Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 85.0% → 99.6%

Time: 14.6s

Alternatives: 24

Speedup: 3.4×

• 65.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m}\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\cosh x\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x_m) (/ y_m x_m)) z_m)))
   (*
    x_s
    (*
     y_s
     (* z_s (if (<= t_0 5e-78) t_0 (/ (* y_m (/ (cosh x_m) z_m)) x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= 5e-78) {
		tmp = t_0;
	} else {
		tmp = (y_m * (cosh(x_m) / z_m)) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (cosh(x_m) * (y_m / x_m)) / z_m
    if (t_0 <= 5d-78) then
        tmp = t_0
    else
        tmp = (y_m * (cosh(x_m) / z_m)) / x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double t_0 = (Math.cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= 5e-78) {
		tmp = t_0;
	} else {
		tmp = (y_m * (Math.cosh(x_m) / z_m)) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	t_0 = (math.cosh(x_m) * (y_m / x_m)) / z_m
	tmp = 0
	if t_0 <= 5e-78:
		tmp = t_0
	else:
		tmp = (y_m * (math.cosh(x_m) / z_m)) / x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	t_0 = Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m)
	tmp = 0.0
	if (t_0 <= 5e-78)
		tmp = t_0;
	else
		tmp = Float64(Float64(y_m * Float64(cosh(x_m) / z_m)) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m)
	t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	tmp = 0.0;
	if (t_0 <= 5e-78)
		tmp = t_0;
	else
		tmp = (y_m * (cosh(x_m) / z_m)) / x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[t$95$0, 5e-78], t$95$0, N[(N[(y$95$m * N[(N[Cosh[x$95$m], $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) < 4.9999999999999996e-78

   1. Initial program 94.7%

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

   if 4.9999999999999996e-78 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 69.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\cosh x\_m}{\frac{x\_m \cdot z\_m}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\cosh x\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 5e-17)
      (/ (cosh x_m) (/ (* x_m z_m) y_m))
      (/ (* y_m (/ (cosh x_m) z_m)) x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-17) {
		tmp = cosh(x_m) / ((x_m * z_m) / y_m);
	} else {
		tmp = (y_m * (cosh(x_m) / z_m)) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5d-17) then
        tmp = cosh(x_m) / ((x_m * z_m) / y_m)
    else
        tmp = (y_m * (cosh(x_m) / z_m)) / x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((Math.cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-17) {
		tmp = Math.cosh(x_m) / ((x_m * z_m) / y_m);
	} else {
		tmp = (y_m * (Math.cosh(x_m) / z_m)) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	tmp = 0
	if ((math.cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-17:
		tmp = math.cosh(x_m) / ((x_m * z_m) / y_m)
	else:
		tmp = (y_m * (math.cosh(x_m) / z_m)) / x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 5e-17)
		tmp = Float64(cosh(x_m) / Float64(Float64(x_m * z_m) / y_m));
	else
		tmp = Float64(Float64(y_m * Float64(cosh(x_m) / z_m)) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-17)
		tmp = cosh(x_m) / ((x_m * z_m) / y_m);
	else
		tmp = (y_m * (cosh(x_m) / z_m)) / x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$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-17], N[(N[Cosh[x$95$m], $MachinePrecision] / N[(N[(x$95$m * z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[Cosh[x$95$m], $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) < 4.9999999999999999e-17

   1. Initial program 94.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. 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. associate-/l/ N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. div-inv N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

   if 4.9999999999999999e-17 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 68.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 3: 94.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m} \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot 0.001388888888888889, x\_m, 0.041666666666666664\right), 0.5\right), 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 5e-78)
      (/
       (*
        (/ 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)
      (/
       (*
        y_m
        (/
         (fma
          (* x_m x_m)
          (fma
           (* x_m x_m)
           (fma (* x_m 0.001388888888888889) x_m 0.041666666666666664)
           0.5)
          1.0)
         z_m))
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-78) {
		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 = (y_m * (fma((x_m * x_m), fma((x_m * x_m), fma((x_m * 0.001388888888888889), x_m, 0.041666666666666664), 0.5), 1.0) / z_m)) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 5e-78)
		tmp = Float64(Float64(Float64(y_m / x_m) * fma(x_m, Float64(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(y_m * Float64(fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), fma(Float64(x_m * 0.001388888888888889), x_m, 0.041666666666666664), 0.5), 1.0) / z_m)) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$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-78], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 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]), $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] * N[(N[(x$95$m * 0.001388888888888889), $MachinePrecision] * x$95$m + 0.041666666666666664), $MachinePrecision] + 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) < 4.9999999999999996e-78

   1. Initial program 94.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. 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 86.5

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

   5. Applied rewrites 86.5%

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

   if 4.9999999999999996e-78 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 69.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 96.2

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

   7. Applied rewrites 96.2%

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

      1. associate-*l* N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 96.2

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

   9. Applied rewrites 96.2%

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

4. Final simplification 91.3%

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

Alternative 4: 94.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, y\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889\right)\right)\right), y\_m\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot 0.001388888888888889, x\_m, 0.041666666666666664\right), 0.5\right), 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 5e-78)
      (/
       (/
        (fma
         (* x_m x_m)
         (* y_m (* x_m (* x_m (* (* x_m x_m) 0.001388888888888889))))
         y_m)
        x_m)
       z_m)
      (/
       (*
        y_m
        (/
         (fma
          (* x_m x_m)
          (fma
           (* x_m x_m)
           (fma (* x_m 0.001388888888888889) x_m 0.041666666666666664)
           0.5)
          1.0)
         z_m))
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-78) {
		tmp = (fma((x_m * x_m), (y_m * (x_m * (x_m * ((x_m * x_m) * 0.001388888888888889)))), y_m) / x_m) / z_m;
	} else {
		tmp = (y_m * (fma((x_m * x_m), fma((x_m * x_m), fma((x_m * 0.001388888888888889), x_m, 0.041666666666666664), 0.5), 1.0) / z_m)) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 5e-78)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), Float64(y_m * Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.001388888888888889)))), 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), fma(Float64(x_m * 0.001388888888888889), x_m, 0.041666666666666664), 0.5), 1.0) / z_m)) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$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-78], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $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] * N[(N[(x$95$m * 0.001388888888888889), $MachinePrecision] * x$95$m + 0.041666666666666664), $MachinePrecision] + 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) < 4.9999999999999996e-78

   1. Initial program 94.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 85.7%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r* N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{720} \cdot {x}^{4}\right) \cdot y}, y\right)}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{y \cdot \left(\frac{1}{720} \cdot {x}^{4}\right)}, y\right)}{x}}{z} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 85.1

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

   10. Applied rewrites 85.1%

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

   if 4.9999999999999996e-78 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 69.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 96.2

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

   7. Applied rewrites 96.2%

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

      1. associate-*l* N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 96.2

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

   9. Applied rewrites 96.2%

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

Alternative 5: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 0.002:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot y\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), y\_m\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889, 0.5\right), 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 0.002)
      (/
       (fma
        (* x_m y_m)
        (*
         x_m
         (fma
          x_m
          (* x_m (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664))
          0.5))
        y_m)
       (* x_m z_m))
      (/
       (*
        y_m
        (/
         (fma
          (* x_m x_m)
          (fma (* x_m x_m) (* (* x_m x_m) 0.001388888888888889) 0.5)
          1.0)
         z_m))
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 0.002) {
		tmp = fma((x_m * y_m), (x_m * fma(x_m, (x_m * fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664)), 0.5)), y_m) / (x_m * z_m);
	} else {
		tmp = (y_m * (fma((x_m * x_m), fma((x_m * x_m), ((x_m * x_m) * 0.001388888888888889), 0.5), 1.0) / z_m)) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 0.002)
		tmp = Float64(fma(Float64(x_m * y_m), Float64(x_m * fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664)), 0.5)), y_m) / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), Float64(Float64(x_m * x_m) * 0.001388888888888889), 0.5), 1.0) / z_m)) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 0.002], N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $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] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 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) < 2e-3

   1. Initial program 94.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

   7. Applied rewrites 76.8%

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

   if 2e-3 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 68.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 96.0

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

   7. Applied rewrites 96.0%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 96.0

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

   10. Applied rewrites 96.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 0.002:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot y, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.001388888888888889, 0.5\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 0.002:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m} \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 0.002)
      (/
       (*
        (/ 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

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 0.002) {
		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 x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 0.002)
		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(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 0.002], 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 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 2e-3

   1. Initial program 94.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   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 82.4

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

   5. Applied rewrites 82.4%

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

   if 2e-3 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 68.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 99.9

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

   4. Applied rewrites 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 91.4

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

   7. Applied rewrites 91.4%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 0.002:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{y\_m \cdot \frac{1}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 5e-17)
      (/ (* y_m (/ 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

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-17) {
		tmp = (y_m * (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 x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 5e-17)
		tmp = Float64(Float64(y_m * Float64(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(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$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-17], N[(N[(y$95$m * N[(1.0 / x$95$m), $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 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4.9999999999999999e-17

   1. Initial program 94.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 58.5

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

   5. Applied rewrites 58.5%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 58.4

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

   7. Applied rewrites 58.4%

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

      1. associate-/r* N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 55.7

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

   9. Applied rewrites 55.7%

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

   if 4.9999999999999999e-17 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 68.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 99.9

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

   4. Applied rewrites 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 91.4

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

   7. Applied rewrites 91.4%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1 \cdot 10^{+46}:\\ \;\;\;\;y\_m \cdot \frac{\cosh x\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot 0.001388888888888889, x\_m, 0.041666666666666664\right), 0.5\right), 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 1.1e+46)
      (* y_m (/ (cosh x_m) (* x_m z_m)))
      (/
       (*
        y_m
        (/
         (fma
          (* x_m x_m)
          (fma
           (* x_m x_m)
           (fma (* x_m 0.001388888888888889) x_m 0.041666666666666664)
           0.5)
          1.0)
         z_m))
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.1e+46) {
		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 * 0.001388888888888889), x_m, 0.041666666666666664), 0.5), 1.0) / z_m)) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 1.1e+46)
		tmp = Float64(y_m * Float64(cosh(x_m) / Float64(x_m * z_m)));
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), fma(Float64(x_m * 0.001388888888888889), x_m, 0.041666666666666664), 0.5), 1.0) / z_m)) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 1.1e+46], N[(y$95$m * N[(N[Cosh[x$95$m], $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $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] * N[(N[(x$95$m * 0.001388888888888889), $MachinePrecision] * x$95$m + 0.041666666666666664), $MachinePrecision] + 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 x < 1.1e46

   1. Initial program 86.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. *-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 85.8

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

   4. Applied rewrites 85.8%

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

   if 1.1e46 < x

   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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. associate-*l* N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 100.0

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

   9. Applied rewrites 100.0%

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

Alternative 9: 92.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, y\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889\right)\right)\right), y\_m\right)}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m, y\_m \cdot \left(x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right)\right), y\_m\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= y_m 4e+68)
      (/
       (/
        (fma
         (* x_m x_m)
         (* y_m (* x_m (* x_m (* (* x_m x_m) 0.001388888888888889))))
         y_m)
        x_m)
       z_m)
      (/
       (/
        (fma
         x_m
         (* y_m (* x_m (fma x_m (* x_m 0.041666666666666664) 0.5)))
         y_m)
        z_m)
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 4e+68) {
		tmp = (fma((x_m * x_m), (y_m * (x_m * (x_m * ((x_m * x_m) * 0.001388888888888889)))), y_m) / x_m) / z_m;
	} else {
		tmp = (fma(x_m, (y_m * (x_m * fma(x_m, (x_m * 0.041666666666666664), 0.5))), y_m) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 4e+68)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), Float64(y_m * Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.001388888888888889)))), y_m) / x_m) / z_m);
	else
		tmp = Float64(Float64(fma(x_m, Float64(y_m * Float64(x_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5))), y_m) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[y$95$m, 4e+68], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(x$95$m * N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $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 y < 3.99999999999999981e68

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 87.6%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r* N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{720} \cdot {x}^{4}\right) \cdot y}, y\right)}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{y \cdot \left(\frac{1}{720} \cdot {x}^{4}\right)}, y\right)}{x}}{z} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 87.5

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

   10. Applied rewrites 87.5%

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

   if 3.99999999999999981e68 < y

   1. Initial program 89.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   7. Applied rewrites 99.8%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, y \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right), y\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, y \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), y\_m, y\_m\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)\right)\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 2.6e+77)
      (/
       (fma
        (*
         (* x_m x_m)
         (fma
          x_m
          (* x_m (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664))
          0.5))
        y_m
        y_m)
       (* x_m z_m))
      (/
       (/ (* y_m (* x_m (* x_m (* (* x_m x_m) 0.041666666666666664)))) z_m)
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 2.6e+77) {
		tmp = fma(((x_m * x_m) * fma(x_m, (x_m * fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664)), 0.5)), y_m, y_m) / (x_m * z_m);
	} else {
		tmp = ((y_m * (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 2.6e+77)
		tmp = Float64(fma(Float64(Float64(x_m * x_m) * fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664)), 0.5)), y_m, y_m) / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664)))) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 2.6e+77], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * y$95$m + y$95$m), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000002e77

   1. Initial program 86.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 78.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

   7. Applied rewrites 79.9%

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

   if 2.6000000000000002e77 < x

   1. Initial program 59.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. unpow3 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow3 N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 59.5

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

   8. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   10. Applied rewrites 100.0%

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

Alternative 11: 89.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, y\_m \cdot \left(x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), y\_m\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)\right)\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 2.6e+77)
      (/
       (fma
        x_m
        (*
         y_m
         (*
          x_m
          (fma
           (* x_m x_m)
           (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
           0.5)))
        y_m)
       (* x_m z_m))
      (/
       (/ (* y_m (* x_m (* x_m (* (* x_m x_m) 0.041666666666666664)))) z_m)
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 2.6e+77) {
		tmp = fma(x_m, (y_m * (x_m * fma((x_m * x_m), fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5))), y_m) / (x_m * z_m);
	} else {
		tmp = ((y_m * (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 2.6e+77)
		tmp = Float64(fma(x_m, Float64(y_m * Float64(x_m * fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5))), y_m) / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664)))) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 2.6e+77], N[(N[(x$95$m * N[(y$95$m * N[(x$95$m * 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]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000002e77

   1. Initial program 86.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

   8. Applied rewrites 79.4%

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

   if 2.6000000000000002e77 < x

   1. Initial program 59.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. unpow3 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow3 N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 59.5

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

   8. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   10. Applied rewrites 100.0%

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

Alternative 12: 89.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)\right)\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 2.6e+77)
      (*
       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)
        (* x_m z_m)))
      (/
       (/ (* y_m (* x_m (* x_m (* (* x_m x_m) 0.041666666666666664)))) z_m)
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 2.6e+77) {
		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) / (x_m * z_m));
	} else {
		tmp = ((y_m * (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 2.6e+77)
		tmp = Float64(y_m * Float64(fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0) / Float64(x_m * z_m)));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664)))) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 2.6e+77], N[(y$95$m * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000002e77

   1. Initial program 86.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 95.2

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 88.9

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

   7. Applied rewrites 88.9%

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

   9. Applied rewrites 79.4%

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

   if 2.6000000000000002e77 < x

   1. Initial program 59.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. unpow3 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow3 N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 59.5

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

   8. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   10. Applied rewrites 100.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 88.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 500000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right)\right), y\_m\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)\right)\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 500000.0)
      (/
       (fma x_m (* x_m (* y_m (fma x_m (* x_m 0.041666666666666664) 0.5))) y_m)
       (* x_m z_m))
      (/
       (/ (* y_m (* x_m (* x_m (* (* x_m x_m) 0.041666666666666664)))) z_m)
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 500000.0) {
		tmp = fma(x_m, (x_m * (y_m * fma(x_m, (x_m * 0.041666666666666664), 0.5))), y_m) / (x_m * z_m);
	} else {
		tmp = ((y_m * (x_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 500000.0)
		tmp = Float64(fma(x_m, Float64(x_m * Float64(y_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5))), y_m) / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664)))) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 500000.0], N[(N[(x$95$m * N[(x$95$m * N[(y$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5e5

   1. Initial program 85.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 76.6%

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

   if 5e5 < x

   1. Initial program 71.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. unpow3 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow3 N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 52.4

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

   8. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   10. Applied rewrites 80.4%

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

Alternative 14: 86.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 6.6 \cdot 10^{+61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right)\right), y\_m\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 6.6e+61)
      (/
       (fma x_m (* x_m (* y_m (fma x_m (* x_m 0.041666666666666664) 0.5))) y_m)
       (* x_m z_m))
      (* y_m (/ (* x_m (* (* x_m x_m) 0.041666666666666664)) z_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 6.6e+61) {
		tmp = fma(x_m, (x_m * (y_m * fma(x_m, (x_m * 0.041666666666666664), 0.5))), y_m) / (x_m * z_m);
	} else {
		tmp = y_m * ((x_m * ((x_m * x_m) * 0.041666666666666664)) / z_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 6.6e+61)
		tmp = Float64(fma(x_m, Float64(x_m * Float64(y_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5))), y_m) / Float64(x_m * z_m));
	else
		tmp = Float64(y_m * Float64(Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664)) / z_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 6.6e+61], N[(N[(x$95$m * N[(x$95$m * N[(y$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 6.5999999999999995e61

   1. Initial program 86.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 74.0%

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

   if 6.5999999999999995e61 < x

   1. Initial program 60.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. unpow3 N/A

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

      16. lower-/.f64 N/A

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

   8. Applied rewrites 91.5%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{+61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot t\_0\right), y\_m\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m \cdot t\_0}{z\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* (* x_m x_m) 0.041666666666666664)))
   (*
    x_s
    (*
     y_s
     (*
      z_s
      (if (<= x_m 1.6e+62)
        (/ (fma x_m (* x_m (* y_m t_0)) y_m) (* x_m z_m))
        (* y_m (/ (* x_m t_0) z_m))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double t_0 = (x_m * x_m) * 0.041666666666666664;
	double tmp;
	if (x_m <= 1.6e+62) {
		tmp = fma(x_m, (x_m * (y_m * t_0)), y_m) / (x_m * z_m);
	} else {
		tmp = y_m * ((x_m * t_0) / z_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	t_0 = Float64(Float64(x_m * x_m) * 0.041666666666666664)
	tmp = 0.0
	if (x_m <= 1.6e+62)
		tmp = Float64(fma(x_m, Float64(x_m * Float64(y_m * t_0)), y_m) / Float64(x_m * z_m));
	else
		tmp = Float64(y_m * Float64(Float64(x_m * t_0) / z_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 1.6e+62], N[(N[(x$95$m * N[(x$95$m * N[(y$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(x$95$m * t$95$0), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.59999999999999992e62

   1. Initial program 86.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 73.7

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

   8. Applied rewrites 73.7%

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

   if 1.59999999999999992e62 < x

   1. Initial program 60.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. unpow3 N/A

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

      16. lower-/.f64 N/A

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

   8. Applied rewrites 91.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right), y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 85.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 7.8 \cdot 10^{+54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, y\_m \cdot \left(x\_m \cdot x\_m\right), y\_m\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 7.8e+54)
      (/ (fma 0.5 (* y_m (* x_m x_m)) y_m) (* x_m z_m))
      (* y_m (/ (* x_m (* (* x_m x_m) 0.041666666666666664)) z_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 7.8e+54) {
		tmp = fma(0.5, (y_m * (x_m * x_m)), y_m) / (x_m * z_m);
	} else {
		tmp = y_m * ((x_m * ((x_m * x_m) * 0.041666666666666664)) / z_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 7.8e+54)
		tmp = Float64(fma(0.5, Float64(y_m * Float64(x_m * x_m)), y_m) / Float64(x_m * z_m));
	else
		tmp = Float64(y_m * Float64(Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664)) / z_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 7.8e+54], N[(N[(0.5 * N[(y$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7.8000000000000005e54

   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 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

      10. associate-*r/ N/A

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

      11. times-frac N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 70.3%

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

   if 7.8000000000000005e54 < x

   1. Initial program 62.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. unpow3 N/A

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

      16. lower-/.f64 N/A

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

   8. Applied rewrites 91.9%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{+54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, y \cdot \left(x \cdot x\right), y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 85.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.017:\\ \;\;\;\;\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot \frac{y\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot 0.041666666666666664\right)\right)\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 0.017)
      (* (fma 0.5 (* x_m x_m) 1.0) (/ y_m (* x_m z_m)))
      (/ (* y_m (* x_m (* x_m (* x_m 0.041666666666666664)))) z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.017) {
		tmp = fma(0.5, (x_m * x_m), 1.0) * (y_m / (x_m * z_m));
	} else {
		tmp = (y_m * (x_m * (x_m * (x_m * 0.041666666666666664)))) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.017)
		tmp = Float64(fma(0.5, Float64(x_m * x_m), 1.0) * Float64(y_m / Float64(x_m * z_m)));
	else
		tmp = Float64(Float64(y_m * Float64(x_m * Float64(x_m * Float64(x_m * 0.041666666666666664)))) / z_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 0.017], N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.017000000000000001

   1. Initial program 85.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 74.2

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

   5. Applied rewrites 74.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 70.5

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

   7. Applied rewrites 70.5%

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

   if 0.017000000000000001 < x

   1. Initial program 72.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. unpow3 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow3 N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 52.4

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

   8. Applied rewrites 52.4%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. unpow3 N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. lower-/.f64 N/A

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

   11. Applied rewrites 75.2%

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

Alternative 18: 85.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.017:\\ \;\;\;\;y\_m \cdot \frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot 0.041666666666666664\right)\right)\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 0.017)
      (* y_m (/ (fma 0.5 (* x_m x_m) 1.0) (* x_m z_m)))
      (/ (* y_m (* x_m (* x_m (* x_m 0.041666666666666664)))) z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.017) {
		tmp = y_m * (fma(0.5, (x_m * x_m), 1.0) / (x_m * z_m));
	} else {
		tmp = (y_m * (x_m * (x_m * (x_m * 0.041666666666666664)))) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.017)
		tmp = Float64(y_m * Float64(fma(0.5, Float64(x_m * x_m), 1.0) / Float64(x_m * z_m)));
	else
		tmp = Float64(Float64(y_m * Float64(x_m * Float64(x_m * Float64(x_m * 0.041666666666666664)))) / z_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 0.017], N[(y$95$m * N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.017000000000000001

   1. Initial program 85.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 74.2

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

   5. Applied rewrites 74.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. clear-num N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. clear-num N/A

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

      13. /-rgt-identity N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 72.0

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

   7. Applied rewrites 72.0%

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

   if 0.017000000000000001 < x

   1. Initial program 72.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. unpow3 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow3 N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 52.4

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

   8. Applied rewrites 52.4%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. unpow3 N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. lower-/.f64 N/A

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

   11. Applied rewrites 75.2%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.017:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 85.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.017:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot 0.041666666666666664\right)\right)\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 0.017)
      (/ y_m (* x_m z_m))
      (/ (* y_m (* x_m (* x_m (* x_m 0.041666666666666664)))) z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.017) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (y_m * (x_m * (x_m * (x_m * 0.041666666666666664)))) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 0.017d0) then
        tmp = y_m / (x_m * z_m)
    else
        tmp = (y_m * (x_m * (x_m * (x_m * 0.041666666666666664d0)))) / z_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.017) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (y_m * (x_m * (x_m * (x_m * 0.041666666666666664)))) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.017:
		tmp = y_m / (x_m * z_m)
	else:
		tmp = (y_m * (x_m * (x_m * (x_m * 0.041666666666666664)))) / z_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.017)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(y_m * Float64(x_m * Float64(x_m * Float64(x_m * 0.041666666666666664)))) / z_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.017)
		tmp = y_m / (x_m * z_m);
	else
		tmp = (y_m * (x_m * (x_m * (x_m * 0.041666666666666664)))) / z_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 0.017], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.017000000000000001

   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 59.6

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

   5. Applied rewrites 59.6%

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

   if 0.017000000000000001 < x

   1. Initial program 72.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. unpow3 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow3 N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 52.4

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

   8. Applied rewrites 52.4%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. unpow3 N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. lower-/.f64 N/A

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

   11. Applied rewrites 75.2%

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

Alternative 20: 85.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.017:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 0.017)
      (/ y_m (* x_m z_m))
      (* y_m (/ (* x_m (* (* x_m x_m) 0.041666666666666664)) z_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.017) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = y_m * ((x_m * ((x_m * x_m) * 0.041666666666666664)) / z_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 0.017d0) then
        tmp = y_m / (x_m * z_m)
    else
        tmp = y_m * ((x_m * ((x_m * x_m) * 0.041666666666666664d0)) / z_m)
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.017) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = y_m * ((x_m * ((x_m * x_m) * 0.041666666666666664)) / z_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.017:
		tmp = y_m / (x_m * z_m)
	else:
		tmp = y_m * ((x_m * ((x_m * x_m) * 0.041666666666666664)) / z_m)
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.017)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(y_m * Float64(Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664)) / z_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.017)
		tmp = y_m / (x_m * z_m);
	else
		tmp = y_m * ((x_m * ((x_m * x_m) * 0.041666666666666664)) / z_m);
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 0.017], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.017000000000000001

   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 59.6

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

   5. Applied rewrites 59.6%

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

   if 0.017000000000000001 < x

   1. Initial program 72.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. unpow3 N/A

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

      16. lower-/.f64 N/A

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

   8. Applied rewrites 73.9%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.017:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 65.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.017:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x\_m \cdot 0.5\right)}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 0.017) (/ y_m (* x_m z_m)) (/ (* y_m (* x_m 0.5)) z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.017) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (y_m * (x_m * 0.5)) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 0.017d0) then
        tmp = y_m / (x_m * z_m)
    else
        tmp = (y_m * (x_m * 0.5d0)) / z_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.017) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (y_m * (x_m * 0.5)) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.017:
		tmp = y_m / (x_m * z_m)
	else:
		tmp = (y_m * (x_m * 0.5)) / z_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.017)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(y_m * Float64(x_m * 0.5)) / z_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.017)
		tmp = y_m / (x_m * z_m);
	else
		tmp = (y_m * (x_m * 0.5)) / z_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 0.017], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.017000000000000001

   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 59.6

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

   5. Applied rewrites 59.6%

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

   if 0.017000000000000001 < x

   1. Initial program 72.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 33.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   10. 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. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 34.4

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

   11. Applied rewrites 34.4%

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

Alternative 22: 65.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.017:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\_m\right) \cdot \frac{0.5}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 0.017) (/ y_m (* x_m z_m)) (* (* x_m y_m) (/ 0.5 z_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.017) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (x_m * y_m) * (0.5 / z_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 0.017d0) then
        tmp = y_m / (x_m * z_m)
    else
        tmp = (x_m * y_m) * (0.5d0 / z_m)
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.017) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (x_m * y_m) * (0.5 / z_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.017:
		tmp = y_m / (x_m * z_m)
	else:
		tmp = (x_m * y_m) * (0.5 / z_m)
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.017)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(x_m * y_m) * Float64(0.5 / z_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.017)
		tmp = y_m / (x_m * z_m);
	else
		tmp = (x_m * y_m) * (0.5 / z_m);
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 0.017], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.017000000000000001

   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 59.6

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

   5. Applied rewrites 59.6%

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

   if 0.017000000000000001 < x

   1. Initial program 72.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 33.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   10. 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. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 34.4

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

   11. Applied rewrites 34.4%

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

       1. associate-*r* N/A

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

       2. associate-/l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 34.4

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

   13. Applied rewrites 34.4%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.017:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 61.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.017:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot 0.5\right) \cdot \frac{y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 0.017) (/ y_m (* x_m z_m)) (* (* x_m 0.5) (/ y_m z_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.017) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (x_m * 0.5) * (y_m / z_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 0.017d0) then
        tmp = y_m / (x_m * z_m)
    else
        tmp = (x_m * 0.5d0) * (y_m / z_m)
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.017) {
		tmp = y_m / (x_m * z_m);
	} else {
		tmp = (x_m * 0.5) * (y_m / z_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.017:
		tmp = y_m / (x_m * z_m)
	else:
		tmp = (x_m * 0.5) * (y_m / z_m)
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.017)
		tmp = Float64(y_m / Float64(x_m * z_m));
	else
		tmp = Float64(Float64(x_m * 0.5) * Float64(y_m / z_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.017)
		tmp = y_m / (x_m * z_m);
	else
		tmp = (x_m * 0.5) * (y_m / z_m);
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 0.017], N[(y$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 0.5), $MachinePrecision] * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.017000000000000001

   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 59.6

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

   5. Applied rewrites 59.6%

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

   if 0.017000000000000001 < x

   1. Initial program 72.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 33.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   10. 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. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 34.4

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

   11. Applied rewrites 34.4%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 20.0

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

   13. Applied rewrites 20.0%

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

Alternative 24: 49.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \frac{y\_m}{x\_m \cdot z\_m}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (* x_s (* y_s (* z_s (/ y_m (* x_m z_m))))))

C

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

Fortran

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

Java

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

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	return x_s * (y_s * (z_s * (y_m / (x_m * z_m))))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	return Float64(x_s * Float64(y_s * Float64(z_s * Float64(y_m / Float64(x_m * z_m)))))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = x_s * (y_s * (z_s * (y_m / (x_m * z_m))));
end

Wolfram

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

Derivation

1. Initial program 81.8%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 46.0

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

5. Applied rewrites 46.0%

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

Developer Target 1: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y z) x) (cosh x))))
   (if (< y -4.618902267687042e-52)
     t_0
     (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y / z) / x) * cosh(x)
    if (y < (-4.618902267687042d-52)) then
        tmp = t_0
    else if (y < 1.038530535935153d-39) then
        tmp = ((cosh(x) * y) / x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * Math.cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((Math.cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / z) / x) * math.cosh(x)
	tmp = 0
	if y < -4.618902267687042e-52:
		tmp = t_0
	elif y < 1.038530535935153e-39:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) / x) * cosh(x))
	tmp = 0.0
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / z) / x) * cosh(x);
	tmp = 0.0;
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -4.618902267687042e-52], t$95$0, If[Less[y, 1.038530535935153e-39], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024216 
(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))