Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 85.0% → 99.8%

Time: 12.8s

Alternatives: 20

Speedup: 6.7×

• 65.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+105}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x\_m \cdot y\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

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 (/ (* (cosh x_m) (/ y_m x_m)) z_m)))
   (*
    z_s
    (*
     y_s
     (* x_s (if (<= t_0 2e+105) t_0 (/ (/ (* (cosh x_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 t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= 2e+105) {
		tmp = t_0;
	} else {
		tmp = ((cosh(x_m) * 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) :: t_0
    real(8) :: tmp
    t_0 = (cosh(x_m) * (y_m / x_m)) / z_m
    if (t_0 <= 2d+105) then
        tmp = t_0
    else
        tmp = ((cosh(x_m) * 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 t_0 = (Math.cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= 2e+105) {
		tmp = t_0;
	} else {
		tmp = ((Math.cosh(x_m) * 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):
	t_0 = (math.cosh(x_m) * (y_m / x_m)) / z_m
	tmp = 0
	if t_0 <= 2e+105:
		tmp = t_0
	else:
		tmp = ((math.cosh(x_m) * 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)
	t_0 = Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m)
	tmp = 0.0
	if (t_0 <= 2e+105)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(cosh(x_m) * 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)
	t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	tmp = 0.0;
	if (t_0 <= 2e+105)
		tmp = t_0;
	else
		tmp = ((cosh(x_m) * 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_] := Block[{t$95$0 = N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 2e+105], t$95$0, N[(N[(N[(N[Cosh[x$95$m], $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.9999999999999999e105

   1. Initial program 97.2%

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

   if 1.9999999999999999e105 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 73.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. cosh-lowering-cosh.f64 99.9%

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

   3. Simplified 99.9%

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

Alternative 2: 95.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-189}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\frac{z\_m}{y\_m}}\\ \mathbf{elif}\;x\_m \leq 9 \cdot 10^{+44}:\\ \;\;\;\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m + y\_m \cdot \left(\left(0.041666666666666664 + \left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 5e-189)
      (/ (/ 1.0 x_m) (/ z_m y_m))
      (if (<= x_m 9e+44)
        (/ (* (cosh x_m) (/ y_m x_m)) z_m)
        (/
         (/
          (+
           y_m
           (*
            y_m
            (*
             (+ 0.041666666666666664 (* (* x_m x_m) 0.001388888888888889))
             (* (* x_m x_m) (* x_m x_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 (x_m <= 5e-189) {
		tmp = (1.0 / x_m) / (z_m / y_m);
	} else if (x_m <= 9e+44) {
		tmp = (cosh(x_m) * (y_m / x_m)) / z_m;
	} else {
		tmp = ((y_m + (y_m * ((0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)) * ((x_m * x_m) * (x_m * x_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 (x_m <= 5d-189) then
        tmp = (1.0d0 / x_m) / (z_m / y_m)
    else if (x_m <= 9d+44) then
        tmp = (cosh(x_m) * (y_m / x_m)) / z_m
    else
        tmp = ((y_m + (y_m * ((0.041666666666666664d0 + ((x_m * x_m) * 0.001388888888888889d0)) * ((x_m * x_m) * (x_m * x_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 (x_m <= 5e-189) {
		tmp = (1.0 / x_m) / (z_m / y_m);
	} else if (x_m <= 9e+44) {
		tmp = (Math.cosh(x_m) * (y_m / x_m)) / z_m;
	} else {
		tmp = ((y_m + (y_m * ((0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)) * ((x_m * x_m) * (x_m * x_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 x_m <= 5e-189:
		tmp = (1.0 / x_m) / (z_m / y_m)
	elif x_m <= 9e+44:
		tmp = (math.cosh(x_m) * (y_m / x_m)) / z_m
	else:
		tmp = ((y_m + (y_m * ((0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)) * ((x_m * x_m) * (x_m * x_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 (x_m <= 5e-189)
		tmp = Float64(Float64(1.0 / x_m) / Float64(z_m / y_m));
	elseif (x_m <= 9e+44)
		tmp = Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m);
	else
		tmp = Float64(Float64(Float64(y_m + Float64(y_m * Float64(Float64(0.041666666666666664 + Float64(Float64(x_m * x_m) * 0.001388888888888889)) * Float64(Float64(x_m * x_m) * Float64(x_m * x_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 (x_m <= 5e-189)
		tmp = (1.0 / x_m) / (z_m / y_m);
	elseif (x_m <= 9e+44)
		tmp = (cosh(x_m) * (y_m / x_m)) / z_m;
	else
		tmp = ((y_m + (y_m * ((0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)) * ((x_m * x_m) * (x_m * x_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[x$95$m, 5e-189], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 9e+44], N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(y$95$m + N[(y$95$m * N[(N[(0.041666666666666664 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 4.9999999999999997e-189

   1. Initial program 86.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 64.9%

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

   5. Simplified 64.9%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

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

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

      7. /-lowering-/.f64 64.1%

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

   7. Applied egg-rr 64.1%

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

   if 4.9999999999999997e-189 < x < 9e44

   1. Initial program 96.3%

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

   if 9e44 < x

   1. Initial program 77.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. cosh-lowering-cosh.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

   12. Applied egg-rr 100.0%

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

   13. Taylor expanded in x around 0

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

   15. Simplified 100.0%

       \[\leadsto \frac{\frac{\color{blue}{y} + \left(\left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot y}{z}}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-189}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{z}{y}}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+44}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y + y \cdot \left(\left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := 0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(0.041666666666666664 + x\_m \cdot \left(x\_m \cdot 0.001388888888888889\right)\right)\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot t\_0\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(1 + x\_m \cdot \left(x\_m \cdot t\_0\right)\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

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
         (+
          0.5
          (*
           (* x_m x_m)
           (+ 0.041666666666666664 (* x_m (* x_m 0.001388888888888889)))))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= y_m 5e-20)
        (/ (/ (* y_m (+ 1.0 (* (* x_m x_m) t_0))) x_m) z_m)
        (/ (/ (* y_m (+ 1.0 (* x_m (* x_m t_0)))) 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 t_0 = 0.5 + ((x_m * x_m) * (0.041666666666666664 + (x_m * (x_m * 0.001388888888888889))));
	double tmp;
	if (y_m <= 5e-20) {
		tmp = ((y_m * (1.0 + ((x_m * x_m) * t_0))) / x_m) / z_m;
	} else {
		tmp = ((y_m * (1.0 + (x_m * (x_m * t_0)))) / 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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + ((x_m * x_m) * (0.041666666666666664d0 + (x_m * (x_m * 0.001388888888888889d0))))
    if (y_m <= 5d-20) then
        tmp = ((y_m * (1.0d0 + ((x_m * x_m) * t_0))) / x_m) / z_m
    else
        tmp = ((y_m * (1.0d0 + (x_m * (x_m * t_0)))) / 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 t_0 = 0.5 + ((x_m * x_m) * (0.041666666666666664 + (x_m * (x_m * 0.001388888888888889))));
	double tmp;
	if (y_m <= 5e-20) {
		tmp = ((y_m * (1.0 + ((x_m * x_m) * t_0))) / x_m) / z_m;
	} else {
		tmp = ((y_m * (1.0 + (x_m * (x_m * t_0)))) / 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):
	t_0 = 0.5 + ((x_m * x_m) * (0.041666666666666664 + (x_m * (x_m * 0.001388888888888889))))
	tmp = 0
	if y_m <= 5e-20:
		tmp = ((y_m * (1.0 + ((x_m * x_m) * t_0))) / x_m) / z_m
	else:
		tmp = ((y_m * (1.0 + (x_m * (x_m * t_0)))) / 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)
	t_0 = Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(0.041666666666666664 + Float64(x_m * Float64(x_m * 0.001388888888888889)))))
	tmp = 0.0
	if (y_m <= 5e-20)
		tmp = Float64(Float64(Float64(y_m * Float64(1.0 + Float64(Float64(x_m * x_m) * t_0))) / x_m) / z_m);
	else
		tmp = Float64(Float64(Float64(y_m * Float64(1.0 + Float64(x_m * Float64(x_m * t_0)))) / 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)
	t_0 = 0.5 + ((x_m * x_m) * (0.041666666666666664 + (x_m * (x_m * 0.001388888888888889))));
	tmp = 0.0;
	if (y_m <= 5e-20)
		tmp = ((y_m * (1.0 + ((x_m * x_m) * t_0))) / x_m) / z_m;
	else
		tmp = ((y_m * (1.0 + (x_m * (x_m * t_0)))) / 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_] := Block[{t$95$0 = 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]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 5e-20], N[(N[(N[(y$95$m * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(y$95$m * N[(1.0 + N[(x$95$m * N[(x$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999999e-20

   1. Initial program 83.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 88.6%

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

   if 4.9999999999999999e-20 < y

   1. Initial program 93.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. cosh-lowering-cosh.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 95.3%

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

   7. Simplified 95.3%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{y \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(0.041666666666666664 + x\_m \cdot \left(x\_m \cdot 0.001388888888888889\right)\right)\right)\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m + y\_m \cdot \left(\left(0.041666666666666664 + \left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= y_m 4.2e-19)
      (/
       (/
        (*
         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
         (*
          y_m
          (*
           (+ 0.041666666666666664 (* (* x_m x_m) 0.001388888888888889))
           (* (* x_m x_m) (* x_m x_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 <= 4.2e-19) {
		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 + (y_m * ((0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)) * ((x_m * x_m) * (x_m * x_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 <= 4.2d-19) 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 + (y_m * ((0.041666666666666664d0 + ((x_m * x_m) * 0.001388888888888889d0)) * ((x_m * x_m) * (x_m * x_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 <= 4.2e-19) {
		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 + (y_m * ((0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)) * ((x_m * x_m) * (x_m * x_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 <= 4.2e-19:
		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 + (y_m * ((0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)) * ((x_m * x_m) * (x_m * x_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 <= 4.2e-19)
		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 + Float64(y_m * Float64(Float64(0.041666666666666664 + Float64(Float64(x_m * x_m) * 0.001388888888888889)) * Float64(Float64(x_m * x_m) * Float64(x_m * x_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 <= 4.2e-19)
		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 + (y_m * ((0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)) * ((x_m * x_m) * (x_m * x_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, 4.2e-19], 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 + N[(y$95$m * N[(N[(0.041666666666666664 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.1999999999999998e-19

   1. Initial program 83.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 88.6%

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

   if 4.1999999999999998e-19 < y

   1. Initial program 93.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. cosh-lowering-cosh.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 95.3%

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

   7. Simplified 95.3%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 95.3%

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

   10. Simplified 95.3%

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

   12. Applied egg-rr 95.3%

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

   13. Taylor expanded in x around 0

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

   15. Simplified 95.3%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{y \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y + y \cdot \left(\left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 7.2 \cdot 10^{-180}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.5 + x\_m \cdot \left(x\_m \cdot 0.041666666666666664\right)\right)\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m + y\_m \cdot \left(\left(0.041666666666666664 + \left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= y_m 7.2e-180)
      (/
       (/
        (*
         y_m
         (+ 1.0 (* (* x_m x_m) (+ 0.5 (* x_m (* x_m 0.041666666666666664))))))
        x_m)
       z_m)
      (/
       (/
        (+
         y_m
         (*
          y_m
          (*
           (+ 0.041666666666666664 (* (* x_m x_m) 0.001388888888888889))
           (* (* x_m x_m) (* x_m x_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 <= 7.2e-180) {
		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 + (y_m * ((0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)) * ((x_m * x_m) * (x_m * x_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 <= 7.2d-180) 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 + (y_m * ((0.041666666666666664d0 + ((x_m * x_m) * 0.001388888888888889d0)) * ((x_m * x_m) * (x_m * x_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 <= 7.2e-180) {
		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 + (y_m * ((0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)) * ((x_m * x_m) * (x_m * x_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 <= 7.2e-180:
		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 + (y_m * ((0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)) * ((x_m * x_m) * (x_m * x_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 <= 7.2e-180)
		tmp = Float64(Float64(Float64(y_m * Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.5 + Float64(x_m * Float64(x_m * 0.041666666666666664)))))) / x_m) / z_m);
	else
		tmp = Float64(Float64(Float64(y_m + Float64(y_m * Float64(Float64(0.041666666666666664 + Float64(Float64(x_m * x_m) * 0.001388888888888889)) * Float64(Float64(x_m * x_m) * Float64(x_m * x_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 <= 7.2e-180)
		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 + (y_m * ((0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)) * ((x_m * x_m) * (x_m * x_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, 7.2e-180], N[(N[(N[(y$95$m * 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]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(y$95$m + N[(y$95$m * N[(N[(0.041666666666666664 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 7.1999999999999998e-180

   1. Initial program 82.8%

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

   3. Taylor expanded in x around 0

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

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

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

   5. Simplified 84.8%

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

   if 7.1999999999999998e-180 < y

   1. Initial program 93.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. cosh-lowering-cosh.f64 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 94.5%

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

   7. Simplified 94.5%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 94.5%

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

   10. Simplified 94.5%

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

   12. Applied egg-rr 94.5%

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

   13. Taylor expanded in x around 0

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

   15. Simplified 94.5%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-180}:\\ \;\;\;\;\frac{\frac{y \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y + y \cdot \left(\left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.5 + x\_m \cdot \left(x\_m \cdot 0.041666666666666664\right)\right)\right)\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{-19}:\\ \;\;\;\;\frac{\frac{t\_0}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

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
         (*
          y_m
          (+
           1.0
           (* (* x_m x_m) (+ 0.5 (* x_m (* x_m 0.041666666666666664))))))))
   (*
    z_s
    (*
     y_s
     (* x_s (if (<= y_m 1e-19) (/ (/ t_0 x_m) z_m) (/ (/ t_0 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 t_0 = y_m * (1.0 + ((x_m * x_m) * (0.5 + (x_m * (x_m * 0.041666666666666664)))));
	double tmp;
	if (y_m <= 1e-19) {
		tmp = (t_0 / x_m) / z_m;
	} else {
		tmp = (t_0 / 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) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + ((x_m * x_m) * (0.5d0 + (x_m * (x_m * 0.041666666666666664d0)))))
    if (y_m <= 1d-19) then
        tmp = (t_0 / x_m) / z_m
    else
        tmp = (t_0 / 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 t_0 = y_m * (1.0 + ((x_m * x_m) * (0.5 + (x_m * (x_m * 0.041666666666666664)))));
	double tmp;
	if (y_m <= 1e-19) {
		tmp = (t_0 / x_m) / z_m;
	} else {
		tmp = (t_0 / 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):
	t_0 = y_m * (1.0 + ((x_m * x_m) * (0.5 + (x_m * (x_m * 0.041666666666666664)))))
	tmp = 0
	if y_m <= 1e-19:
		tmp = (t_0 / x_m) / z_m
	else:
		tmp = (t_0 / 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)
	t_0 = Float64(y_m * Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.5 + Float64(x_m * Float64(x_m * 0.041666666666666664))))))
	tmp = 0.0
	if (y_m <= 1e-19)
		tmp = Float64(Float64(t_0 / x_m) / z_m);
	else
		tmp = Float64(Float64(t_0 / 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)
	t_0 = y_m * (1.0 + ((x_m * x_m) * (0.5 + (x_m * (x_m * 0.041666666666666664)))));
	tmp = 0.0;
	if (y_m <= 1e-19)
		tmp = (t_0 / x_m) / z_m;
	else
		tmp = (t_0 / 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_] := Block[{t$95$0 = N[(y$95$m * 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]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1e-19], N[(N[(t$95$0 / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(t$95$0 / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.9999999999999998e-20

   1. Initial program 83.8%

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

   3. Taylor expanded in x around 0

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

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

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

   5. Simplified 85.5%

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

   if 9.9999999999999998e-20 < y

   1. Initial program 93.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. cosh-lowering-cosh.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 95.3%

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

   7. Simplified 95.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-19}:\\ \;\;\;\;\frac{\frac{y \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.15 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.5 + x\_m \cdot \left(x\_m \cdot 0.041666666666666664\right)\right)\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot 0.5\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 1.15e+42)
      (/
       (/
        (*
         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))) 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 <= 1.15e+42) {
		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))) / 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 <= 1.15d+42) 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))) / 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 <= 1.15e+42) {
		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))) / 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 <= 1.15e+42:
		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))) / 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 <= 1.15e+42)
		tmp = Float64(Float64(Float64(y_m * Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.5 + Float64(x_m * Float64(x_m * 0.041666666666666664)))))) / x_m) / z_m);
	else
		tmp = Float64(Float64(Float64(y_m * Float64(1.0 + 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 (y_m <= 1.15e+42)
		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))) / 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, 1.15e+42], N[(N[(N[(y$95$m * 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]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(y$95$m * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.15e42

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

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

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

   5. Simplified 86.4%

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

   if 1.15e42 < y

   1. Initial program 92.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. cosh-lowering-cosh.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 92.2%

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

   7. Simplified 92.2%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{y \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4.5 \cdot 10^{+67}:\\ \;\;\;\;y\_m \cdot \frac{1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.5 + x\_m \cdot \left(x\_m \cdot 0.041666666666666664\right)\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 4.5e+67)
      (*
       y_m
       (/
        (+ 1.0 (* (* x_m x_m) (+ 0.5 (* x_m (* x_m 0.041666666666666664)))))
        (* x_m z_m)))
      (/
       (/ (* y_m (* (* x_m x_m) (* 0.041666666666666664 (* 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 <= 4.5e+67) {
		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 * ((x_m * x_m) * (0.041666666666666664 * (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 <= 4.5d+67) 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 * ((x_m * x_m) * (0.041666666666666664d0 * (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 <= 4.5e+67) {
		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 * ((x_m * x_m) * (0.041666666666666664 * (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 <= 4.5e+67:
		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 * ((x_m * x_m) * (0.041666666666666664 * (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 <= 4.5e+67)
		tmp = Float64(y_m * Float64(Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.5 + Float64(x_m * Float64(x_m * 0.041666666666666664))))) / Float64(x_m * z_m)));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(Float64(x_m * x_m) * Float64(0.041666666666666664 * Float64(x_m * 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 <= 4.5e+67)
		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 * ((x_m * x_m) * (0.041666666666666664 * (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, 4.5e+67], N[(y$95$m * 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] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.041666666666666664 * N[(x$95$m * x$95$m), $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 < 4.4999999999999998e67

   1. Initial program 89.1%

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

   3. Taylor expanded in x around 0

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

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

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

   5. Simplified 82.3%

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

      1. associate-/l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 82.0%

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

   7. Applied egg-rr 82.0%

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

   if 4.4999999999999998e67 < x

   1. Initial program 76.3%

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

   3. Taylor expanded in x around 0

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+67}:\\ \;\;\;\;y \cdot \frac{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.8 \cdot 10^{+20}:\\ \;\;\;\;y\_m \cdot \frac{\frac{1}{x\_m} + x\_m \cdot 0.5}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{x\_m}}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 2.8e+20)
      (* y_m (/ (+ (/ 1.0 x_m) (* x_m 0.5)) z_m))
      (/
       (/ (* y_m (* (* x_m x_m) (* 0.041666666666666664 (* 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 <= 2.8e+20) {
		tmp = y_m * (((1.0 / x_m) + (x_m * 0.5)) / z_m);
	} else {
		tmp = ((y_m * ((x_m * x_m) * (0.041666666666666664 * (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 <= 2.8d+20) then
        tmp = y_m * (((1.0d0 / x_m) + (x_m * 0.5d0)) / z_m)
    else
        tmp = ((y_m * ((x_m * x_m) * (0.041666666666666664d0 * (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 <= 2.8e+20) {
		tmp = y_m * (((1.0 / x_m) + (x_m * 0.5)) / z_m);
	} else {
		tmp = ((y_m * ((x_m * x_m) * (0.041666666666666664 * (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 <= 2.8e+20:
		tmp = y_m * (((1.0 / x_m) + (x_m * 0.5)) / z_m)
	else:
		tmp = ((y_m * ((x_m * x_m) * (0.041666666666666664 * (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 <= 2.8e+20)
		tmp = Float64(y_m * Float64(Float64(Float64(1.0 / x_m) + Float64(x_m * 0.5)) / z_m));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(Float64(x_m * x_m) * Float64(0.041666666666666664 * Float64(x_m * 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 <= 2.8e+20)
		tmp = y_m * (((1.0 / x_m) + (x_m * 0.5)) / z_m);
	else
		tmp = ((y_m * ((x_m * x_m) * (0.041666666666666664 * (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, 2.8e+20], N[(y$95$m * N[(N[(N[(1.0 / x$95$m), $MachinePrecision] + N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.041666666666666664 * N[(x$95$m * x$95$m), $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 < 2.8e20

   1. Initial program 88.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-rgt-identity N/A

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

      8. associate-/l* N/A

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

      9. associate-*r/ N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. associate-/l* N/A

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

      16. *-inverses N/A

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

      17. *-rgt-identity N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. *-commutative N/A

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

      21. distribute-lft-out N/A

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

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

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

   5. Simplified 74.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 78.8%

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

   7. Applied egg-rr 78.8%

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

   if 2.8e20 < x

   1. Initial program 79.1%

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

   3. Taylor expanded in x around 0

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

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

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

   5. Simplified 90.0%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 90.0%

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

   8. Simplified 90.0%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+20}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x} + x \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 9 \cdot 10^{+21}:\\ \;\;\;\;y\_m \cdot \frac{\frac{1}{x\_m} + x\_m \cdot 0.5}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot \left(y\_m \cdot \left(0.041666666666666664 \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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+21)
      (* y_m (/ (+ (/ 1.0 x_m) (* x_m 0.5)) z_m))
      (/
       (/ (* x_m (* y_m (* 0.041666666666666664 (* x_m (* x_m x_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 (x_m <= 9e+21) {
		tmp = y_m * (((1.0 / x_m) + (x_m * 0.5)) / z_m);
	} else {
		tmp = ((x_m * (y_m * (0.041666666666666664 * (x_m * (x_m * x_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 (x_m <= 9d+21) then
        tmp = y_m * (((1.0d0 / x_m) + (x_m * 0.5d0)) / z_m)
    else
        tmp = ((x_m * (y_m * (0.041666666666666664d0 * (x_m * (x_m * x_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 (x_m <= 9e+21) {
		tmp = y_m * (((1.0 / x_m) + (x_m * 0.5)) / z_m);
	} else {
		tmp = ((x_m * (y_m * (0.041666666666666664 * (x_m * (x_m * x_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 x_m <= 9e+21:
		tmp = y_m * (((1.0 / x_m) + (x_m * 0.5)) / z_m)
	else:
		tmp = ((x_m * (y_m * (0.041666666666666664 * (x_m * (x_m * x_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 (x_m <= 9e+21)
		tmp = Float64(y_m * Float64(Float64(Float64(1.0 / x_m) + Float64(x_m * 0.5)) / z_m));
	else
		tmp = Float64(Float64(Float64(x_m * Float64(y_m * Float64(0.041666666666666664 * Float64(x_m * Float64(x_m * x_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 (x_m <= 9e+21)
		tmp = y_m * (((1.0 / x_m) + (x_m * 0.5)) / z_m);
	else
		tmp = ((x_m * (y_m * (0.041666666666666664 * (x_m * (x_m * x_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[x$95$m, 9e+21], N[(y$95$m * N[(N[(N[(1.0 / x$95$m), $MachinePrecision] + N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * N[(y$95$m * N[(0.041666666666666664 * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $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 < 9e21

   1. Initial program 88.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-rgt-identity N/A

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

      8. associate-/l* N/A

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

      9. associate-*r/ N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. associate-/l* N/A

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

      16. *-inverses N/A

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

      17. *-rgt-identity N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. *-commutative N/A

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

      21. distribute-lft-out N/A

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

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

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

   5. Simplified 74.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 78.8%

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

   7. Applied egg-rr 78.8%

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

   if 9e21 < x

   1. Initial program 79.1%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. cosh-lowering-cosh.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in x around 0

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

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

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

   10. Simplified 78.3%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

       4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       9. associate-*l* N/A

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

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

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

       11. *-lowering-*.f64 78.3%

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

   12. Applied egg-rr 78.3%

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

   13. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{24} \cdot \frac{{x}^{4} \cdot y}{z}\right)}, x\right) \]
   14. 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. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \frac{y}{z}\right), x\right) \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. associate-/l* N/A

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

       8. associate-*r* N/A

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

       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. associate-*r/ N/A

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

       11. associate-*r/ N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. *-commutative N/A

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

       15. associate-*r* N/A

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

   15. Simplified 88.5%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x} + x \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(y \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.4 \cdot 10^{+20}:\\ \;\;\;\;y\_m \cdot \frac{\frac{1}{x\_m} + x\_m \cdot 0.5}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(0.041666666666666664 \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 2.4e+20)
      (* y_m (/ (+ (/ 1.0 x_m) (* x_m 0.5)) z_m))
      (/ (* y_m (* 0.041666666666666664 (* 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 <= 2.4e+20) {
		tmp = y_m * (((1.0 / x_m) + (x_m * 0.5)) / z_m);
	} else {
		tmp = (y_m * (0.041666666666666664 * (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 <= 2.4d+20) then
        tmp = y_m * (((1.0d0 / x_m) + (x_m * 0.5d0)) / z_m)
    else
        tmp = (y_m * (0.041666666666666664d0 * (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 <= 2.4e+20) {
		tmp = y_m * (((1.0 / x_m) + (x_m * 0.5)) / z_m);
	} else {
		tmp = (y_m * (0.041666666666666664 * (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 <= 2.4e+20:
		tmp = y_m * (((1.0 / x_m) + (x_m * 0.5)) / z_m)
	else:
		tmp = (y_m * (0.041666666666666664 * (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 <= 2.4e+20)
		tmp = Float64(y_m * Float64(Float64(Float64(1.0 / x_m) + Float64(x_m * 0.5)) / z_m));
	else
		tmp = Float64(Float64(y_m * Float64(0.041666666666666664 * 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 <= 2.4e+20)
		tmp = y_m * (((1.0 / x_m) + (x_m * 0.5)) / z_m);
	else
		tmp = (y_m * (0.041666666666666664 * (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, 2.4e+20], N[(y$95$m * N[(N[(N[(1.0 / x$95$m), $MachinePrecision] + N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(0.041666666666666664 * 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 < 2.4e20

   1. Initial program 88.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-rgt-identity N/A

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

      8. associate-/l* N/A

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

      9. associate-*r/ N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. associate-/l* N/A

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

      16. *-inverses N/A

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

      17. *-rgt-identity N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. *-commutative N/A

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

      21. distribute-lft-out N/A

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

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

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

   5. Simplified 74.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 78.8%

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

   7. Applied egg-rr 78.8%

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

   if 2.4e20 < x

   1. Initial program 79.1%

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

   3. Taylor expanded in x around 0

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

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

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

   5. Simplified 90.0%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 87.1%

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

   8. Simplified 87.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+20}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x} + x \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.65 \cdot 10^{+20}:\\ \;\;\;\;y\_m \cdot \frac{\frac{1}{x\_m} + x\_m \cdot 0.5}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(y\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 2.65e+20)
      (* y_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 <= 2.65e+20) {
		tmp = y_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 <= 2.65d+20) then
        tmp = y_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 <= 2.65e+20) {
		tmp = y_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 <= 2.65e+20:
		tmp = y_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 <= 2.65e+20)
		tmp = Float64(y_m * Float64(Float64(Float64(1.0 / 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 <= 2.65e+20)
		tmp = y_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, 2.65e+20], N[(y$95$m * N[(N[(N[(1.0 / x$95$m), $MachinePrecision] + N[(x$95$m * 0.5), $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 < 2.65e20

   1. Initial program 88.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-rgt-identity N/A

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

      8. associate-/l* N/A

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

      9. associate-*r/ N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. associate-/l* N/A

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

      16. *-inverses N/A

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

      17. *-rgt-identity N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. *-commutative N/A

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

      21. distribute-lft-out N/A

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

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

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

   5. Simplified 74.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 78.8%

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

   7. Applied egg-rr 78.8%

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

   if 2.65e20 < x

   1. Initial program 79.1%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. cosh-lowering-cosh.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in x around 0

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

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

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

   10. Simplified 78.3%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]
   12. 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 87.1%

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

   13. Simplified 87.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{+20}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x} + x \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(y \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 7.6 \cdot 10^{+38}:\\ \;\;\;\;y\_m \cdot \frac{\frac{1}{x\_m} + x\_m \cdot 0.5}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\left(0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \frac{y\_m}{z\_m}\right)\\ \end{array}\right)\right) \end{array} \]

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 7.6e+38)
      (* y_m (/ (+ (/ 1.0 x_m) (* x_m 0.5)) z_m))
      (* x_m (* (* 0.041666666666666664 (* x_m x_m)) (/ 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 <= 7.6e+38) {
		tmp = y_m * (((1.0 / x_m) + (x_m * 0.5)) / z_m);
	} else {
		tmp = x_m * ((0.041666666666666664 * (x_m * x_m)) * (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 <= 7.6d+38) then
        tmp = y_m * (((1.0d0 / x_m) + (x_m * 0.5d0)) / z_m)
    else
        tmp = x_m * ((0.041666666666666664d0 * (x_m * x_m)) * (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 <= 7.6e+38) {
		tmp = y_m * (((1.0 / x_m) + (x_m * 0.5)) / z_m);
	} else {
		tmp = x_m * ((0.041666666666666664 * (x_m * x_m)) * (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 <= 7.6e+38:
		tmp = y_m * (((1.0 / x_m) + (x_m * 0.5)) / z_m)
	else:
		tmp = x_m * ((0.041666666666666664 * (x_m * x_m)) * (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 <= 7.6e+38)
		tmp = Float64(y_m * Float64(Float64(Float64(1.0 / x_m) + Float64(x_m * 0.5)) / z_m));
	else
		tmp = Float64(x_m * Float64(Float64(0.041666666666666664 * Float64(x_m * x_m)) * 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 <= 7.6e+38)
		tmp = y_m * (((1.0 / x_m) + (x_m * 0.5)) / z_m);
	else
		tmp = x_m * ((0.041666666666666664 * (x_m * x_m)) * (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, 7.6e+38], N[(y$95$m * N[(N[(N[(1.0 / x$95$m), $MachinePrecision] + N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7.5999999999999996e38

   1. Initial program 88.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-rgt-identity N/A

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

      8. associate-/l* N/A

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

      9. associate-*r/ N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + \frac{\left(\left(\frac{1}{2} \cdot y\right) \cdot x\right) \cdot x}{x}\right), z\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + \left(\left(\frac{1}{2} \cdot y\right) \cdot x\right) \cdot \frac{x}{x}\right), z\right) \]

      16. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + \left(\left(\frac{1}{2} \cdot y\right) \cdot x\right) \cdot 1\right), z\right) \]

      17. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot y\right) \cdot x\right), z\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + \left(y \cdot \frac{1}{2}\right) \cdot x\right), z\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + y \cdot \left(\frac{1}{2} \cdot x\right)\right), z\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + y \cdot \left(x \cdot \frac{1}{2}\right)\right), z\right) \]

      21. distribute-lft-out N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \left(\frac{1}{x} + x \cdot \frac{1}{2}\right)\right), z\right) \]

      22. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{x} + x \cdot \frac{1}{2}\right)\right), z\right) \]

   5. Simplified 73.2%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{x} + x \cdot 0.5\right)}}{z} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{x} + x \cdot \frac{1}{2}}{z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{1}{x} + x \cdot \frac{1}{2}}{z} \cdot \color{blue}{y} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{x} + x \cdot \frac{1}{2}}{z}\right), \color{blue}{y}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{x} + x \cdot \frac{1}{2}\right), z\right), y\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{x}\right), \left(x \cdot \frac{1}{2}\right)\right), z\right), y\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(x \cdot \frac{1}{2}\right)\right), z\right), y\right) \]

      7. *-lowering-*.f64 78.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), z\right), y\right) \]

   7. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{x} + x \cdot 0.5}{z} \cdot y} \]

   if 7.5999999999999996e38 < x

   1. Initial program 78.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \cosh x}{z} \]

      2. associate-/l* N/A

         \[\leadsto \frac{y}{x} \cdot \color{blue}{\frac{\cosh x}{z}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{y \cdot \frac{\cosh x}{z}}{\color{blue}{x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{\cosh x}{z}\right), \color{blue}{x}\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot \cosh x}{z}\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cosh x \cdot y}{z}\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\cosh x \cdot y\right), z\right), x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\cosh x, y\right), z\right), x\right) \]

      9. cosh-lowering-cosh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), y\right), z\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x \cdot y}{z}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, y\right), z\right), x\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), y\right), z\right), x\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), y\right), z\right), x\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right)\right)\right), y\right), z\right), x\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right)\right)\right), y\right), z\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      17. *-lowering-*.f64 98.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

   7. Simplified 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\right)} \cdot y}{z}}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   10. Simplified 81.8%

       \[\leadsto \color{blue}{\frac{\frac{y}{z} + \left(x \cdot x\right) \cdot \left(\frac{y}{z} \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{x}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} \cdot \frac{y}{z} + \frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)} \]

   13. Simplified 80.4%

       \[\leadsto \color{blue}{x \cdot \left(\frac{y}{z} \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \]

   14. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]
   15. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right) \]

       4. *-lowering-*.f64 80.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right) \]

   16. Simplified 80.4%

       \[\leadsto x \cdot \left(\frac{y}{z} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.6 \cdot 10^{+38}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x} + x \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(0.041666666666666664 \cdot \left(x \cdot x\right)\right) \cdot \frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\left(0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \frac{y\_m}{z\_m}\right)\\ \end{array}\right)\right) \end{array} \]

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 2.2)
      (/ y_m (* x_m z_m))
      (* x_m (* (* 0.041666666666666664 (* x_m x_m)) (/ 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 <= 2.2) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = x_m * ((0.041666666666666664 * (x_m * x_m)) * (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 <= 2.2d0) then
        tmp = y_m / (x_m * z_m)
    else
        tmp = x_m * ((0.041666666666666664d0 * (x_m * x_m)) * (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 <= 2.2) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = x_m * ((0.041666666666666664 * (x_m * x_m)) * (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 <= 2.2:
		tmp = y_m / (x_m * z_m)
	else:
		tmp = x_m * ((0.041666666666666664 * (x_m * x_m)) * (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 <= 2.2)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(x_m * Float64(Float64(0.041666666666666664 * Float64(x_m * x_m)) * 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 <= 2.2)
		tmp = y_m / (x_m * z_m);
	else
		tmp = x_m * ((0.041666666666666664 * (x_m * x_m)) * (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, 2.2], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 88.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 68.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 68.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
   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 71.7%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{z}\right)\right) \]

   7. Applied egg-rr 71.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 2.2000000000000002 < x

   1. Initial program 80.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \cosh x}{z} \]

      2. associate-/l* N/A

         \[\leadsto \frac{y}{x} \cdot \color{blue}{\frac{\cosh x}{z}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{y \cdot \frac{\cosh x}{z}}{\color{blue}{x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{\cosh x}{z}\right), \color{blue}{x}\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot \cosh x}{z}\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cosh x \cdot y}{z}\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\cosh x \cdot y\right), z\right), x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\cosh x, y\right), z\right), x\right) \]

      9. cosh-lowering-cosh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), y\right), z\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x \cdot y}{z}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, y\right), z\right), x\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), y\right), z\right), x\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), y\right), z\right), x\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right)\right)\right), y\right), z\right), x\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right)\right)\right), y\right), z\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      17. *-lowering-*.f64 88.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

   7. Simplified 88.3%

      \[\leadsto \frac{\frac{\color{blue}{\left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\right)} \cdot y}{z}}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   10. Simplified 72.3%

       \[\leadsto \color{blue}{\frac{\frac{y}{z} + \left(x \cdot x\right) \cdot \left(\frac{y}{z} \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{x}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} \cdot \frac{y}{z} + \frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)} \]

   13. Simplified 71.0%

       \[\leadsto \color{blue}{x \cdot \left(\frac{y}{z} \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \]

   14. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]
   15. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right) \]

       4. *-lowering-*.f64 71.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right) \]

   16. Simplified 71.0%

       \[\leadsto x \cdot \left(\frac{y}{z} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(0.041666666666666664 \cdot \left(x \cdot x\right)\right) \cdot \frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 78.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(0.041666666666666664 \cdot \frac{y\_m}{z\_m}\right)\right)\right)\\ \end{array}\right)\right) \end{array} \]

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 2.2)
      (/ y_m (* x_m z_m))
      (* x_m (* x_m (* x_m (* 0.041666666666666664 (/ 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 <= 2.2) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = x_m * (x_m * (x_m * (0.041666666666666664 * (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 <= 2.2d0) then
        tmp = y_m / (x_m * z_m)
    else
        tmp = x_m * (x_m * (x_m * (0.041666666666666664d0 * (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 <= 2.2) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = x_m * (x_m * (x_m * (0.041666666666666664 * (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 <= 2.2:
		tmp = y_m / (x_m * z_m)
	else:
		tmp = x_m * (x_m * (x_m * (0.041666666666666664 * (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 <= 2.2)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(x_m * Float64(x_m * Float64(x_m * Float64(0.041666666666666664 * 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 <= 2.2)
		tmp = y_m / (x_m * z_m);
	else
		tmp = x_m * (x_m * (x_m * (0.041666666666666664 * (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, 2.2], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(0.041666666666666664 * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 88.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 68.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 68.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
   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 71.7%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{z}\right)\right) \]

   7. Applied egg-rr 71.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 2.2000000000000002 < x

   1. Initial program 80.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \cosh x}{z} \]

      2. associate-/l* N/A

         \[\leadsto \frac{y}{x} \cdot \color{blue}{\frac{\cosh x}{z}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{y \cdot \frac{\cosh x}{z}}{\color{blue}{x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{\cosh x}{z}\right), \color{blue}{x}\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot \cosh x}{z}\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cosh x \cdot y}{z}\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\cosh x \cdot y\right), z\right), x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\cosh x, y\right), z\right), x\right) \]

      9. cosh-lowering-cosh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), y\right), z\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x \cdot y}{z}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, y\right), z\right), x\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), y\right), z\right), x\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), y\right), z\right), x\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right)\right)\right), y\right), z\right), x\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right)\right)\right), y\right), z\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      17. *-lowering-*.f64 88.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

   7. Simplified 88.3%

      \[\leadsto \frac{\frac{\color{blue}{\left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\right)} \cdot y}{z}}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   10. Simplified 72.3%

       \[\leadsto \color{blue}{\frac{\frac{y}{z} + \left(x \cdot x\right) \cdot \left(\frac{y}{z} \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{x}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} \cdot \frac{y}{z} + \frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)} \]

   13. Simplified 71.0%

       \[\leadsto \color{blue}{x \cdot \left(\frac{y}{z} \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \]

   14. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z}\right)}\right) \]
   15. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \color{blue}{\frac{y}{z}}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{\frac{y}{z}}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left({x}^{2} \cdot \frac{1}{24}\right) \cdot \frac{\color{blue}{y}}{z}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{y}{z}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot \frac{y}{z}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot \frac{y}{z}\right)\right)}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot \frac{y}{z}\right)\right)}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot \frac{y}{z}\right)}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{y}{z}\right)}\right)\right)\right)\right) \]

       10. /-lowering-/.f64 69.9%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right)\right)\right)\right) \]

   16. Simplified 69.9%

       \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(0.041666666666666664 \cdot \frac{y}{z}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 66.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x\_m \cdot 0.5\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 1.4) (/ 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 <= 1.4) {
		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 <= 1.4d0) 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 <= 1.4) {
		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 <= 1.4:
		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 <= 1.4)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(y_m * Float64(x_m * 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 <= 1.4)
		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, 1.4], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 88.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 68.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 68.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
   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 71.7%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{z}\right)\right) \]

   7. Applied egg-rr 71.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 1.3999999999999999 < x

   1. Initial program 80.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}\right)}, z\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}{x}\right), z\right) \]

      2. distribute-rgt1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot y}{x}\right), z\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot y}{x}\right), z\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{x}\right), z\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{y}{x}\right), z\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{x}\right), z\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot 1}{x} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{x}\right), z\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{x}\right), z\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + \frac{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}{x}\right), z\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + \frac{\frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}\right), z\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + \frac{\frac{1}{2} \cdot \left(y \cdot {x}^{2}\right)}{x}\right), z\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + \frac{\left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}{x}\right), z\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + \frac{\left(\frac{1}{2} \cdot y\right) \cdot \left(x \cdot x\right)}{x}\right), z\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + \frac{\left(\left(\frac{1}{2} \cdot y\right) \cdot x\right) \cdot x}{x}\right), z\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + \left(\left(\frac{1}{2} \cdot y\right) \cdot x\right) \cdot \frac{x}{x}\right), z\right) \]

      16. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + \left(\left(\frac{1}{2} \cdot y\right) \cdot x\right) \cdot 1\right), z\right) \]

      17. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot y\right) \cdot x\right), z\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + \left(y \cdot \frac{1}{2}\right) \cdot x\right), z\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + y \cdot \left(\frac{1}{2} \cdot x\right)\right), z\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{1}{x} + y \cdot \left(x \cdot \frac{1}{2}\right)\right), z\right) \]

      21. distribute-lft-out N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \left(\frac{1}{x} + x \cdot \frac{1}{2}\right)\right), z\right) \]

      22. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{x} + x \cdot \frac{1}{2}\right)\right), z\right) \]

   5. Simplified 42.4%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{x} + x \cdot 0.5\right)}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{\color{blue}{z}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot x\right) \cdot y}{z} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot y\right), \color{blue}{z}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \left(\frac{1}{2} \cdot x\right)\right), z\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot x\right)\right), z\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(x \cdot \frac{1}{2}\right)\right), z\right) \]

      7. *-lowering-*.f64 42.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), z\right) \]

   8. Simplified 42.4%

      \[\leadsto \color{blue}{\frac{y \cdot \left(x \cdot 0.5\right)}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 62.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(y\_m \cdot \frac{0.5}{z\_m}\right)\\ \end{array}\right)\right) \end{array} \]

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 1.4) (/ y_m (* x_m z_m)) (* x_m (* y_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 <= 1.4) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = x_m * (y_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 <= 1.4d0) then
        tmp = y_m / (x_m * z_m)
    else
        tmp = x_m * (y_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 <= 1.4) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = x_m * (y_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 <= 1.4:
		tmp = y_m / (x_m * z_m)
	else:
		tmp = x_m * (y_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 <= 1.4)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(x_m * Float64(y_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 <= 1.4)
		tmp = y_m / (x_m * z_m);
	else
		tmp = x_m * (y_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, 1.4], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m * N[(0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 88.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 68.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 68.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
   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 71.7%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{z}\right)\right) \]

   7. Applied egg-rr 71.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 1.3999999999999999 < x

   1. Initial program 80.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \cosh x}{z} \]

      2. associate-/l* N/A

         \[\leadsto \frac{y}{x} \cdot \color{blue}{\frac{\cosh x}{z}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{y \cdot \frac{\cosh x}{z}}{\color{blue}{x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \frac{\cosh x}{z}\right), \color{blue}{x}\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot \cosh x}{z}\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cosh x \cdot y}{z}\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\cosh x \cdot y\right), z\right), x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\cosh x, y\right), z\right), x\right) \]

      9. cosh-lowering-cosh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), y\right), z\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x \cdot y}{z}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, y\right), z\right), x\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), y\right), z\right), x\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), y\right), z\right), x\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right)\right)\right), y\right), z\right), x\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right)\right)\right), y\right), z\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

      17. *-lowering-*.f64 88.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right), y\right), z\right), x\right) \]

   7. Simplified 88.3%

      \[\leadsto \frac{\frac{\color{blue}{\left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\right)} \cdot y}{z}}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}\right), \color{blue}{x}\right) \]

   10. Simplified 72.3%

       \[\leadsto \color{blue}{\frac{\frac{y}{z} + \left(x \cdot x\right) \cdot \left(\frac{y}{z} \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{x}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} \cdot \frac{y}{z} + \frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)} \]

   13. Simplified 71.0%

       \[\leadsto \color{blue}{x \cdot \left(\frac{y}{z} \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \]

   14. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   15. 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. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{1}{2} \cdot y}{\color{blue}{z}}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y \cdot \frac{1}{2}}{z}\right)\right) \]

       8. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{\frac{\frac{1}{2}}{z}}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{2}}{z}\right)}\right)\right) \]

       10. /-lowering-/.f64 33.3%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{z}\right)\right)\right) \]

   16. Simplified 33.3%

       \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{z}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 56.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= z_m 8e+22) (/ (/ y_m z_m) x_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 (z_m <= 8e+22) {
		tmp = (y_m / z_m) / x_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 (z_m <= 8d+22) then
        tmp = (y_m / z_m) / x_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 (z_m <= 8e+22) {
		tmp = (y_m / z_m) / x_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 z_m <= 8e+22:
		tmp = (y_m / z_m) / x_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 (z_m <= 8e+22)
		tmp = Float64(Float64(y_m / z_m) / x_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 (z_m <= 8e+22)
		tmp = (y_m / z_m) / x_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[z$95$m, 8e+22], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$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 z < 8e22

   1. Initial program 85.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 56.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, z\right), x\right) \]

   5. Simplified 56.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

   if 8e22 < z

   1. Initial program 87.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 48.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 48.0%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
   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 59.4%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{z}\right)\right) \]

   7. Applied egg-rr 59.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 52.3% accurate, 10.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= z_m 2e+32) (/ (/ 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 (z_m <= 2e+32) {
		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 (z_m <= 2d+32) 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 (z_m <= 2e+32) {
		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 z_m <= 2e+32:
		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 (z_m <= 2e+32)
		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 (z_m <= 2e+32)
		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[z$95$m, 2e+32], 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 z < 2.00000000000000011e32

   1. Initial program 86.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}, z\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 52.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 52.3%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   if 2.00000000000000011e32 < z

   1. Initial program 85.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}, z\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 48.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

   5. Simplified 48.5%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
   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 60.9%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{z}\right)\right) \]

   7. Applied egg-rr 60.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 49.1% 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 86.2%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
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 51.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right) \]

5. Simplified 51.4%

   \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
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 53.6%

      \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{z}\right)\right) \]

7. Applied egg-rr 53.6%

   \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
8. Add Preprocessing

Developer Target 1: 97.1% 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 2024158 
(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))