cdot 0.001388888888888889\right)\right)\right)\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m} \cdot \left(\left(1 + x\_m \cdot \left(x\_m \cdot 0.5\right)\right) + x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)\right)\right)}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= y_m 1.12e+115)
      (/
       (/
        (*
         y_m
         (+
          1.0
          (*
           (* x_m x_m)
           (+
            0.5
            (*
             (* x_m x_m)
             (+ 0.041666666666666664 (* x_m (* x_m 0.001388888888888889))))))))
        x_m)
       z_m)
      (/
       (*
        (/ y_m z_m)
        (+
         (+ 1.0 (* x_m (* x_m 0.5)))
         (* x_m (* x_m (* (* x_m x_m) 0.041666666666666664)))))
       x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1.12e+115) {
		tmp = ((y_m * (1.0 + ((x_m * x_m) * (0.5 + ((x_m * x_m) * (0.041666666666666664 + (x_m * (x_m * 0.001388888888888889)))))))) / x_m) / z_m;
	} else {
		tmp = ((y_m / z_m) * ((1.0 + (x_m * (x_m * 0.5))) + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))))) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 1.12d+115) then
        tmp = ((y_m * (1.0d0 + ((x_m * x_m) * (0.5d0 + ((x_m * x_m) * (0.041666666666666664d0 + (x_m * (x_m * 0.001388888888888889d0)))))))) / x_m) / z_m
    else
        tmp = ((y_m / z_m) * ((1.0d0 + (x_m * (x_m * 0.5d0))) + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664d0))))) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1.12e+115) {
		tmp = ((y_m * (1.0 + ((x_m * x_m) * (0.5 + ((x_m * x_m) * (0.041666666666666664 + (x_m * (x_m * 0.001388888888888889)))))))) / x_m) / z_m;
	} else {
		tmp = ((y_m / z_m) * ((1.0 + (x_m * (x_m * 0.5))) + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))))) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 1.12e+115:
		tmp = ((y_m * (1.0 + ((x_m * x_m) * (0.5 + ((x_m * x_m) * (0.041666666666666664 + (x_m * (x_m * 0.001388888888888889)))))))) / x_m) / z_m
	else:
		tmp = ((y_m / z_m) * ((1.0 + (x_m * (x_m * 0.5))) + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))))) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 1.12e+115)
		tmp = Float64(Float64(Float64(y_m * Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(0.041666666666666664 + Float64(x_m * Float64(x_m * 0.001388888888888889)))))))) / x_m) / z_m);
	else
		tmp = Float64(Float64(Float64(y_m / z_m) * Float64(Float64(1.0 + Float64(x_m * Float64(x_m * 0.5))) + Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664))))) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 1.12e+115)
		tmp = ((y_m * (1.0 + ((x_m * x_m) * (0.5 + ((x_m * x_m) * (0.041666666666666664 + (x_m * (x_m * 0.001388888888888889)))))))) / x_m) / z_m;
	else
		tmp = ((y_m / z_m) * ((1.0 + (x_m * (x_m * 0.5))) + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))))) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1.12e+115], N[(N[(N[(y$95$m * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.041666666666666664 + N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(y$95$m / z$95$m), $MachinePrecision] * N[(N[(1.0 + N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.12e115

   1. Initial program 86.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}\right)}, z\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)\right), x\right), z\right) \]

   5. Simplified 93.6%

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

   if 1.12e115 < y

   1. Initial program 78.6%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   5. Simplified 93.7%

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

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left(1 + \left(\frac{1}{2} \cdot \left(x \cdot x\right) + \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right) + \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right), x\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

      15. *-lowering-*.f64 93.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 93.7%

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

Alternative 4: 90.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := 1 + x\_m \cdot \left(x\_m \cdot 0.5\right)\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.12 \cdot 10^{+115}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(t\_0 + x\_m \cdot \left(0.041666666666666664 \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m} \cdot \left(t\_0 + x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)\right)\right)}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x_m (* x_m 0.5)))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= y_m 1.12e+115)
        (/
         (/
          (* y_m (+ t_0 (* x_m (* 0.041666666666666664 (* x_m (* x_m x_m))))))
          x_m)
         z_m)
        (/
         (*
          (/ y_m z_m)
          (+ t_0 (* x_m (* x_m (* (* x_m x_m) 0.041666666666666664)))))
         x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = 1.0 + (x_m * (x_m * 0.5));
	double tmp;
	if (y_m <= 1.12e+115) {
		tmp = ((y_m * (t_0 + (x_m * (0.041666666666666664 * (x_m * (x_m * x_m)))))) / x_m) / z_m;
	} else {
		tmp = ((y_m / z_m) * (t_0 + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))))) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x_m * (x_m * 0.5d0))
    if (y_m <= 1.12d+115) then
        tmp = ((y_m * (t_0 + (x_m * (0.041666666666666664d0 * (x_m * (x_m * x_m)))))) / x_m) / z_m
    else
        tmp = ((y_m / z_m) * (t_0 + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664d0))))) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = 1.0 + (x_m * (x_m * 0.5));
	double tmp;
	if (y_m <= 1.12e+115) {
		tmp = ((y_m * (t_0 + (x_m * (0.041666666666666664 * (x_m * (x_m * x_m)))))) / x_m) / z_m;
	} else {
		tmp = ((y_m / z_m) * (t_0 + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))))) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	t_0 = 1.0 + (x_m * (x_m * 0.5))
	tmp = 0
	if y_m <= 1.12e+115:
		tmp = ((y_m * (t_0 + (x_m * (0.041666666666666664 * (x_m * (x_m * x_m)))))) / x_m) / z_m
	else:
		tmp = ((y_m / z_m) * (t_0 + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))))) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(1.0 + Float64(x_m * Float64(x_m * 0.5)))
	tmp = 0.0
	if (y_m <= 1.12e+115)
		tmp = Float64(Float64(Float64(y_m * Float64(t_0 + Float64(x_m * Float64(0.041666666666666664 * Float64(x_m * Float64(x_m * x_m)))))) / x_m) / z_m);
	else
		tmp = Float64(Float64(Float64(y_m / z_m) * Float64(t_0 + Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664))))) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	t_0 = 1.0 + (x_m * (x_m * 0.5));
	tmp = 0.0;
	if (y_m <= 1.12e+115)
		tmp = ((y_m * (t_0 + (x_m * (0.041666666666666664 * (x_m * (x_m * x_m)))))) / x_m) / z_m;
	else
		tmp = ((y_m / z_m) * (t_0 + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))))) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(1.0 + N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1.12e+115], N[(N[(N[(y$95$m * N[(t$95$0 + N[(x$95$m * N[(0.041666666666666664 * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(y$95$m / z$95$m), $MachinePrecision] * N[(t$95$0 + N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.12e115

   1. Initial program 86.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}\right)}, z\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)\right), x\right), z\right) \]

   5. Simplified 93.6%

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

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \left(\left(x \cdot x\right) \cdot \frac{1}{2} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right), x\right), z\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right), z\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(\left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right), z\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{720}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{720}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{720}\right)\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

   7. Applied egg-rr 93.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(y + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)\right)}, x\right), z\right) \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)\right), x\right), z\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right) \cdot {x}^{2}\right), x\right), z\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot y + \frac{1}{2} \cdot y\right) \cdot {x}^{2}\right), x\right), z\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + \left(y \cdot \left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)\right) \cdot {x}^{2}\right), x\right), z\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + \left(y \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right), x\right), z\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + y \cdot \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}\right)\right), x\right), z\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + y \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), x\right), z\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), x\right), z\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right) \cdot y\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right), y\right), x\right), z\right) \]

   10. Simplified 90.1%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{24}\right)\right), y\right), x\right), z\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \left(\frac{1}{2} \cdot \left(x \cdot x\right) + \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       3. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right), y\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right)\right), y\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x\right) \cdot x\right)\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} \cdot x\right)\right)\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right)\right), y\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x\right)\right)\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right)\right), y\right), x\right), z\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot \left(x \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right), y\right), x\right), z\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right), y\right), x\right), z\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(x \cdot \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot x\right)\right)\right), y\right), x\right), z\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \left(\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \left(\left(\frac{1}{24} \cdot \left(x \cdot x\right)\right) \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)\right), y\right), x\right), z\right) \]

       17. pow3 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{3}\right)\right)\right), y\right), x\right), z\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \left({x}^{3}\right)\right)\right)\right), y\right), x\right), z\right) \]

       19. cube-unmult N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right), y\right), x\right), z\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right)\right), y\right), x\right), z\right) \]

       21. *-lowering-*.f64 90.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), y\right), x\right), z\right) \]

   12. Applied egg-rr 90.1%

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

   if 1.12e115 < y

   1. Initial program 78.6%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   5. Simplified 93.7%

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

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left(1 + \left(\frac{1}{2} \cdot \left(x \cdot x\right) + \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right) + \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right), x\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

      15. *-lowering-*.f64 93.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 93.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.12 \cdot 10^{+115}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\left(1 + x \cdot \left(x \cdot 0.5\right)\right) + x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \left(\left(1 + x \cdot \left(x \cdot 0.5\right)\right) + x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.25 \cdot 10^{+115}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(1 + x\_m \cdot \left(x\_m \cdot \left(0.5 + x\_m \cdot \left(x\_m \cdot 0.041666666666666664\right)\right)\right)\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m} \cdot \left(\left(1 + x\_m \cdot \left(x\_m \cdot 0.5\right)\right) + x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)\right)\right)}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= y_m 1.25e+115)
      (/
       (/
        (*
         y_m
         (+ 1.0 (* x_m (* x_m (+ 0.5 (* x_m (* x_m 0.041666666666666664)))))))
        x_m)
       z_m)
      (/
       (*
        (/ y_m z_m)
        (+
         (+ 1.0 (* x_m (* x_m 0.5)))
         (* x_m (* x_m (* (* x_m x_m) 0.041666666666666664)))))
       x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1.25e+115) {
		tmp = ((y_m * (1.0 + (x_m * (x_m * (0.5 + (x_m * (x_m * 0.041666666666666664))))))) / x_m) / z_m;
	} else {
		tmp = ((y_m / z_m) * ((1.0 + (x_m * (x_m * 0.5))) + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))))) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 1.25d+115) then
        tmp = ((y_m * (1.0d0 + (x_m * (x_m * (0.5d0 + (x_m * (x_m * 0.041666666666666664d0))))))) / x_m) / z_m
    else
        tmp = ((y_m / z_m) * ((1.0d0 + (x_m * (x_m * 0.5d0))) + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664d0))))) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1.25e+115) {
		tmp = ((y_m * (1.0 + (x_m * (x_m * (0.5 + (x_m * (x_m * 0.041666666666666664))))))) / x_m) / z_m;
	} else {
		tmp = ((y_m / z_m) * ((1.0 + (x_m * (x_m * 0.5))) + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))))) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 1.25e+115:
		tmp = ((y_m * (1.0 + (x_m * (x_m * (0.5 + (x_m * (x_m * 0.041666666666666664))))))) / x_m) / z_m
	else:
		tmp = ((y_m / z_m) * ((1.0 + (x_m * (x_m * 0.5))) + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))))) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 1.25e+115)
		tmp = Float64(Float64(Float64(y_m * Float64(1.0 + Float64(x_m * Float64(x_m * Float64(0.5 + Float64(x_m * Float64(x_m * 0.041666666666666664))))))) / x_m) / z_m);
	else
		tmp = Float64(Float64(Float64(y_m / z_m) * Float64(Float64(1.0 + Float64(x_m * Float64(x_m * 0.5))) + Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664))))) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 1.25e+115)
		tmp = ((y_m * (1.0 + (x_m * (x_m * (0.5 + (x_m * (x_m * 0.041666666666666664))))))) / x_m) / z_m;
	else
		tmp = ((y_m / z_m) * ((1.0 + (x_m * (x_m * 0.5))) + (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))))) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1.25e+115], N[(N[(N[(y$95$m * N[(1.0 + N[(x$95$m * N[(x$95$m * N[(0.5 + N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(y$95$m / z$95$m), $MachinePrecision] * N[(N[(1.0 + N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.25000000000000002e115

   1. Initial program 86.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}\right)}, z\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)\right), x\right), z\right) \]

   5. Simplified 93.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}\right)}, z\right) \]
   7. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot 1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}\right), z\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot 1 + \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right) \cdot {x}^{2}}{x}\right), z\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot 1 + \left(\frac{1}{2} \cdot y + \frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right) \cdot {x}^{2}}{x}\right), z\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot 1 + \left(\frac{1}{2} \cdot y + \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot y\right) \cdot {x}^{2}}{x}\right), z\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot 1 + \left(y \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right) \cdot {x}^{2}}{x}\right), z\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot 1 + y \cdot \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}\right)}{x}\right), z\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot 1 + y \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}{x}\right), z\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}{x}\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), x\right), z\right) \]

   8. Simplified 90.1%

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

   if 1.25000000000000002e115 < y

   1. Initial program 78.6%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   5. Simplified 93.7%

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

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left(1 + \left(\frac{1}{2} \cdot \left(x \cdot x\right) + \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right) + \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right), x\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

      15. *-lowering-*.f64 93.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 93.7%

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

Alternative 6: 86.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{+41}:\\ \;\;\;\;\frac{y\_m}{x\_m} \cdot \frac{1 + x\_m \cdot \left(x\_m \cdot 0.5\right)}{z\_m}\\ \mathbf{elif}\;x\_m \leq 3.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m} \cdot \left(x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot 0.041666666666666664\right)\right)\right)\right)}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(y\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 4e+41)
      (* (/ y_m x_m) (/ (+ 1.0 (* x_m (* x_m 0.5))) z_m))
      (if (<= x_m 3.5e+102)
        (/
         (* (/ y_m z_m) (* x_m (* x_m (* x_m (* x_m 0.041666666666666664)))))
         x_m)
        (/ (* 0.041666666666666664 (* y_m (* x_m (* x_m x_m)))) z_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 4e+41) {
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * 0.5))) / z_m);
	} else if (x_m <= 3.5e+102) {
		tmp = ((y_m / z_m) * (x_m * (x_m * (x_m * (x_m * 0.041666666666666664))))) / x_m;
	} else {
		tmp = (0.041666666666666664 * (y_m * (x_m * (x_m * x_m)))) / z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 4d+41) then
        tmp = (y_m / x_m) * ((1.0d0 + (x_m * (x_m * 0.5d0))) / z_m)
    else if (x_m <= 3.5d+102) then
        tmp = ((y_m / z_m) * (x_m * (x_m * (x_m * (x_m * 0.041666666666666664d0))))) / x_m
    else
        tmp = (0.041666666666666664d0 * (y_m * (x_m * (x_m * x_m)))) / z_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 4e+41) {
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * 0.5))) / z_m);
	} else if (x_m <= 3.5e+102) {
		tmp = ((y_m / z_m) * (x_m * (x_m * (x_m * (x_m * 0.041666666666666664))))) / x_m;
	} else {
		tmp = (0.041666666666666664 * (y_m * (x_m * (x_m * x_m)))) / z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 4e+41:
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * 0.5))) / z_m)
	elif x_m <= 3.5e+102:
		tmp = ((y_m / z_m) * (x_m * (x_m * (x_m * (x_m * 0.041666666666666664))))) / x_m
	else:
		tmp = (0.041666666666666664 * (y_m * (x_m * (x_m * x_m)))) / z_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 4e+41)
		tmp = Float64(Float64(y_m / x_m) * Float64(Float64(1.0 + Float64(x_m * Float64(x_m * 0.5))) / z_m));
	elseif (x_m <= 3.5e+102)
		tmp = Float64(Float64(Float64(y_m / z_m) * Float64(x_m * Float64(x_m * Float64(x_m * Float64(x_m * 0.041666666666666664))))) / x_m);
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(y_m * Float64(x_m * Float64(x_m * x_m)))) / z_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 4e+41)
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * 0.5))) / z_m);
	elseif (x_m <= 3.5e+102)
		tmp = ((y_m / z_m) * (x_m * (x_m * (x_m * (x_m * 0.041666666666666664))))) / x_m;
	else
		tmp = (0.041666666666666664 * (y_m * (x_m * (x_m * x_m)))) / z_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 4e+41], N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(N[(1.0 + N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 3.5e+102], N[(N[(N[(y$95$m / z$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(0.041666666666666664 * N[(y$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 4.00000000000000002e41

   1. Initial program 88.1%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{x} \cdot \cosh x}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{x} \cdot \cosh x\right), \color{blue}{z}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\cosh x \cdot \frac{1}{x}\right), z\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{\cosh x}{x}\right), z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cosh x, x\right), z\right)\right) \]

      10. cosh-lowering-cosh.f64 92.6%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), x\right), z\right)\right) \]

   4. Applied egg-rr 92.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{2} \cdot \frac{{x}^{2}}{z} + \frac{1}{z}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{\frac{1}{2} \cdot {x}^{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{{x}^{2} \cdot \frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{z} + \frac{1}{z}}{x}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}}{x}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}\right), \color{blue}{x}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      9. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{1}{z}\right), x\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z}\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      16. /-lowering-/.f64 79.3%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, z\right)\right), x\right)\right) \]

   7. Simplified 79.3%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. remove-double-div N/A

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

      6. inv-pow N/A

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

      7. pow-prod-down N/A

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

      8. div-inv N/A

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

      9. inv-pow N/A

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

      10. clear-num N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{z}\right), \color{blue}{\left(\frac{y}{x}\right)}\right) \]

      12. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 + \frac{1}{2} \cdot \left(x \cdot x\right)}{z}\right), \left(\frac{\color{blue}{y}}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right), z\right), \left(\frac{\color{blue}{y}}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x\right) \cdot x\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} \cdot x\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      20. /-lowering-/.f64 75.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), z\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 75.6%

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

   if 4.00000000000000002e41 < x < 3.50000000000000011e102

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   5. Simplified 70.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left(\frac{1}{24} \cdot {x}^{\left(2 \cdot 2\right)}\right)\right), x\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left(\frac{1}{24} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}\right)\right), x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right), x\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left(x \cdot \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot x\right)\right)\right)\right)\right), x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

      15. *-lowering-*.f64 70.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), x\right) \]

   8. Simplified 70.1%

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

   if 3.50000000000000011e102 < x

   1. Initial program 63.3%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   5. Simplified 86.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)\right), \color{blue}{z}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \left({x}^{3} \cdot y\right)\right), z\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \left(y \cdot {x}^{3}\right)\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left({x}^{3}\right)\right)\right), z\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left(x \cdot \left(x \cdot x\right)\right)\right)\right), z\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left(x \cdot {x}^{2}\right)\right)\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right)\right), z\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right), z\right) \]

      10. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), z\right) \]

   8. Simplified 100.0%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+41}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1 + x \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(y \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.12 \cdot 10^{+115}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(1 + x\_m \cdot \left(x\_m \cdot \left(0.5 + x\_m \cdot \left(x\_m \cdot 0.041666666666666664\right)\right)\right)\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.5 + \left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= y_m 1.12e+115)
      (/
       (/
        (*
         y_m
         (+ 1.0 (* x_m (* x_m (+ 0.5 (* x_m (* x_m 0.041666666666666664)))))))
        x_m)
       z_m)
      (/
       (*
        y_m
        (/
         (+ 1.0 (* (* x_m x_m) (+ 0.5 (* (* x_m x_m) 0.041666666666666664))))
         z_m))
       x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1.12e+115) {
		tmp = ((y_m * (1.0 + (x_m * (x_m * (0.5 + (x_m * (x_m * 0.041666666666666664))))))) / x_m) / z_m;
	} else {
		tmp = (y_m * ((1.0 + ((x_m * x_m) * (0.5 + ((x_m * x_m) * 0.041666666666666664)))) / z_m)) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 1.12d+115) then
        tmp = ((y_m * (1.0d0 + (x_m * (x_m * (0.5d0 + (x_m * (x_m * 0.041666666666666664d0))))))) / x_m) / z_m
    else
        tmp = (y_m * ((1.0d0 + ((x_m * x_m) * (0.5d0 + ((x_m * x_m) * 0.041666666666666664d0)))) / z_m)) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1.12e+115) {
		tmp = ((y_m * (1.0 + (x_m * (x_m * (0.5 + (x_m * (x_m * 0.041666666666666664))))))) / x_m) / z_m;
	} else {
		tmp = (y_m * ((1.0 + ((x_m * x_m) * (0.5 + ((x_m * x_m) * 0.041666666666666664)))) / z_m)) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 1.12e+115:
		tmp = ((y_m * (1.0 + (x_m * (x_m * (0.5 + (x_m * (x_m * 0.041666666666666664))))))) / x_m) / z_m
	else:
		tmp = (y_m * ((1.0 + ((x_m * x_m) * (0.5 + ((x_m * x_m) * 0.041666666666666664)))) / z_m)) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 1.12e+115)
		tmp = Float64(Float64(Float64(y_m * Float64(1.0 + Float64(x_m * Float64(x_m * Float64(0.5 + Float64(x_m * Float64(x_m * 0.041666666666666664))))))) / x_m) / z_m);
	else
		tmp = Float64(Float64(y_m * Float64(Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.5 + Float64(Float64(x_m * x_m) * 0.041666666666666664)))) / z_m)) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 1.12e+115)
		tmp = ((y_m * (1.0 + (x_m * (x_m * (0.5 + (x_m * (x_m * 0.041666666666666664))))))) / x_m) / z_m;
	else
		tmp = (y_m * ((1.0 + ((x_m * x_m) * (0.5 + ((x_m * x_m) * 0.041666666666666664)))) / z_m)) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1.12e+115], N[(N[(N[(y$95$m * N[(1.0 + N[(x$95$m * N[(x$95$m * N[(0.5 + N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y$95$m * N[(N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.12e115

   1. Initial program 86.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}\right)}, z\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)\right), x\right), z\right) \]

   5. Simplified 93.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}\right)}, z\right) \]
   7. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot 1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}\right), z\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot 1 + \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right) \cdot {x}^{2}}{x}\right), z\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot 1 + \left(\frac{1}{2} \cdot y + \frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right) \cdot {x}^{2}}{x}\right), z\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot 1 + \left(\frac{1}{2} \cdot y + \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot y\right) \cdot {x}^{2}}{x}\right), z\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot 1 + \left(y \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right) \cdot {x}^{2}}{x}\right), z\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot 1 + y \cdot \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}\right)}{x}\right), z\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot 1 + y \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}{x}\right), z\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}{x}\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), x\right), z\right) \]

   8. Simplified 90.1%

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

   if 1.12e115 < y

   1. Initial program 78.6%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   5. Simplified 93.7%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)}{z}\right), x\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)}{z}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)}{z}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right), z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right), z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right), z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right), z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), z\right)\right), x\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right), z\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right), z\right)\right), x\right) \]

      11. *-lowering-*.f64 93.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right), z\right)\right), x\right) \]

   7. Applied egg-rr 93.6%

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

Alternative 8: 88.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4.3 \cdot 10^{-117}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.5 + \left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 4.3e-117)
      (/ y_m (* x_m z_m))
      (/
       (*
        y_m
        (/
         (+ 1.0 (* (* x_m x_m) (+ 0.5 (* (* x_m x_m) 0.041666666666666664))))
         z_m))
       x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 4.3e-117) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (y_m * ((1.0 + ((x_m * x_m) * (0.5 + ((x_m * x_m) * 0.041666666666666664)))) / z_m)) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 4.3d-117) then
        tmp = y_m / (x_m * z_m)
    else
        tmp = (y_m * ((1.0d0 + ((x_m * x_m) * (0.5d0 + ((x_m * x_m) * 0.041666666666666664d0)))) / z_m)) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 4.3e-117) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (y_m * ((1.0 + ((x_m * x_m) * (0.5 + ((x_m * x_m) * 0.041666666666666664)))) / z_m)) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 4.3e-117:
		tmp = y_m / (x_m * z_m)
	else:
		tmp = (y_m * ((1.0 + ((x_m * x_m) * (0.5 + ((x_m * x_m) * 0.041666666666666664)))) / z_m)) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 4.3e-117)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(y_m * Float64(Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.5 + Float64(Float64(x_m * x_m) * 0.041666666666666664)))) / z_m)) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 4.3e-117)
		tmp = y_m / (x_m * z_m);
	else
		tmp = (y_m * ((1.0 + ((x_m * x_m) * (0.5 + ((x_m * x_m) * 0.041666666666666664)))) / z_m)) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 4.3e-117], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.3e-117

   1. Initial program 85.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}, z\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 58.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 58.2%

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

      1. associate-/l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(z \cdot x\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(y, \left(x \cdot \color{blue}{z}\right)\right) \]

      4. *-lowering-*.f64 60.0%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{z}\right)\right) \]

   7. Applied egg-rr 60.0%

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

   if 4.3e-117 < x

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   5. Simplified 84.6%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)}{z}\right), x\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)}{z}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)}{z}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right), z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right), z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right), z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right), z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), z\right)\right), x\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right), z\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right), z\right)\right), x\right) \]

      11. *-lowering-*.f64 92.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right), z\right)\right), x\right) \]

   7. Applied egg-rr 92.7%

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

Alternative 9: 88.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.3 \cdot 10^{+63}:\\ \;\;\;\;\frac{y\_m}{x\_m} \cdot \frac{1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.5 + x\_m \cdot \left(x\_m \cdot 0.041666666666666664\right)\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)\right)\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 2.3e+63)
      (*
       (/ y_m x_m)
       (/
        (+ 1.0 (* (* x_m x_m) (+ 0.5 (* x_m (* x_m 0.041666666666666664)))))
        z_m))
      (/
       (/ (* y_m (* x_m (* x_m (* (* x_m x_m) 0.041666666666666664)))) x_m)
       z_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 2.3e+63) {
		tmp = (y_m / x_m) * ((1.0 + ((x_m * x_m) * (0.5 + (x_m * (x_m * 0.041666666666666664))))) / z_m);
	} else {
		tmp = ((y_m * (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / x_m) / z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 2.3d+63) then
        tmp = (y_m / x_m) * ((1.0d0 + ((x_m * x_m) * (0.5d0 + (x_m * (x_m * 0.041666666666666664d0))))) / z_m)
    else
        tmp = ((y_m * (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664d0)))) / x_m) / z_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 2.3e+63) {
		tmp = (y_m / x_m) * ((1.0 + ((x_m * x_m) * (0.5 + (x_m * (x_m * 0.041666666666666664))))) / z_m);
	} else {
		tmp = ((y_m * (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / x_m) / z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 2.3e+63:
		tmp = (y_m / x_m) * ((1.0 + ((x_m * x_m) * (0.5 + (x_m * (x_m * 0.041666666666666664))))) / z_m)
	else:
		tmp = ((y_m * (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / x_m) / z_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 2.3e+63)
		tmp = Float64(Float64(y_m / x_m) * Float64(Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.5 + Float64(x_m * Float64(x_m * 0.041666666666666664))))) / z_m));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664)))) / x_m) / z_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 2.3e+63)
		tmp = (y_m / x_m) * ((1.0 + ((x_m * x_m) * (0.5 + (x_m * (x_m * 0.041666666666666664))))) / z_m);
	else
		tmp = ((y_m * (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / x_m) / z_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 2.3e+63], N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.5 + N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.29999999999999993e63

   1. Initial program 88.3%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   5. Simplified 80.7%

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

      1. associate-/l* N/A

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

      2. frac-times N/A

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

      3. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{z \cdot x}{y \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(z \cdot x\right), \color{blue}{\left(y \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot z\right), \left(\color{blue}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), \left(\color{blue}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 77.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 77.8%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{24}\right)}{z}\right), \color{blue}{\left(\frac{y}{x}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{24}\right)\right), z\right), \left(\frac{\color{blue}{y}}{x}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      14. /-lowering-/.f64 80.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), z\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 80.7%

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

   if 2.29999999999999993e63 < x

   1. Initial program 68.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}\right)}, z\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)\right), x\right), z\right) \]

   5. Simplified 100.0%

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

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \left(\left(x \cdot x\right) \cdot \frac{1}{2} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right), x\right), z\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right), z\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(\left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right), z\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{720}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{720}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{720}\right)\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(y + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)\right)}, x\right), z\right) \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)\right), x\right), z\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right) \cdot {x}^{2}\right), x\right), z\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot y + \frac{1}{2} \cdot y\right) \cdot {x}^{2}\right), x\right), z\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + \left(y \cdot \left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)\right) \cdot {x}^{2}\right), x\right), z\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + \left(y \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right), x\right), z\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + y \cdot \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}\right)\right), x\right), z\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + y \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), x\right), z\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), x\right), z\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right) \cdot y\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right), y\right), x\right), z\right) \]

   10. Simplified 97.6%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right)}, y\right), x\right), z\right) \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{24} \cdot {x}^{\left(3 + 1\right)}\right), y\right), x\right), z\right) \]

       2. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{24} \cdot \left({x}^{3} \cdot x\right)\right), y\right), x\right), z\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x\right), y\right), x\right), z\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)\right), y\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{3}\right)\right), y\right), x\right), z\right) \]

       6. unpow3 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left({x}^{2} \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right)\right), y\right), x\right), z\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right), y\right), x\right), z\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right), y\right), x\right), z\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right), y\right), x\right), z\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{24}\right)\right)\right), y\right), x\right), z\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right), y\right), x\right), z\right) \]

       14. *-lowering-*.f64 97.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right), y\right), x\right), z\right) \]

   13. Simplified 97.6%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{+63}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.96:\\ \;\;\;\;\frac{\frac{1}{z\_m}}{\frac{x\_m}{y\_m}}\\ \mathbf{elif}\;x\_m \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(0.041666666666666664 \cdot \frac{y\_m}{z\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\left(x\_m \cdot x\_m\right) \cdot 0.5}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 0.96)
      (/ (/ 1.0 z_m) (/ x_m y_m))
      (if (<= x_m 1.3e+154)
        (* (* x_m (* x_m x_m)) (* 0.041666666666666664 (/ y_m z_m)))
        (* y_m (/ (/ (* (* x_m x_m) 0.5) z_m) x_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.96) {
		tmp = (1.0 / z_m) / (x_m / y_m);
	} else if (x_m <= 1.3e+154) {
		tmp = (x_m * (x_m * x_m)) * (0.041666666666666664 * (y_m / z_m));
	} else {
		tmp = y_m * ((((x_m * x_m) * 0.5) / z_m) / x_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.96d0) then
        tmp = (1.0d0 / z_m) / (x_m / y_m)
    else if (x_m <= 1.3d+154) then
        tmp = (x_m * (x_m * x_m)) * (0.041666666666666664d0 * (y_m / z_m))
    else
        tmp = y_m * ((((x_m * x_m) * 0.5d0) / z_m) / x_m)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.96) {
		tmp = (1.0 / z_m) / (x_m / y_m);
	} else if (x_m <= 1.3e+154) {
		tmp = (x_m * (x_m * x_m)) * (0.041666666666666664 * (y_m / z_m));
	} else {
		tmp = y_m * ((((x_m * x_m) * 0.5) / z_m) / x_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.96:
		tmp = (1.0 / z_m) / (x_m / y_m)
	elif x_m <= 1.3e+154:
		tmp = (x_m * (x_m * x_m)) * (0.041666666666666664 * (y_m / z_m))
	else:
		tmp = y_m * ((((x_m * x_m) * 0.5) / z_m) / x_m)
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.96)
		tmp = Float64(Float64(1.0 / z_m) / Float64(x_m / y_m));
	elseif (x_m <= 1.3e+154)
		tmp = Float64(Float64(x_m * Float64(x_m * x_m)) * Float64(0.041666666666666664 * Float64(y_m / z_m)));
	else
		tmp = Float64(y_m * Float64(Float64(Float64(Float64(x_m * x_m) * 0.5) / z_m) / x_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.96)
		tmp = (1.0 / z_m) / (x_m / y_m);
	elseif (x_m <= 1.3e+154)
		tmp = (x_m * (x_m * x_m)) * (0.041666666666666664 * (y_m / z_m));
	else
		tmp = y_m * ((((x_m * x_m) * 0.5) / z_m) / x_m);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 0.96], N[(N[(1.0 / z$95$m), $MachinePrecision] / N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 1.3e+154], N[(N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 0.95999999999999996

   1. Initial program 87.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}, z\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 64.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 64.6%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{z}\right), \color{blue}{\left(\frac{x}{y}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, z\right), \left(\frac{\color{blue}{x}}{y}\right)\right) \]

      7. /-lowering-/.f64 65.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, z\right), \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   7. Applied egg-rr 65.1%

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

   if 0.95999999999999996 < x < 1.29999999999999994e154

   1. Initial program 89.7%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   5. Simplified 70.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} \cdot \frac{y}{z} + \frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z} + \color{blue}{\frac{1}{24} \cdot \frac{y}{z}}\right) \]

      2. metadata-eval N/A

         \[\leadsto {x}^{3} \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \frac{y}{{x}^{2} \cdot z} + \frac{1}{24} \cdot \frac{y}{z}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto {x}^{3} \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)\right) + \color{blue}{\frac{1}{24}} \cdot \frac{y}{z}\right) \]

      4. metadata-eval N/A

         \[\leadsto {x}^{3} \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{24}\right)\right) \cdot \frac{\color{blue}{y}}{z}\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto {x}^{3} \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{24} \cdot \frac{y}{z}\right)\right)\right) \]

      6. distribute-neg-in N/A

         \[\leadsto {x}^{3} \cdot \left(\mathsf{neg}\left(\left(\frac{-1}{2} \cdot \frac{y}{{x}^{2} \cdot z} + \frac{-1}{24} \cdot \frac{y}{z}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-1}{2} \cdot \frac{y}{{x}^{2} \cdot z} + \frac{-1}{24} \cdot \frac{y}{z}\right)\right)\right)}\right) \]

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. distribute-neg-in N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

   8. Simplified 57.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \color{blue}{\left(\frac{1}{24} \cdot \frac{y}{z}\right)}\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

       2. /-lowering-/.f64 57.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   11. Simplified 57.1%

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

   if 1.29999999999999994e154 < x

   1. Initial program 55.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{x} \cdot \cosh x}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{x} \cdot \cosh x\right), \color{blue}{z}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\cosh x \cdot \frac{1}{x}\right), z\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{\cosh x}{x}\right), z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cosh x, x\right), z\right)\right) \]

      10. cosh-lowering-cosh.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), x\right), z\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{2} \cdot \frac{{x}^{2}}{z} + \frac{1}{z}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{\frac{1}{2} \cdot {x}^{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{{x}^{2} \cdot \frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{z} + \frac{1}{z}}{x}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}}{x}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}\right), \color{blue}{x}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      9. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{1}{z}\right), x\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z}\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      16. /-lowering-/.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, z\right)\right), x\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{{x}^{2}}{z}\right)}, x\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot {x}^{2}}{z}\right), x\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {x}^{2}\right), z\right), x\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right), z\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right), z\right), x\right)\right) \]

      5. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), z\right), x\right)\right) \]

   10. Simplified 100.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.96:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{x}{y}}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.041666666666666664 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\left(x \cdot x\right) \cdot 0.5}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.9 \cdot 10^{+21}:\\ \;\;\;\;\frac{y\_m}{x\_m} \cdot \frac{1 + x\_m \cdot \left(x\_m \cdot 0.5\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)\right)\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 3.9e+21)
      (* (/ y_m x_m) (/ (+ 1.0 (* x_m (* x_m 0.5))) z_m))
      (/
       (/ (* y_m (* x_m (* x_m (* (* x_m x_m) 0.041666666666666664)))) x_m)
       z_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 3.9e+21) {
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * 0.5))) / z_m);
	} else {
		tmp = ((y_m * (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / x_m) / z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 3.9d+21) then
        tmp = (y_m / x_m) * ((1.0d0 + (x_m * (x_m * 0.5d0))) / z_m)
    else
        tmp = ((y_m * (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664d0)))) / x_m) / z_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 3.9e+21) {
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * 0.5))) / z_m);
	} else {
		tmp = ((y_m * (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / x_m) / z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 3.9e+21:
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * 0.5))) / z_m)
	else:
		tmp = ((y_m * (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / x_m) / z_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 3.9e+21)
		tmp = Float64(Float64(y_m / x_m) * Float64(Float64(1.0 + Float64(x_m * Float64(x_m * 0.5))) / z_m));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664)))) / x_m) / z_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 3.9e+21)
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * 0.5))) / z_m);
	else
		tmp = ((y_m * (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / x_m) / z_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 3.9e+21], N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(N[(1.0 + N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.9e21

   1. Initial program 87.9%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{x} \cdot \cosh x}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{x} \cdot \cosh x\right), \color{blue}{z}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\cosh x \cdot \frac{1}{x}\right), z\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{\cosh x}{x}\right), z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cosh x, x\right), z\right)\right) \]

      10. cosh-lowering-cosh.f64 92.5%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), x\right), z\right)\right) \]

   4. Applied egg-rr 92.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{2} \cdot \frac{{x}^{2}}{z} + \frac{1}{z}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{\frac{1}{2} \cdot {x}^{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{{x}^{2} \cdot \frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{z} + \frac{1}{z}}{x}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}}{x}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}\right), \color{blue}{x}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      9. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{1}{z}\right), x\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z}\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      16. /-lowering-/.f64 79.9%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, z\right)\right), x\right)\right) \]

   7. Simplified 79.9%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. remove-double-div N/A

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

      6. inv-pow N/A

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

      7. pow-prod-down N/A

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

      8. div-inv N/A

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

      9. inv-pow N/A

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

      10. clear-num N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{z}\right), \color{blue}{\left(\frac{y}{x}\right)}\right) \]

      12. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 + \frac{1}{2} \cdot \left(x \cdot x\right)}{z}\right), \left(\frac{\color{blue}{y}}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right), z\right), \left(\frac{\color{blue}{y}}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x\right) \cdot x\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} \cdot x\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      20. /-lowering-/.f64 76.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), z\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 76.2%

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

   if 3.9e21 < x

   1. Initial program 73.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}\right)}, z\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)\right), x\right), z\right) \]

   5. Simplified 93.9%

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

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \left(\left(x \cdot x\right) \cdot \frac{1}{2} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right), x\right), z\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), x\right), z\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right), z\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(\left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right), x\right), z\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{720}\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{720}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{720}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{720}\right)\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(x \cdot x\right)\right)\right)\right)\right), x\right), z\right) \]

   7. Applied egg-rr 93.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(y + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)\right)}, x\right), z\right) \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)\right), x\right), z\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right) \cdot {x}^{2}\right), x\right), z\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot y + \frac{1}{2} \cdot y\right) \cdot {x}^{2}\right), x\right), z\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + \left(y \cdot \left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)\right) \cdot {x}^{2}\right), x\right), z\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + \left(y \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right), x\right), z\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + y \cdot \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}\right)\right), x\right), z\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot 1 + y \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), x\right), z\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), x\right), z\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right) \cdot y\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right), y\right), x\right), z\right) \]

   10. Simplified 83.3%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right)}, y\right), x\right), z\right) \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{24} \cdot {x}^{\left(3 + 1\right)}\right), y\right), x\right), z\right) \]

       2. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{24} \cdot \left({x}^{3} \cdot x\right)\right), y\right), x\right), z\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x\right), y\right), x\right), z\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)\right), y\right), x\right), z\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{3}\right)\right), y\right), x\right), z\right) \]

       6. unpow3 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left({x}^{2} \cdot x\right)\right)\right), y\right), x\right), z\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right)\right), y\right), x\right), z\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right), y\right), x\right), z\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right), y\right), x\right), z\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right), y\right), x\right), z\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{24}\right)\right)\right), y\right), x\right), z\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right), y\right), x\right), z\right) \]

       14. *-lowering-*.f64 83.3%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right), y\right), x\right), z\right) \]

   13. Simplified 83.3%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{+21}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1 + x \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 9 \cdot 10^{+20}:\\ \;\;\;\;\frac{y\_m}{x\_m} \cdot \frac{1 + x\_m \cdot \left(x\_m \cdot 0.5\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot \left(y\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)\right)\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 9e+20)
      (* (/ y_m x_m) (/ (+ 1.0 (* x_m (* x_m 0.5))) z_m))
      (/
       (/ (* x_m (* y_m (* x_m (* (* x_m x_m) 0.041666666666666664)))) z_m)
       x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 9e+20) {
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * 0.5))) / z_m);
	} else {
		tmp = ((x_m * (y_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 9d+20) then
        tmp = (y_m / x_m) * ((1.0d0 + (x_m * (x_m * 0.5d0))) / z_m)
    else
        tmp = ((x_m * (y_m * (x_m * ((x_m * x_m) * 0.041666666666666664d0)))) / z_m) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 9e+20) {
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * 0.5))) / z_m);
	} else {
		tmp = ((x_m * (y_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 9e+20:
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * 0.5))) / z_m)
	else:
		tmp = ((x_m * (y_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / z_m) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 9e+20)
		tmp = Float64(Float64(y_m / x_m) * Float64(Float64(1.0 + Float64(x_m * Float64(x_m * 0.5))) / z_m));
	else
		tmp = Float64(Float64(Float64(x_m * Float64(y_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664)))) / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 9e+20)
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * 0.5))) / z_m);
	else
		tmp = ((x_m * (y_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / z_m) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 9e+20], N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(N[(1.0 + N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * N[(y$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9e20

   1. Initial program 87.9%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{x} \cdot \cosh x}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{x} \cdot \cosh x\right), \color{blue}{z}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\cosh x \cdot \frac{1}{x}\right), z\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{\cosh x}{x}\right), z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cosh x, x\right), z\right)\right) \]

      10. cosh-lowering-cosh.f64 92.5%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), x\right), z\right)\right) \]

   4. Applied egg-rr 92.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{2} \cdot \frac{{x}^{2}}{z} + \frac{1}{z}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{\frac{1}{2} \cdot {x}^{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{{x}^{2} \cdot \frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{z} + \frac{1}{z}}{x}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}}{x}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}\right), \color{blue}{x}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      9. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{1}{z}\right), x\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z}\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      16. /-lowering-/.f64 79.9%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, z\right)\right), x\right)\right) \]

   7. Simplified 79.9%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. remove-double-div N/A

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

      6. inv-pow N/A

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

      7. pow-prod-down N/A

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

      8. div-inv N/A

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

      9. inv-pow N/A

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

      10. clear-num N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{z}\right), \color{blue}{\left(\frac{y}{x}\right)}\right) \]

      12. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 + \frac{1}{2} \cdot \left(x \cdot x\right)}{z}\right), \left(\frac{\color{blue}{y}}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right), z\right), \left(\frac{\color{blue}{y}}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x\right) \cdot x\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} \cdot x\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      20. /-lowering-/.f64 76.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), z\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 76.2%

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

   if 9e20 < x

   1. Initial program 73.9%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   5. Simplified 76.6%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right) + 1\right)\right), x\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right), 1\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right), 1\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right), 1\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right), 1\right)\right), x\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right), 1\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right), 1\right)\right), x\right) \]

      8. *-lowering-*.f64 76.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right), 1\right)\right), x\right) \]

   7. Applied egg-rr 76.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{24} \cdot \frac{{x}^{4} \cdot y}{z}\right)}, x\right) \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{24} \cdot \left({x}^{4} \cdot \frac{y}{z}\right)\right), x\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{24} \cdot \left({x}^{\left(2 \cdot 2\right)} \cdot \frac{y}{z}\right)\right), x\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{24} \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \frac{y}{z}\right)\right), x\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \frac{y}{z}\right)\right)\right), x\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{24} \cdot \left({x}^{2} \cdot \frac{{x}^{2} \cdot y}{z}\right)\right), x\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \frac{{x}^{2} \cdot y}{z}\right), x\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot y\right)}{z}\right), x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left({x}^{2} \cdot \frac{1}{24}\right) \cdot \left({x}^{2} \cdot y\right)}{z}\right), x\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)}{z}\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)\right), z\right), x\right) \]

   10. Simplified 83.3%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+20}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1 + x \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(y \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 9 \cdot 10^{+20}:\\ \;\;\;\;\frac{y\_m}{x\_m} \cdot \frac{1 + x\_m \cdot \left(x\_m \cdot 0.5\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(y\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 9e+20)
      (* (/ y_m x_m) (/ (+ 1.0 (* x_m (* x_m 0.5))) z_m))
      (/ (* 0.041666666666666664 (* y_m (* x_m (* x_m x_m)))) z_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 9e+20) {
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * 0.5))) / z_m);
	} else {
		tmp = (0.041666666666666664 * (y_m * (x_m * (x_m * x_m)))) / z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 9d+20) then
        tmp = (y_m / x_m) * ((1.0d0 + (x_m * (x_m * 0.5d0))) / z_m)
    else
        tmp = (0.041666666666666664d0 * (y_m * (x_m * (x_m * x_m)))) / z_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 9e+20) {
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * 0.5))) / z_m);
	} else {
		tmp = (0.041666666666666664 * (y_m * (x_m * (x_m * x_m)))) / z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 9e+20:
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * 0.5))) / z_m)
	else:
		tmp = (0.041666666666666664 * (y_m * (x_m * (x_m * x_m)))) / z_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 9e+20)
		tmp = Float64(Float64(y_m / x_m) * Float64(Float64(1.0 + Float64(x_m * Float64(x_m * 0.5))) / z_m));
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(y_m * Float64(x_m * Float64(x_m * x_m)))) / z_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 9e+20)
		tmp = (y_m / x_m) * ((1.0 + (x_m * (x_m * 0.5))) / z_m);
	else
		tmp = (0.041666666666666664 * (y_m * (x_m * (x_m * x_m)))) / z_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 9e+20], N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(N[(1.0 + N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(0.041666666666666664 * N[(y$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9e20

   1. Initial program 87.9%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{x} \cdot \cosh x}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{x} \cdot \cosh x\right), \color{blue}{z}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\cosh x \cdot \frac{1}{x}\right), z\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{\cosh x}{x}\right), z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cosh x, x\right), z\right)\right) \]

      10. cosh-lowering-cosh.f64 92.5%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), x\right), z\right)\right) \]

   4. Applied egg-rr 92.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{2} \cdot \frac{{x}^{2}}{z} + \frac{1}{z}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{\frac{1}{2} \cdot {x}^{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{{x}^{2} \cdot \frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{z} + \frac{1}{z}}{x}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}}{x}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}\right), \color{blue}{x}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      9. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{1}{z}\right), x\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z}\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      16. /-lowering-/.f64 79.9%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, z\right)\right), x\right)\right) \]

   7. Simplified 79.9%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. remove-double-div N/A

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

      6. inv-pow N/A

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

      7. pow-prod-down N/A

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

      8. div-inv N/A

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

      9. inv-pow N/A

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

      10. clear-num N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{z}\right), \color{blue}{\left(\frac{y}{x}\right)}\right) \]

      12. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 + \frac{1}{2} \cdot \left(x \cdot x\right)}{z}\right), \left(\frac{\color{blue}{y}}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right), z\right), \left(\frac{\color{blue}{y}}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x\right) \cdot x\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} \cdot x\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), z\right), \left(\frac{y}{x}\right)\right) \]

      20. /-lowering-/.f64 76.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), z\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 76.2%

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

   if 9e20 < x

   1. Initial program 73.9%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   5. Simplified 76.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)\right), \color{blue}{z}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \left({x}^{3} \cdot y\right)\right), z\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \left(y \cdot {x}^{3}\right)\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left({x}^{3}\right)\right)\right), z\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left(x \cdot \left(x \cdot x\right)\right)\right)\right), z\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left(x \cdot {x}^{2}\right)\right)\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right)\right), z\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right), z\right) \]

      10. *-lowering-*.f64 77.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), z\right) \]

   8. Simplified 77.1%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+20}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1 + x \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(y \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{-120}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \mathbf{elif}\;x\_m \leq 0.96:\\ \;\;\;\;\frac{1}{z\_m \cdot \frac{x\_m}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{0.5}{z\_m}\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 3.6e-120)
      (/ y_m (* x_m z_m))
      (if (<= x_m 0.96)
        (/ 1.0 (* z_m (/ x_m y_m)))
        (* y_m (* x_m (/ 0.5 z_m)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 3.6e-120) {
		tmp = y_m / (x_m * z_m);
	} else if (x_m <= 0.96) {
		tmp = 1.0 / (z_m * (x_m / y_m));
	} else {
		tmp = y_m * (x_m * (0.5 / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 3.6d-120) then
        tmp = y_m / (x_m * z_m)
    else if (x_m <= 0.96d0) then
        tmp = 1.0d0 / (z_m * (x_m / y_m))
    else
        tmp = y_m * (x_m * (0.5d0 / z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 3.6e-120) {
		tmp = y_m / (x_m * z_m);
	} else if (x_m <= 0.96) {
		tmp = 1.0 / (z_m * (x_m / y_m));
	} else {
		tmp = y_m * (x_m * (0.5 / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 3.6e-120:
		tmp = y_m / (x_m * z_m)
	elif x_m <= 0.96:
		tmp = 1.0 / (z_m * (x_m / y_m))
	else:
		tmp = y_m * (x_m * (0.5 / z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 3.6e-120)
		tmp = Float64(y_m / Float64(x_m * z_m));
	elseif (x_m <= 0.96)
		tmp = Float64(1.0 / Float64(z_m * Float64(x_m / y_m)));
	else
		tmp = Float64(y_m * Float64(x_m * Float64(0.5 / z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 3.6e-120)
		tmp = y_m / (x_m * z_m);
	elseif (x_m <= 0.96)
		tmp = 1.0 / (z_m * (x_m / y_m));
	else
		tmp = y_m * (x_m * (0.5 / z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 3.6e-120], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 0.96], N[(1.0 / N[(z$95$m * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 3.6000000000000003e-120

   1. Initial program 85.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}, z\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 58.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 58.2%

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

      1. associate-/l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(z \cdot x\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(y, \left(x \cdot \color{blue}{z}\right)\right) \]

      4. *-lowering-*.f64 60.0%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{z}\right)\right) \]

   7. Applied egg-rr 60.0%

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

   if 3.6000000000000003e-120 < x < 0.95999999999999996

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}, z\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 96.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 96.5%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{y} \cdot z\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{x}{y}\right), \color{blue}{z}\right)\right) \]

      7. /-lowering-/.f64 96.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), z\right)\right) \]

   7. Applied egg-rr 96.5%

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

   if 0.95999999999999996 < x

   1. Initial program 75.5%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{x} \cdot \cosh x}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{x} \cdot \cosh x\right), \color{blue}{z}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\cosh x \cdot \frac{1}{x}\right), z\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{\cosh x}{x}\right), z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cosh x, x\right), z\right)\right) \]

      10. cosh-lowering-cosh.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), x\right), z\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{2} \cdot \frac{{x}^{2}}{z} + \frac{1}{z}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{\frac{1}{2} \cdot {x}^{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{{x}^{2} \cdot \frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{z} + \frac{1}{z}}{x}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}}{x}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}\right), \color{blue}{x}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      9. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{1}{z}\right), x\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z}\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      16. /-lowering-/.f64 64.6%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, z\right)\right), x\right)\right) \]

   7. Simplified 64.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{z}\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x \cdot \frac{\frac{1}{2} \cdot 1}{z}\right)\right) \]

      5. associate-*r/ N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(\frac{\frac{1}{2}}{z}\right)\right)\right) \]

      9. /-lowering-/.f64 44.9%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{z}\right)\right)\right) \]

   10. Simplified 44.9%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-120}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\frac{1}{z \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 86.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.96:\\ \;\;\;\;\frac{\frac{1}{z\_m}}{\frac{x\_m}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(y\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 0.96)
      (/ (/ 1.0 z_m) (/ x_m y_m))
      (/ (* 0.041666666666666664 (* y_m (* x_m (* x_m x_m)))) z_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.96) {
		tmp = (1.0 / z_m) / (x_m / y_m);
	} else {
		tmp = (0.041666666666666664 * (y_m * (x_m * (x_m * x_m)))) / z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.96d0) then
        tmp = (1.0d0 / z_m) / (x_m / y_m)
    else
        tmp = (0.041666666666666664d0 * (y_m * (x_m * (x_m * x_m)))) / z_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.96) {
		tmp = (1.0 / z_m) / (x_m / y_m);
	} else {
		tmp = (0.041666666666666664 * (y_m * (x_m * (x_m * x_m)))) / z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.96:
		tmp = (1.0 / z_m) / (x_m / y_m)
	else:
		tmp = (0.041666666666666664 * (y_m * (x_m * (x_m * x_m)))) / z_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.96)
		tmp = Float64(Float64(1.0 / z_m) / Float64(x_m / y_m));
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(y_m * Float64(x_m * Float64(x_m * x_m)))) / z_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.96)
		tmp = (1.0 / z_m) / (x_m / y_m);
	else
		tmp = (0.041666666666666664 * (y_m * (x_m * (x_m * x_m)))) / z_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 0.96], N[(N[(1.0 / z$95$m), $MachinePrecision] / N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(0.041666666666666664 * N[(y$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.95999999999999996

   1. Initial program 87.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}, z\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 64.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 64.6%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{z}\right), \color{blue}{\left(\frac{x}{y}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, z\right), \left(\frac{\color{blue}{x}}{y}\right)\right) \]

      7. /-lowering-/.f64 65.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, z\right), \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   7. Applied egg-rr 65.1%

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

   if 0.95999999999999996 < x

   1. Initial program 75.5%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   5. Simplified 74.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)\right), \color{blue}{z}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \left({x}^{3} \cdot y\right)\right), z\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \left(y \cdot {x}^{3}\right)\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left({x}^{3}\right)\right)\right), z\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left(x \cdot \left(x \cdot x\right)\right)\right)\right), z\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left(x \cdot {x}^{2}\right)\right)\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right)\right), z\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right), z\right) \]

      10. *-lowering-*.f64 74.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), z\right) \]

   8. Simplified 74.6%

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

Alternative 16: 79.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.96:\\ \;\;\;\;\frac{\frac{1}{z\_m}}{\frac{x\_m}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\left(x\_m \cdot x\_m\right) \cdot 0.5}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 0.96)
      (/ (/ 1.0 z_m) (/ x_m y_m))
      (* y_m (/ (/ (* (* x_m x_m) 0.5) z_m) x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.96) {
		tmp = (1.0 / z_m) / (x_m / y_m);
	} else {
		tmp = y_m * ((((x_m * x_m) * 0.5) / z_m) / x_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.96d0) then
        tmp = (1.0d0 / z_m) / (x_m / y_m)
    else
        tmp = y_m * ((((x_m * x_m) * 0.5d0) / z_m) / x_m)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.96) {
		tmp = (1.0 / z_m) / (x_m / y_m);
	} else {
		tmp = y_m * ((((x_m * x_m) * 0.5) / z_m) / x_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.96:
		tmp = (1.0 / z_m) / (x_m / y_m)
	else:
		tmp = y_m * ((((x_m * x_m) * 0.5) / z_m) / x_m)
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.96)
		tmp = Float64(Float64(1.0 / z_m) / Float64(x_m / y_m));
	else
		tmp = Float64(y_m * Float64(Float64(Float64(Float64(x_m * x_m) * 0.5) / z_m) / x_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.96)
		tmp = (1.0 / z_m) / (x_m / y_m);
	else
		tmp = y_m * ((((x_m * x_m) * 0.5) / z_m) / x_m);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 0.96], N[(N[(1.0 / z$95$m), $MachinePrecision] / N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.95999999999999996

   1. Initial program 87.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}, z\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 64.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 64.6%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{z}\right), \color{blue}{\left(\frac{x}{y}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, z\right), \left(\frac{\color{blue}{x}}{y}\right)\right) \]

      7. /-lowering-/.f64 65.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, z\right), \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   7. Applied egg-rr 65.1%

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

   if 0.95999999999999996 < x

   1. Initial program 75.5%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{x} \cdot \cosh x}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{x} \cdot \cosh x\right), \color{blue}{z}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\cosh x \cdot \frac{1}{x}\right), z\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{\cosh x}{x}\right), z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cosh x, x\right), z\right)\right) \]

      10. cosh-lowering-cosh.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), x\right), z\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{2} \cdot \frac{{x}^{2}}{z} + \frac{1}{z}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{\frac{1}{2} \cdot {x}^{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{{x}^{2} \cdot \frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{z} + \frac{1}{z}}{x}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}}{x}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}\right), \color{blue}{x}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      9. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{1}{z}\right), x\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z}\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      16. /-lowering-/.f64 64.6%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, z\right)\right), x\right)\right) \]

   7. Simplified 64.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{{x}^{2}}{z}\right)}, x\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot {x}^{2}}{z}\right), x\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {x}^{2}\right), z\right), x\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right), z\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right), z\right), x\right)\right) \]

      5. *-lowering-*.f64 64.6%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), z\right), x\right)\right) \]

   10. Simplified 64.6%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.96:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\left(x \cdot x\right) \cdot 0.5}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.96:\\ \;\;\;\;\frac{\frac{1}{z\_m}}{\frac{x\_m}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{0.5}{z\_m}\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 0.96)
      (/ (/ 1.0 z_m) (/ x_m y_m))
      (* y_m (* x_m (/ 0.5 z_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.96) {
		tmp = (1.0 / z_m) / (x_m / y_m);
	} else {
		tmp = y_m * (x_m * (0.5 / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.96d0) then
        tmp = (1.0d0 / z_m) / (x_m / y_m)
    else
        tmp = y_m * (x_m * (0.5d0 / z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.96) {
		tmp = (1.0 / z_m) / (x_m / y_m);
	} else {
		tmp = y_m * (x_m * (0.5 / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.96:
		tmp = (1.0 / z_m) / (x_m / y_m)
	else:
		tmp = y_m * (x_m * (0.5 / z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.96)
		tmp = Float64(Float64(1.0 / z_m) / Float64(x_m / y_m));
	else
		tmp = Float64(y_m * Float64(x_m * Float64(0.5 / z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.96)
		tmp = (1.0 / z_m) / (x_m / y_m);
	else
		tmp = y_m * (x_m * (0.5 / z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 0.96], N[(N[(1.0 / z$95$m), $MachinePrecision] / N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.95999999999999996

   1. Initial program 87.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}, z\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 64.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 64.6%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{z}\right), \color{blue}{\left(\frac{x}{y}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, z\right), \left(\frac{\color{blue}{x}}{y}\right)\right) \]

      7. /-lowering-/.f64 65.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, z\right), \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   7. Applied egg-rr 65.1%

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

   if 0.95999999999999996 < x

   1. Initial program 75.5%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{x} \cdot \cosh x}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{x} \cdot \cosh x\right), \color{blue}{z}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\cosh x \cdot \frac{1}{x}\right), z\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{\cosh x}{x}\right), z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cosh x, x\right), z\right)\right) \]

      10. cosh-lowering-cosh.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), x\right), z\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{2} \cdot \frac{{x}^{2}}{z} + \frac{1}{z}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{\frac{1}{2} \cdot {x}^{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{{x}^{2} \cdot \frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{z} + \frac{1}{z}}{x}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}}{x}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}\right), \color{blue}{x}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      9. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{1}{z}\right), x\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z}\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      16. /-lowering-/.f64 64.6%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, z\right)\right), x\right)\right) \]

   7. Simplified 64.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{z}\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x \cdot \frac{\frac{1}{2} \cdot 1}{z}\right)\right) \]

      5. associate-*r/ N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(\frac{\frac{1}{2}}{z}\right)\right)\right) \]

      9. /-lowering-/.f64 44.9%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{z}\right)\right)\right) \]

   10. Simplified 44.9%

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

Alternative 18: 65.2% accurate, 8.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.96:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{0.5}{z\_m}\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 0.96) (/ (/ y_m x_m) z_m) (* y_m (* x_m (/ 0.5 z_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.96) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = y_m * (x_m * (0.5 / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.96d0) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = y_m * (x_m * (0.5d0 / z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.96) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = y_m * (x_m * (0.5 / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.96:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = y_m * (x_m * (0.5 / z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.96)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(y_m * Float64(x_m * Float64(0.5 / z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.96)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = y_m * (x_m * (0.5 / z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 0.96], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.95999999999999996

   1. Initial program 87.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}, z\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 64.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 64.6%

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

   if 0.95999999999999996 < x

   1. Initial program 75.5%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{x} \cdot \cosh x}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{x} \cdot \cosh x\right), \color{blue}{z}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\cosh x \cdot \frac{1}{x}\right), z\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{\cosh x}{x}\right), z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cosh x, x\right), z\right)\right) \]

      10. cosh-lowering-cosh.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), x\right), z\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{2} \cdot \frac{{x}^{2}}{z} + \frac{1}{z}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{\frac{1}{2} \cdot {x}^{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{\frac{{x}^{2} \cdot \frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2}}{z} + \frac{1}{z}}{x}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{z} + \frac{1}{z}}{x}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{{x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}}{x}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}\right), \color{blue}{x}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z} + \frac{1}{z}\right), x\right)\right) \]

      9. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{1}{z}\right), x\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z}\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{z}\right)\right), x\right)\right) \]

      16. /-lowering-/.f64 64.6%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, z\right)\right), x\right)\right) \]

   7. Simplified 64.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{z}\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x \cdot \frac{\frac{1}{2} \cdot 1}{z}\right)\right) \]

      5. associate-*r/ N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(\frac{\frac{1}{2}}{z}\right)\right)\right) \]

      9. /-lowering-/.f64 44.9%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{z}\right)\right)\right) \]

   10. Simplified 44.9%

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

Alternative 19: 61.1% accurate, 8.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.96:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(0.5 \cdot \frac{y\_m}{z\_m}\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 0.96) (/ (/ y_m x_m) z_m) (* x_m (* 0.5 (/ y_m z_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.96) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = x_m * (0.5 * (y_m / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.96d0) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = x_m * (0.5d0 * (y_m / z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.96) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = x_m * (0.5 * (y_m / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.96:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = x_m * (0.5 * (y_m / z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.96)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(x_m * Float64(0.5 * Float64(y_m / z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.96)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = x_m * (0.5 * (y_m / z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 0.96], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(x$95$m * N[(0.5 * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.95999999999999996

   1. Initial program 87.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}, z\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 64.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 64.6%

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

   if 0.95999999999999996 < x

   1. Initial program 75.5%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   5. Simplified 74.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} \cdot \frac{y}{z} + \frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z} + \color{blue}{\frac{1}{24} \cdot \frac{y}{z}}\right) \]

      2. metadata-eval N/A

         \[\leadsto {x}^{3} \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \frac{y}{{x}^{2} \cdot z} + \frac{1}{24} \cdot \frac{y}{z}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto {x}^{3} \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)\right) + \color{blue}{\frac{1}{24}} \cdot \frac{y}{z}\right) \]

      4. metadata-eval N/A

         \[\leadsto {x}^{3} \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{24}\right)\right) \cdot \frac{\color{blue}{y}}{z}\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto {x}^{3} \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{24} \cdot \frac{y}{z}\right)\right)\right) \]

      6. distribute-neg-in N/A

         \[\leadsto {x}^{3} \cdot \left(\mathsf{neg}\left(\left(\frac{-1}{2} \cdot \frac{y}{{x}^{2} \cdot z} + \frac{-1}{24} \cdot \frac{y}{z}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-1}{2} \cdot \frac{y}{{x}^{2} \cdot z} + \frac{-1}{24} \cdot \frac{y}{z}\right)\right)\right)}\right) \]

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. distribute-neg-in N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

   8. Simplified 66.4%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

       7. /-lowering-/.f64 31.3%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   11. Simplified 31.3%

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

Alternative 20: 55.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (* y_s (* x_s (if (<= y_m 5e+18) (/ (/ y_m x_m) z_m) (/ (/ y_m z_m) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 5e+18) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (y_m / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 5d+18) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = (y_m / z_m) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 5e+18) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (y_m / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 5e+18:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = (y_m / z_m) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 5e+18)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(Float64(y_m / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 5e+18)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = (y_m / z_m) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 5e+18], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5e18

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}, z\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 55.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 55.3%

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

   if 5e18 < y

   1. Initial program 84.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{z}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 66.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, z\right), x\right) \]

   5. Simplified 66.6%

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

Alternative 21: 51.5% accurate, 10.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (* y_s (* x_s (if (<= y_m 3e-62) (/ (/ y_m x_m) z_m) (/ y_m (* x_m z_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 3e-62) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = y_m / (x_m * z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 3d-62) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = y_m / (x_m * z_m)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 3e-62) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = y_m / (x_m * z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 3e-62:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = y_m / (x_m * z_m)
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 3e-62)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(y_m / Float64(x_m * z_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 3e-62)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = y_m / (x_m * z_m);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 3e-62], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.0000000000000001e-62

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}, z\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 53.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 53.4%

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

   if 3.0000000000000001e-62 < y

   1. Initial program 89.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}, z\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 54.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 54.0%

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

      1. associate-/l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(z \cdot x\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(y, \left(x \cdot \color{blue}{z}\right)\right) \]

      4. *-lowering-*.f64 64.2%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{z}\right)\right) \]

   7. Applied egg-rr 64.2%

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

Alternative 22: 48.3% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = z_s * (y_s * (x_s * (y_m / (x_m * z_m))));
end

Wolfram

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

Derivation

1. Initial program 85.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}, z\right) \]
4. Step-by-step derivation

   1. /-lowering-/.f64 53.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

5. Simplified 53.6%

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

   1. associate-/l/ N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(z \cdot x\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(y, \left(x \cdot \color{blue}{z}\right)\right) \]

   4. *-lowering-*.f64 53.0%

      \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{z}\right)\right) \]

7. Applied egg-rr 53.0%

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

Developer Target 1: 97.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y z) x) (cosh x))))
   (if (< y -4.618902267687042e-52)
     t_0
     (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y / z) / x) * cosh(x)
    if (y < (-4.618902267687042d-52)) then
        tmp = t_0
    else if (y < 1.038530535935153d-39) then
        tmp = ((cosh(x) * y) / x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * Math.cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((Math.cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / z) / x) * math.cosh(x)
	tmp = 0
	if y < -4.618902267687042e-52:
		tmp = t_0
	elif y < 1.038530535935153e-39:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) / x) * cosh(x))
	tmp = 0.0
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / z) / x) * cosh(x);
	tmp = 0.0;
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -4.618902267687042e-52], t$95$0, If[Less[y, 1.038530535935153e-39], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024192 
(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))