Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.8% → 99.8%

Time: 14.4s

Alternatives: 20

Speedup: 3.4×

• 63.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\cosh x\_m}{z\_m}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 4 \cdot 10^{+189}:\\ \;\;\;\;\frac{y\_m}{x\_m} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot t\_0}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

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) z_m)))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 4e+189)
        (* (/ y_m x_m) t_0)
        (/ (* y_m t_0) 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) / z_m;
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 4e+189) {
		tmp = (y_m / x_m) * t_0;
	} else {
		tmp = (y_m * t_0) / 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) / z_m
    if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 4d+189) then
        tmp = (y_m / x_m) * t_0
    else
        tmp = (y_m * t_0) / 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) / z_m;
	double tmp;
	if (((Math.cosh(x_m) * (y_m / x_m)) / z_m) <= 4e+189) {
		tmp = (y_m / x_m) * t_0;
	} else {
		tmp = (y_m * t_0) / 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) / z_m
	tmp = 0
	if ((math.cosh(x_m) * (y_m / x_m)) / z_m) <= 4e+189:
		tmp = (y_m / x_m) * t_0
	else:
		tmp = (y_m * t_0) / 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(cosh(x_m) / z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 4e+189)
		tmp = Float64(Float64(y_m / x_m) * t_0);
	else
		tmp = Float64(Float64(y_m * t_0) / 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) / z_m;
	tmp = 0.0;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 4e+189)
		tmp = (y_m / x_m) * t_0;
	else
		tmp = (y_m * t_0) / 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[Cosh[x$95$m], $MachinePrecision] / z$95$m), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 4e+189], N[(N[(y$95$m / x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(y$95$m * t$95$0), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4.0000000000000001e189

   1. Initial program 97.8%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 97.8

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

   4. Applied egg-rr 97.8%

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

   if 4.0000000000000001e189 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 61.7%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 4 \cdot 10^{+189}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.2% accurate, 0.5× speedup

Math

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

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 (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) INFINITY)
      (* (/ y_m x_m) (/ (cosh x_m) z_m))
      (/
       (* y_m (/ (* x_m (* x_m (* (* x_m x_m) 0.041666666666666664))) z_m))
       x_m))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 95.1%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 95.1

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

   4. Applied egg-rr 95.1%

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

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

   1. Initial program 0.0%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{y \cdot \frac{\color{blue}{\frac{1}{24} \cdot {x}^{4}}}{z}}{x} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 100.0

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

   10. Simplified 100.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq \infty:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{y\_m}{x\_m} \cdot \frac{t\_0}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot t\_0}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

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
         (fma
          (* x_m x_m)
          (fma
           (* x_m x_m)
           (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
           0.5)
          1.0)))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 5e-69)
        (* (/ y_m x_m) (/ t_0 z_m))
        (/ (/ (* y_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 = fma((x_m * x_m), fma((x_m * x_m), fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0);
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-69) {
		tmp = (y_m / x_m) * (t_0 / z_m);
	} else {
		tmp = ((y_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 = fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 5e-69)
		tmp = Float64(Float64(y_m / x_m) * Float64(t_0 / z_m));
	else
		tmp = Float64(Float64(Float64(y_m * t_0) / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(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[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5e-69], N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(t$95$0 / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * t$95$0), $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) < 5.00000000000000033e-69

   1. Initial program 97.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 89.0

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

   5. Simplified 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 89.1%

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

   if 5.00000000000000033e-69 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 67.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 62.2

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

   5. Simplified 62.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

   7. Applied egg-rr 95.6%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 10^{+197}:\\ \;\;\;\;\frac{y\_m}{x\_m} \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y\_m \cdot \left(x\_m \cdot x\_m\right), \mathsf{fma}\left(x\_m, 0.001388888888888889 \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right), 0.5\right), y\_m\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 1e+197)
      (*
       (/ y_m x_m)
       (/
        (fma
         (* x_m x_m)
         (fma
          (* x_m x_m)
          (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
          0.5)
         1.0)
        z_m))
      (/
       (/
        (fma
         (* y_m (* x_m x_m))
         (fma x_m (* 0.001388888888888889 (* x_m (* x_m x_m))) 0.5)
         y_m)
        z_m)
       x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 1e+197) {
		tmp = (y_m / x_m) * (fma((x_m * x_m), fma((x_m * x_m), fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0) / z_m);
	} else {
		tmp = (fma((y_m * (x_m * x_m)), fma(x_m, (0.001388888888888889 * (x_m * (x_m * x_m))), 0.5), y_m) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 1e+197)
		tmp = Float64(Float64(y_m / x_m) * Float64(fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0) / z_m));
	else
		tmp = Float64(Float64(fma(Float64(y_m * Float64(x_m * x_m)), fma(x_m, Float64(0.001388888888888889 * Float64(x_m * Float64(x_m * x_m))), 0.5), y_m) / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 1e+197], N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * N[(0.001388888888888889 * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $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) < 9.9999999999999995e196

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 90.0

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

   5. Simplified 90.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied egg-rr 90.0%

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

   if 9.9999999999999995e196 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 61.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 56.0

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

   5. Simplified 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

   7. Applied egg-rr 88.7%

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

   9. Applied egg-rr 93.8%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 93.8

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

   12. Simplified 93.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 10^{+197}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, 0.001388888888888889 \cdot \left(x \cdot \left(x \cdot x\right)\right), 0.5\right), y\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 4 \cdot 10^{+52}:\\ \;\;\;\;\frac{y\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 4e+52)
      (/
       (* y_m (fma x_m (* x_m (fma x_m (* x_m 0.041666666666666664) 0.5)) 1.0))
       (* x_m z_m))
      (/
       (*
        y_m
        (/
         (fma (* x_m x_m) (fma (* x_m x_m) 0.041666666666666664 0.5) 1.0)
         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 (((cosh(x_m) * (y_m / x_m)) / z_m) <= 4e+52) {
		tmp = (y_m * fma(x_m, (x_m * fma(x_m, (x_m * 0.041666666666666664), 0.5)), 1.0)) / (x_m * z_m);
	} else {
		tmp = (y_m * (fma((x_m * x_m), fma((x_m * x_m), 0.041666666666666664, 0.5), 1.0) / 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 (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 4e+52)
		tmp = Float64(Float64(y_m * fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5)), 1.0)) / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), 1.0) / z_m)) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 4e+52], N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4e52

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 87.4

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

   5. Simplified 87.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   7. Applied egg-rr 77.7%

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

   if 4e52 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 65.4%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 89.9

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

   7. Simplified 89.9%

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

Alternative 6: 93.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 10^{+207}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y\_m \cdot \left(x\_m \cdot x\_m\right), \mathsf{fma}\left(x\_m, 0.001388888888888889 \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right), 0.5\right), y\_m\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= (* (cosh x_m) (/ y_m x_m)) 1e+207)
      (/
       (/
        (*
         y_m
         (fma x_m (* x_m (fma x_m (* x_m 0.041666666666666664) 0.5)) 1.0))
        x_m)
       z_m)
      (/
       (/
        (fma
         (* y_m (* x_m x_m))
         (fma x_m (* 0.001388888888888889 (* x_m (* x_m x_m))) 0.5)
         y_m)
        z_m)
       x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 1e+207) {
		tmp = ((y_m * fma(x_m, (x_m * fma(x_m, (x_m * 0.041666666666666664), 0.5)), 1.0)) / x_m) / z_m;
	} else {
		tmp = (fma((y_m * (x_m * x_m)), fma(x_m, (0.001388888888888889 * (x_m * (x_m * x_m))), 0.5), y_m) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 1e+207)
		tmp = Float64(Float64(Float64(y_m * fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5)), 1.0)) / x_m) / z_m);
	else
		tmp = Float64(Float64(fma(Float64(y_m * Float64(x_m * x_m)), fma(x_m, Float64(0.001388888888888889 * Float64(x_m * Float64(x_m * x_m))), 0.5), y_m) / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 1e+207], N[(N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[(y$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * N[(0.001388888888888889 * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $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 (cosh.f64 x) (/.f64 y x)) < 1e207

   1. Initial program 96.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 87.8

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

   5. Simplified 87.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 88.5

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 88.5

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

   7. Applied egg-rr 88.5%

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

   if 1e207 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 62.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 53.0

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

   5. Simplified 53.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

   7. Applied egg-rr 86.3%

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

   9. Applied egg-rr 88.0%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 88.0

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

   12. Simplified 88.0%

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

Alternative 7: 92.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= (* (cosh x_m) (/ y_m x_m)) 5e+283)
      (/
       (/
        (*
         y_m
         (fma x_m (* x_m (fma x_m (* x_m 0.041666666666666664) 0.5)) 1.0))
        x_m)
       z_m)
      (/
       (*
        y_m
        (/
         (fma (* x_m x_m) (fma (* x_m x_m) 0.041666666666666664 0.5) 1.0)
         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 ((cosh(x_m) * (y_m / x_m)) <= 5e+283) {
		tmp = ((y_m * fma(x_m, (x_m * fma(x_m, (x_m * 0.041666666666666664), 0.5)), 1.0)) / x_m) / z_m;
	} else {
		tmp = (y_m * (fma((x_m * x_m), fma((x_m * x_m), 0.041666666666666664, 0.5), 1.0) / 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 (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 5e+283)
		tmp = Float64(Float64(Float64(y_m * fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5)), 1.0)) / x_m) / z_m);
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), 1.0) / z_m)) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 5e+283], N[(N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.0000000000000004e283

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 88.4

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

   5. Simplified 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 89.0

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 89.0

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

   7. Applied egg-rr 89.0%

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

   if 5.0000000000000004e283 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 59.5%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 88.1

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

   7. Simplified 88.1%

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

Alternative 8: 92.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m} \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= (* (cosh x_m) (/ y_m x_m)) 5e+283)
      (/
       (*
        (/ y_m x_m)
        (fma (* x_m x_m) (fma x_m (* x_m 0.041666666666666664) 0.5) 1.0))
       z_m)
      (/
       (*
        y_m
        (/
         (fma (* x_m x_m) (fma (* x_m x_m) 0.041666666666666664 0.5) 1.0)
         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 tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 5e+283) {
		tmp = ((y_m / x_m) * fma((x_m * x_m), fma(x_m, (x_m * 0.041666666666666664), 0.5), 1.0)) / z_m;
	} else {
		tmp = (y_m * (fma((x_m * x_m), fma((x_m * x_m), 0.041666666666666664, 0.5), 1.0) / 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)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 5e+283)
		tmp = Float64(Float64(Float64(y_m / x_m) * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.041666666666666664), 0.5), 1.0)) / z_m);
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), 1.0) / z_m)) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 5e+283], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.0000000000000004e283

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 88.4

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

   5. Simplified 88.4%

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

   if 5.0000000000000004e283 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 59.5%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 88.1

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

   7. Simplified 88.1%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.95 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{elif}\;x\_m \leq 2.25 \cdot 10^{+48}:\\ \;\;\;\;\frac{\cosh x\_m \cdot y\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 1.95e-78)
      (/ (/ y_m x_m) z_m)
      (if (<= x_m 2.25e+48)
        (/ (* (cosh x_m) y_m) (* x_m z_m))
        (/
         (/
          (*
           y_m
           (fma
            (* x_m x_m)
            (fma
             (* x_m x_m)
             (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
             0.5)
            1.0))
          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 <= 1.95e-78) {
		tmp = (y_m / x_m) / z_m;
	} else if (x_m <= 2.25e+48) {
		tmp = (cosh(x_m) * y_m) / (x_m * z_m);
	} else {
		tmp = ((y_m * fma((x_m * x_m), fma((x_m * x_m), fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0)) / 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 <= 1.95e-78)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	elseif (x_m <= 2.25e+48)
		tmp = Float64(Float64(cosh(x_m) * y_m) / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(Float64(y_m * fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0)) / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(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, 1.95e-78], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], If[LessEqual[x$95$m, 2.25e+48], N[(N[(N[Cosh[x$95$m], $MachinePrecision] * y$95$m), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.9500000000000001e-78

   1. Initial program 84.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 59.3

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

   5. Simplified 59.3%

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

   if 1.9500000000000001e-78 < x < 2.24999999999999998e48

   1. Initial program 96.2%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 96.5

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

   4. Applied egg-rr 96.5%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 96.3

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

   6. Applied egg-rr 96.3%

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

   if 2.24999999999999998e48 < x

   1. Initial program 71.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 69.7

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

   5. Simplified 69.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+48}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{elif}\;x\_m \leq 2.25 \cdot 10^{+48}:\\ \;\;\;\;y\_m \cdot \frac{\cosh x\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 4.2e-94)
      (/ (/ y_m x_m) z_m)
      (if (<= x_m 2.25e+48)
        (* y_m (/ (cosh x_m) (* x_m z_m)))
        (/
         (/
          (*
           y_m
           (fma
            (* x_m x_m)
            (fma
             (* x_m x_m)
             (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
             0.5)
            1.0))
          z_m)
         x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 4.2e-94) {
		tmp = (y_m / x_m) / z_m;
	} else if (x_m <= 2.25e+48) {
		tmp = y_m * (cosh(x_m) / (x_m * z_m));
	} else {
		tmp = ((y_m * fma((x_m * x_m), fma((x_m * x_m), fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0)) / 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)
	tmp = 0.0
	if (x_m <= 4.2e-94)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	elseif (x_m <= 2.25e+48)
		tmp = Float64(y_m * Float64(cosh(x_m) / Float64(x_m * z_m)));
	else
		tmp = Float64(Float64(Float64(y_m * fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0)) / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(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, 4.2e-94], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], If[LessEqual[x$95$m, 2.25e+48], N[(y$95$m * N[(N[Cosh[x$95$m], $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[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 4.2000000000000002e-94

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 59.0

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

   5. Simplified 59.0%

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

   if 4.2000000000000002e-94 < x < 2.24999999999999998e48

   1. Initial program 96.4%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.4

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

   4. Applied egg-rr 96.4%

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

   if 2.24999999999999998e48 < x

   1. Initial program 71.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 69.7

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

   5. Simplified 69.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

   7. Applied egg-rr 100.0%

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

Alternative 11: 92.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.85 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889\right)\right), y\_m \cdot \left(x\_m \cdot x\_m\right), y\_m\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 1.85e+92)
      (/
       (/
        (fma
         (* x_m (* x_m (* (* x_m x_m) 0.001388888888888889)))
         (* y_m (* x_m x_m))
         y_m)
        x_m)
       z_m)
      (/
       (*
        y_m
        (/
         (fma (* x_m x_m) (fma (* x_m x_m) 0.041666666666666664 0.5) 1.0)
         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 <= 1.85e+92) {
		tmp = (fma((x_m * (x_m * ((x_m * x_m) * 0.001388888888888889))), (y_m * (x_m * x_m)), y_m) / x_m) / z_m;
	} else {
		tmp = (y_m * (fma((x_m * x_m), fma((x_m * x_m), 0.041666666666666664, 0.5), 1.0) / 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 <= 1.85e+92)
		tmp = Float64(Float64(fma(Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.001388888888888889))), Float64(y_m * Float64(x_m * x_m)), y_m) / x_m) / z_m);
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), 1.0) / z_m)) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(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, 1.85e+92], N[(N[(N[(N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.84999999999999999e92

   1. Initial program 81.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 74.1

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

   5. Simplified 74.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

   7. Applied egg-rr 90.8%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 90.6

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

   10. Simplified 90.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. /-rgt-identity N/A

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

   12. Applied egg-rr 89.3%

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

   if 1.84999999999999999e92 < y

   1. Initial program 86.3%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.1

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

   7. Simplified 91.1%

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

Alternative 12: 87.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{elif}\;x\_m \leq 1.66 \cdot 10^{+99}:\\ \;\;\;\;\frac{y\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(0.041666666666666664 \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 1.8e-78)
      (/ (/ y_m x_m) z_m)
      (if (<= x_m 1.66e+99)
        (/
         (*
          y_m
          (fma x_m (* x_m (fma x_m (* x_m 0.041666666666666664) 0.5)) 1.0))
         (* x_m 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 <= 1.8e-78) {
		tmp = (y_m / x_m) / z_m;
	} else if (x_m <= 1.66e+99) {
		tmp = (y_m * fma(x_m, (x_m * fma(x_m, (x_m * 0.041666666666666664), 0.5)), 1.0)) / (x_m * z_m);
	} else {
		tmp = (y_m * (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 <= 1.8e-78)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	elseif (x_m <= 1.66e+99)
		tmp = Float64(Float64(y_m * fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5)), 1.0)) / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(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

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, 1.8e-78], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], If[LessEqual[x$95$m, 1.66e+99], N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 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 3 regimes

2. if x < 1.8000000000000001e-78

   1. Initial program 84.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 59.3

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

   5. Simplified 59.3%

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

   if 1.8000000000000001e-78 < x < 1.6599999999999999e99

   1. Initial program 92.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 63.6

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

   5. Simplified 63.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   7. Applied egg-rr 68.4%

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

   if 1.6599999999999999e99 < x

   1. Initial program 68.5%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   10. Simplified 100.0%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{elif}\;x \leq 1.66 \cdot 10^{+99}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 88.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0058:\\ \;\;\;\;\frac{1}{z\_m \cdot \frac{x\_m}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 0.0058)
      (/ 1.0 (* z_m (/ x_m y_m)))
      (/
       (* y_m (/ (* x_m (* x_m (* (* x_m x_m) 0.041666666666666664))) z_m))
       x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.0058) {
		tmp = 1.0 / (z_m * (x_m / y_m));
	} else {
		tmp = (y_m * ((x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))) / z_m)) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.0058d0) then
        tmp = 1.0d0 / (z_m * (x_m / y_m))
    else
        tmp = (y_m * ((x_m * (x_m * ((x_m * x_m) * 0.041666666666666664d0))) / z_m)) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.0058) {
		tmp = 1.0 / (z_m * (x_m / y_m));
	} else {
		tmp = (y_m * ((x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))) / z_m)) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.0058:
		tmp = 1.0 / (z_m * (x_m / y_m))
	else:
		tmp = (y_m * ((x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))) / z_m)) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.0058)
		tmp = Float64(1.0 / Float64(z_m * Float64(x_m / y_m)));
	else
		tmp = Float64(Float64(y_m * Float64(Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664))) / z_m)) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.0058)
		tmp = 1.0 / (z_m * (x_m / y_m));
	else
		tmp = (y_m * ((x_m * (x_m * ((x_m * x_m) * 0.041666666666666664))) / z_m)) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 0.0058], N[(1.0 / N[(z$95$m * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0058

   1. Initial program 85.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 78.7

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

   5. Simplified 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

   7. Applied egg-rr 87.9%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 62.6

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

   10. Simplified 62.6%

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

   if 0.0058 < x

   1. Initial program 75.3%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.2

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

   7. Simplified 86.2%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{y \cdot \frac{\color{blue}{\frac{1}{24} \cdot {x}^{4}}}{z}}{x} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 86.2

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

   10. Simplified 86.2%

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

Alternative 14: 85.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0058:\\ \;\;\;\;\frac{1}{z\_m \cdot \frac{x\_m}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(0.041666666666666664 \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 0.0058)
      (/ 1.0 (* z_m (/ x_m y_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 <= 0.0058) {
		tmp = 1.0 / (z_m * (x_m / y_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 <= 0.0058d0) then
        tmp = 1.0d0 / (z_m * (x_m / y_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 <= 0.0058) {
		tmp = 1.0 / (z_m * (x_m / y_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 <= 0.0058:
		tmp = 1.0 / (z_m * (x_m / y_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 <= 0.0058)
		tmp = Float64(1.0 / Float64(z_m * Float64(x_m / y_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 <= 0.0058)
		tmp = 1.0 / (z_m * (x_m / y_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, 0.0058], N[(1.0 / N[(z$95$m * N[(x$95$m / y$95$m), $MachinePrecision]), $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 < 0.0058

   1. Initial program 85.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 78.7

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

   5. Simplified 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \cdot \frac{y}{x}}{z} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}}{z} \]

      9. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{z}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{z}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}}} \]

   7. Applied egg-rr 87.9%

      \[\leadsto \color{blue}{\frac{1}{z \cdot \frac{x}{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{z \cdot \color{blue}{\frac{x}{y}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 62.6

         \[\leadsto \frac{1}{z \cdot \color{blue}{\frac{x}{y}}} \]

   10. Simplified 62.6%

       \[\leadsto \frac{1}{z \cdot \color{blue}{\frac{x}{y}}} \]

   if 0.0058 < x

   1. Initial program 75.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{1}{\frac{x}{y}}}}{z} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x}{\frac{x}{y}}}}{z} \]

      4. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{1}{\frac{x}{y}}}}{z} \]

      5. clear-num N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{y}{x}}}{z} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

      7. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}} \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\cosh x \cdot y}{z}}{x}} \]

      9. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot y\right) \cdot \frac{1}{z}}}{x} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{z}}{x}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(y \cdot \cosh x\right)} \cdot \frac{1}{z}}{x} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\color{blue}{y \cdot \left(\cosh x \cdot \frac{1}{z}\right)}}{x} \]

      13. div-inv N/A

          \[\leadsto \frac{y \cdot \color{blue}{\frac{\cosh x}{z}}}{x} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{y \cdot \frac{\cosh x}{z}}}{x} \]

      15. lower-/.f64 100.0

          \[\leadsto \frac{y \cdot \color{blue}{\frac{\cosh x}{z}}}{x} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{\cosh x}{z}}{x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot \frac{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}}{z}}{x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{z}}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{z}}{x} \]

      3. unpow2 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{z}}{x} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{z}}{x} \]

      5. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{z}}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{z}}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{z}}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{z}}{x} \]

      9. lower-*.f64 86.2

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{z}}{x} \]

   7. Simplified 86.2%

      \[\leadsto \frac{y \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}}{z}}{x} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}{z}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot y}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot y}}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)} \cdot y}{z} \]

      6. cube-mult N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right) \cdot y}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right) \cdot y}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right) \cdot y}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot y}{z} \]

      10. lower-*.f64 82.5

          \[\leadsto \frac{\left(0.041666666666666664 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot y}{z} \]

   10. Simplified 82.5%

       \[\leadsto \color{blue}{\frac{\left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot y}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;\frac{1}{z \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 83.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0058:\\ \;\;\;\;\frac{1}{z\_m \cdot \frac{x\_m}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(y\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 0.0058)
      (/ 1.0 (* z_m (/ x_m y_m)))
      (/ (* x_m (* y_m (* (* x_m x_m) 0.041666666666666664))) z_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.0058) {
		tmp = 1.0 / (z_m * (x_m / y_m));
	} else {
		tmp = (x_m * (y_m * ((x_m * x_m) * 0.041666666666666664))) / z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.0058d0) then
        tmp = 1.0d0 / (z_m * (x_m / y_m))
    else
        tmp = (x_m * (y_m * ((x_m * x_m) * 0.041666666666666664d0))) / z_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.0058) {
		tmp = 1.0 / (z_m * (x_m / y_m));
	} else {
		tmp = (x_m * (y_m * ((x_m * x_m) * 0.041666666666666664))) / z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.0058:
		tmp = 1.0 / (z_m * (x_m / y_m))
	else:
		tmp = (x_m * (y_m * ((x_m * x_m) * 0.041666666666666664))) / z_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.0058)
		tmp = Float64(1.0 / Float64(z_m * Float64(x_m / y_m)));
	else
		tmp = Float64(Float64(x_m * Float64(y_m * Float64(Float64(x_m * x_m) * 0.041666666666666664))) / z_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.0058)
		tmp = 1.0 / (z_m * (x_m / y_m));
	else
		tmp = (x_m * (y_m * ((x_m * x_m) * 0.041666666666666664))) / z_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 0.0058], N[(1.0 / N[(z$95$m * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0058

   1. Initial program 85.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      15. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      16. lower-*.f64 78.7

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 78.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right)} + \frac{1}{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right)}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right)\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \cdot \frac{y}{x}}{z} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}}{z} \]

      9. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{z}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{z}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}}} \]

   7. Applied egg-rr 87.9%

      \[\leadsto \color{blue}{\frac{1}{z \cdot \frac{x}{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{z \cdot \color{blue}{\frac{x}{y}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 62.6

         \[\leadsto \frac{1}{z \cdot \color{blue}{\frac{x}{y}}} \]

   10. Simplified 62.6%

       \[\leadsto \frac{1}{z \cdot \color{blue}{\frac{x}{y}}} \]

   if 0.0058 < x

   1. Initial program 75.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 60.4

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 60.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot y\right) \cdot \frac{1}{24}}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(y \cdot \frac{1}{24}\right)}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{{x}^{3} \cdot \color{blue}{\left(\frac{1}{24} \cdot y\right)}}{z} \]

      4. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{24} \cdot y\right)}{z} \]

      5. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{24} \cdot y\right)}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} \cdot y\right)\right)}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot y\right) \cdot {x}^{2}\right)}}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\left(\frac{1}{24} \cdot y\right) \cdot {x}^{2}\right)}}{z} \]

      9. *-commutative N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(y \cdot \frac{1}{24}\right)} \cdot {x}^{2}\right)}{z} \]

      10. associate-*l* N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)}}{z} \]

      12. *-commutative N/A

          \[\leadsto \frac{x \cdot \left(y \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}\right)}{z} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \left(y \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}\right)}{z} \]

      14. unpow2 N/A

          \[\leadsto \frac{x \cdot \left(y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24}\right)\right)}{z} \]

      15. lower-*.f64 75.0

          \[\leadsto \frac{x \cdot \left(y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.041666666666666664\right)\right)}{z} \]

   8. Simplified 75.0%

      \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 65.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0058:\\ \;\;\;\;\frac{1}{z\_m \cdot \frac{x\_m}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \left(y\_m \cdot 0.5\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 0.0058)
      (/ 1.0 (* z_m (/ x_m y_m)))
      (* (/ x_m z_m) (* y_m 0.5)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.0058) {
		tmp = 1.0 / (z_m * (x_m / y_m));
	} else {
		tmp = (x_m / z_m) * (y_m * 0.5);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.0058d0) then
        tmp = 1.0d0 / (z_m * (x_m / y_m))
    else
        tmp = (x_m / z_m) * (y_m * 0.5d0)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.0058) {
		tmp = 1.0 / (z_m * (x_m / y_m));
	} else {
		tmp = (x_m / z_m) * (y_m * 0.5);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.0058:
		tmp = 1.0 / (z_m * (x_m / y_m))
	else:
		tmp = (x_m / z_m) * (y_m * 0.5)
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.0058)
		tmp = Float64(1.0 / Float64(z_m * Float64(x_m / y_m)));
	else
		tmp = Float64(Float64(x_m / z_m) * Float64(y_m * 0.5));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.0058)
		tmp = 1.0 / (z_m * (x_m / y_m));
	else
		tmp = (x_m / z_m) * (y_m * 0.5);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 0.0058], N[(1.0 / N[(z$95$m * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0058

   1. Initial program 85.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      15. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      16. lower-*.f64 78.7

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 78.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right)} + \frac{1}{2}\right)\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right)}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right)\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \cdot \frac{y}{x}}{z} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}}{z} \]

      9. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{z}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{z}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}}} \]

   7. Applied egg-rr 87.9%

      \[\leadsto \color{blue}{\frac{1}{z \cdot \frac{x}{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{z \cdot \color{blue}{\frac{x}{y}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 62.6

         \[\leadsto \frac{1}{z \cdot \color{blue}{\frac{x}{y}}} \]

   10. Simplified 62.6%

       \[\leadsto \frac{1}{z \cdot \color{blue}{\frac{x}{y}}} \]

   if 0.0058 < x

   1. Initial program 75.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{{x}^{2} \cdot y}{z} \cdot \frac{1}{2}} + \frac{y}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z}} + \frac{y}{z}}{x} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{y}{z}\right)} + \frac{y}{z}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{z}} + \frac{y}{z}}{x} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{y}{z}}}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{z}}{x} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot {x}^{2}}{x} \cdot \frac{y}{z}} \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot y}{x \cdot z}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot y}{x \cdot z} \]

      10. distribute-rgt1-in N/A

          \[\leadsto \frac{\color{blue}{y + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x \cdot z} \]

      11. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}}{x \cdot z} \]

      12. *-commutative N/A

          \[\leadsto \frac{y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot {x}^{2}\right)}}{x \cdot z} \]

      13. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}}{x \cdot z} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y + \left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}{x \cdot z}} \]

   5. Simplified 47.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 0.5 \cdot \left(x \cdot x\right), y\right)}{x \cdot z}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{z}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(y \cdot x\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot x}}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{2}\right)} \cdot x}{z} \]

      7. lower-*.f64 41.4

         \[\leadsto \frac{\color{blue}{\left(y \cdot 0.5\right)} \cdot x}{z} \]

   8. Simplified 41.4%

      \[\leadsto \color{blue}{\frac{\left(y \cdot 0.5\right) \cdot x}{z}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{2}\right)} \cdot x}{z} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y \cdot \frac{1}{2}\right) \cdot \frac{x}{z}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(y \cdot \frac{1}{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(y \cdot \frac{1}{2}\right)} \]

      5. lower-/.f64 41.3

         \[\leadsto \color{blue}{\frac{x}{z}} \cdot \left(y \cdot 0.5\right) \]

   10. Applied egg-rr 41.3%

       \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(y \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 65.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0058:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \left(y\_m \cdot 0.5\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 0.0058) (/ (/ y_m x_m) z_m) (* (/ x_m z_m) (* y_m 0.5)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.0058) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (x_m / z_m) * (y_m * 0.5);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.0058d0) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = (x_m / z_m) * (y_m * 0.5d0)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.0058) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (x_m / z_m) * (y_m * 0.5);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.0058:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = (x_m / z_m) * (y_m * 0.5)
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.0058)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(Float64(x_m / z_m) * Float64(y_m * 0.5));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.0058)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = (x_m / z_m) * (y_m * 0.5);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 0.0058], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0058

   1. Initial program 85.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
   4. Step-by-step derivation

      1. lower-/.f64 62.3

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   5. Simplified 62.3%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   if 0.0058 < x

   1. Initial program 75.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{{x}^{2} \cdot y}{z} \cdot \frac{1}{2}} + \frac{y}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z}} + \frac{y}{z}}{x} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{y}{z}\right)} + \frac{y}{z}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{z}} + \frac{y}{z}}{x} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{y}{z}}}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{z}}{x} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot {x}^{2}}{x} \cdot \frac{y}{z}} \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot y}{x \cdot z}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot y}{x \cdot z} \]

      10. distribute-rgt1-in N/A

          \[\leadsto \frac{\color{blue}{y + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x \cdot z} \]

      11. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}}{x \cdot z} \]

      12. *-commutative N/A

          \[\leadsto \frac{y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot {x}^{2}\right)}}{x \cdot z} \]

      13. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}}{x \cdot z} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y + \left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}{x \cdot z}} \]

   5. Simplified 47.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 0.5 \cdot \left(x \cdot x\right), y\right)}{x \cdot z}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{z}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(y \cdot x\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot x}}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{2}\right)} \cdot x}{z} \]

      7. lower-*.f64 41.4

         \[\leadsto \frac{\color{blue}{\left(y \cdot 0.5\right)} \cdot x}{z} \]

   8. Simplified 41.4%

      \[\leadsto \color{blue}{\frac{\left(y \cdot 0.5\right) \cdot x}{z}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{2}\right)} \cdot x}{z} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y \cdot \frac{1}{2}\right) \cdot \frac{x}{z}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(y \cdot \frac{1}{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(y \cdot \frac{1}{2}\right)} \]

      5. lower-/.f64 41.3

         \[\leadsto \color{blue}{\frac{x}{z}} \cdot \left(y \cdot 0.5\right) \]

   10. Applied egg-rr 41.3%

       \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(y \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 66.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0058:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \left(y\_m \cdot 0.5\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 0.0058) (/ y_m (* x_m z_m)) (* (/ x_m z_m) (* y_m 0.5)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.0058) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (x_m / z_m) * (y_m * 0.5);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.0058d0) then
        tmp = y_m / (x_m * z_m)
    else
        tmp = (x_m / z_m) * (y_m * 0.5d0)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.0058) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (x_m / z_m) * (y_m * 0.5);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.0058:
		tmp = y_m / (x_m * z_m)
	else:
		tmp = (x_m / z_m) * (y_m * 0.5)
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.0058)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(x_m / z_m) * Float64(y_m * 0.5));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.0058)
		tmp = y_m / (x_m * z_m);
	else
		tmp = (x_m / z_m) * (y_m * 0.5);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 0.0058], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0058

   1. Initial program 85.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. lower-*.f64 60.9

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

   5. Simplified 60.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 0.0058 < x

   1. Initial program 75.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{{x}^{2} \cdot y}{z} \cdot \frac{1}{2}} + \frac{y}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z}} + \frac{y}{z}}{x} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{y}{z}\right)} + \frac{y}{z}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{z}} + \frac{y}{z}}{x} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{y}{z}}}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{z}}{x} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot {x}^{2}}{x} \cdot \frac{y}{z}} \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot y}{x \cdot z}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot y}{x \cdot z} \]

      10. distribute-rgt1-in N/A

          \[\leadsto \frac{\color{blue}{y + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x \cdot z} \]

      11. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}}{x \cdot z} \]

      12. *-commutative N/A

          \[\leadsto \frac{y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot {x}^{2}\right)}}{x \cdot z} \]

      13. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}}{x \cdot z} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y + \left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}{x \cdot z}} \]

   5. Simplified 47.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 0.5 \cdot \left(x \cdot x\right), y\right)}{x \cdot z}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{z}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(y \cdot x\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot x}}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{2}\right)} \cdot x}{z} \]

      7. lower-*.f64 41.4

         \[\leadsto \frac{\color{blue}{\left(y \cdot 0.5\right)} \cdot x}{z} \]

   8. Simplified 41.4%

      \[\leadsto \color{blue}{\frac{\left(y \cdot 0.5\right) \cdot x}{z}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{2}\right)} \cdot x}{z} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y \cdot \frac{1}{2}\right) \cdot \frac{x}{z}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(y \cdot \frac{1}{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(y \cdot \frac{1}{2}\right)} \]

      5. lower-/.f64 41.3

         \[\leadsto \color{blue}{\frac{x}{z}} \cdot \left(y \cdot 0.5\right) \]

   10. Applied egg-rr 41.3%

       \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(y \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 62.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0058:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m \cdot 0.5}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 0.0058) (/ 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 <= 0.0058) {
		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 <= 0.0058d0) 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 <= 0.0058) {
		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 <= 0.0058:
		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 <= 0.0058)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(x_m * Float64(Float64(y_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 <= 0.0058)
		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, 0.0058], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y$95$m * 0.5), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0058

   1. Initial program 85.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. lower-*.f64 60.9

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

   5. Simplified 60.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 0.0058 < x

   1. Initial program 75.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{{x}^{2} \cdot y}{z} \cdot \frac{1}{2}} + \frac{y}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z}} + \frac{y}{z}}{x} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{y}{z}\right)} + \frac{y}{z}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{z}} + \frac{y}{z}}{x} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{y}{z}}}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{z}}{x} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot {x}^{2}}{x} \cdot \frac{y}{z}} \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot y}{x \cdot z}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot y}{x \cdot z} \]

      10. distribute-rgt1-in N/A

          \[\leadsto \frac{\color{blue}{y + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x \cdot z} \]

      11. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}}{x \cdot z} \]

      12. *-commutative N/A

          \[\leadsto \frac{y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot {x}^{2}\right)}}{x \cdot z} \]

      13. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}}{x \cdot z} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y + \left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}{x \cdot z}} \]

   5. Simplified 47.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 0.5 \cdot \left(x \cdot x\right), y\right)}{x \cdot z}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{z}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(y \cdot x\right)}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot x}}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{2}\right)} \cdot x}{z} \]

      7. lower-*.f64 41.4

         \[\leadsto \frac{\color{blue}{\left(y \cdot 0.5\right)} \cdot x}{z} \]

   8. Simplified 41.4%

      \[\leadsto \color{blue}{\frac{\left(y \cdot 0.5\right) \cdot x}{z}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{2}\right)} \cdot x}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \frac{1}{2}\right)}}{z} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y \cdot \frac{1}{2}}{z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{y \cdot \frac{1}{2}}{z}} \]

      5. lower-/.f64 36.4

         \[\leadsto x \cdot \color{blue}{\frac{y \cdot 0.5}{z}} \]

   10. Applied egg-rr 36.4%

       \[\leadsto \color{blue}{x \cdot \frac{y \cdot 0.5}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 49.9% accurate, 7.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \frac{y\_m}{x\_m \cdot z\_m}\right)\right) \end{array} \]

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 82.1%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   2. lower-*.f64 45.3

      \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

5. Simplified 45.3%

   \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
6. Add Preprocessing

Developer Target 1: 97.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y z) x) (cosh x))))
   (if (< y -4.618902267687042e-52)
     t_0
     (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y / z) / x) * cosh(x)
    if (y < (-4.618902267687042d-52)) then
        tmp = t_0
    else if (y < 1.038530535935153d-39) then
        tmp = ((cosh(x) * y) / x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * Math.cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((Math.cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / z) / x) * math.cosh(x)
	tmp = 0
	if y < -4.618902267687042e-52:
		tmp = t_0
	elif y < 1.038530535935153e-39:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) / x) * cosh(x))
	tmp = 0.0
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / z) / x) * cosh(x);
	tmp = 0.0;
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -4.618902267687042e-52], t$95$0, If[Less[y, 1.038530535935153e-39], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024200 
(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))