Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.7% → 99.8%

Time: 16.0s

Alternatives: 30

Speedup: 2.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \cosh x\_m \cdot \frac{y\_m}{x\_m}\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\cosh x\_m}{x\_m}}{z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (cosh x_m) (/ y_m x_m))))
   (*
    x_s
    (* y_s (if (<= t_0 2e+306) (/ t_0 z) (* y_m (/ (/ (cosh x_m) x_m) z)))))))

C

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 x_m, double y_m, double z) {
	double t_0 = cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 2e+306) {
		tmp = t_0 / z;
	} else {
		tmp = y_m * ((cosh(x_m) / x_m) / z);
	}
	return x_s * (y_s * tmp);
}

Fortran

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, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x_m) * (y_m / x_m)
    if (t_0 <= 2d+306) then
        tmp = t_0 / z
    else
        tmp = y_m * ((cosh(x_m) / x_m) / z)
    end if
    code = x_s * (y_s * tmp)
end function

Java

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 x_m, double y_m, double z) {
	double t_0 = Math.cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 2e+306) {
		tmp = t_0 / z;
	} else {
		tmp = y_m * ((Math.cosh(x_m) / x_m) / z);
	}
	return x_s * (y_s * tmp);
}

Python

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, x_m, y_m, z):
	t_0 = math.cosh(x_m) * (y_m / x_m)
	tmp = 0
	if t_0 <= 2e+306:
		tmp = t_0 / z
	else:
		tmp = y_m * ((math.cosh(x_m) / x_m) / z)
	return x_s * (y_s * tmp)

Julia

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, x_m, y_m, z)
	t_0 = Float64(cosh(x_m) * Float64(y_m / x_m))
	tmp = 0.0
	if (t_0 <= 2e+306)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(y_m * Float64(Float64(cosh(x_m) / x_m) / z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

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, x_m, y_m, z)
	t_0 = cosh(x_m) * (y_m / x_m);
	tmp = 0.0;
	if (t_0 <= 2e+306)
		tmp = t_0 / z;
	else
		tmp = y_m * ((cosh(x_m) / x_m) / z);
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$0, 2e+306], N[(t$95$0 / z), $MachinePrecision], N[(y$95$m * N[(N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000003e306

   1. Initial program 97.0%

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

   if 2.00000000000000003e306 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 62.1%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

Alternative 2: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\cosh x\_m}{z}\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{y\_m}{x\_m} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot t\_0}{x\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (cosh x_m) z)))
   (*
    x_s
    (*
     y_s
     (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 5e-80)
       (* (/ y_m x_m) t_0)
       (/ (* y_m t_0) x_m))))))

C

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 x_m, double y_m, double z) {
	double t_0 = cosh(x_m) / z;
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 5e-80) {
		tmp = (y_m / x_m) * t_0;
	} else {
		tmp = (y_m * t_0) / x_m;
	}
	return x_s * (y_s * tmp);
}

Fortran

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, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x_m) / z
    if (((cosh(x_m) * (y_m / x_m)) / z) <= 5d-80) then
        tmp = (y_m / x_m) * t_0
    else
        tmp = (y_m * t_0) / x_m
    end if
    code = x_s * (y_s * tmp)
end function

Java

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 x_m, double y_m, double z) {
	double t_0 = Math.cosh(x_m) / z;
	double tmp;
	if (((Math.cosh(x_m) * (y_m / x_m)) / z) <= 5e-80) {
		tmp = (y_m / x_m) * t_0;
	} else {
		tmp = (y_m * t_0) / x_m;
	}
	return x_s * (y_s * tmp);
}

Python

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, x_m, y_m, z):
	t_0 = math.cosh(x_m) / z
	tmp = 0
	if ((math.cosh(x_m) * (y_m / x_m)) / z) <= 5e-80:
		tmp = (y_m / x_m) * t_0
	else:
		tmp = (y_m * t_0) / x_m
	return x_s * (y_s * tmp)

Julia

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, x_m, y_m, z)
	t_0 = Float64(cosh(x_m) / z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 5e-80)
		tmp = Float64(Float64(y_m / x_m) * t_0);
	else
		tmp = Float64(Float64(y_m * t_0) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

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, x_m, y_m, z)
	t_0 = cosh(x_m) / z;
	tmp = 0.0;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 5e-80)
		tmp = (y_m / x_m) * t_0;
	else
		tmp = (y_m * t_0) / x_m;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Cosh[x$95$m], $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e-80], N[(N[(y$95$m / x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(y$95$m * t$95$0), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5e-80

   1. Initial program 94.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 94.3

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

   4. Applied egg-rr 94.3%

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

   if 5e-80 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 74.4%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 10^{+279}:\\ \;\;\;\;\frac{y\_m}{x\_m} \cdot \frac{\cosh x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\cosh x\_m}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 1e+279)
     (* (/ y_m x_m) (/ (cosh x_m) z))
     (* y_m (/ (/ (cosh x_m) x_m) z))))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 1e+279) {
		tmp = (y_m / x_m) * (cosh(x_m) / z);
	} else {
		tmp = y_m * ((cosh(x_m) / x_m) / z);
	}
	return x_s * (y_s * tmp);
}

Fortran

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, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((cosh(x_m) * (y_m / x_m)) / z) <= 1d+279) then
        tmp = (y_m / x_m) * (cosh(x_m) / z)
    else
        tmp = y_m * ((cosh(x_m) / x_m) / z)
    end if
    code = x_s * (y_s * tmp)
end function

Java

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 x_m, double y_m, double z) {
	double tmp;
	if (((Math.cosh(x_m) * (y_m / x_m)) / z) <= 1e+279) {
		tmp = (y_m / x_m) * (Math.cosh(x_m) / z);
	} else {
		tmp = y_m * ((Math.cosh(x_m) / x_m) / z);
	}
	return x_s * (y_s * tmp);
}

Python

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, x_m, y_m, z):
	tmp = 0
	if ((math.cosh(x_m) * (y_m / x_m)) / z) <= 1e+279:
		tmp = (y_m / x_m) * (math.cosh(x_m) / z)
	else:
		tmp = y_m * ((math.cosh(x_m) / x_m) / z)
	return x_s * (y_s * tmp)

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 1e+279)
		tmp = Float64(Float64(y_m / x_m) * Float64(cosh(x_m) / z));
	else
		tmp = Float64(y_m * Float64(Float64(cosh(x_m) / x_m) / z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

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, x_m, y_m, z)
	tmp = 0.0;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 1e+279)
		tmp = (y_m / x_m) * (cosh(x_m) / z);
	else
		tmp = y_m * ((cosh(x_m) / x_m) / z);
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+279], N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(N[Cosh[x$95$m], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.00000000000000006e279

   1. Initial program 95.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 95.2

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

   4. Applied egg-rr 95.2%

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

   if 1.00000000000000006e279 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 65.8%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 10^{+279}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot \left(x\_m \cdot y\_m\right), \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), y\_m\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\cosh x\_m}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (cosh x_m) (/ y_m x_m)) 2e+306)
     (/
      (/
       (fma
        (* x_m (* x_m y_m))
        (fma
         (* x_m x_m)
         (fma x_m (* x_m 0.001388888888888889) 0.041666666666666664)
         0.5)
        y_m)
       x_m)
      z)
     (* y_m (/ (/ (cosh x_m) x_m) z))))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 2e+306) {
		tmp = (fma((x_m * (x_m * y_m)), fma((x_m * x_m), fma(x_m, (x_m * 0.001388888888888889), 0.041666666666666664), 0.5), y_m) / x_m) / z;
	} else {
		tmp = y_m * ((cosh(x_m) / x_m) / z);
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 2e+306)
		tmp = Float64(Float64(fma(Float64(x_m * Float64(x_m * y_m)), fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.001388888888888889), 0.041666666666666664), 0.5), y_m) / x_m) / z);
	else
		tmp = Float64(y_m * Float64(Float64(cosh(x_m) / x_m) / z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 2e+306], N[(N[(N[(N[(x$95$m * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000003e306

   1. Initial program 97.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), y\right)}{x \cdot z}} \]
   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 egg-rr 91.6%

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

   if 2.00000000000000003e306 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 62.1%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

Alternative 5: 91.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 5 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot \left(x\_m \cdot y\_m\right), \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), y\_m\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x\_m}{y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}}}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 5e+135)
     (/
      (/
       (fma
        (* x_m (* x_m y_m))
        (fma
         (* x_m x_m)
         (fma x_m (* x_m 0.001388888888888889) 0.041666666666666664)
         0.5)
        y_m)
       x_m)
      z)
     (/
      1.0
      (/
       x_m
       (*
        y_m
        (/
         (fma
          (* x_m x_m)
          (fma
           x_m
           (* x_m (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664))
           0.5)
          1.0)
         z))))))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 5e+135) {
		tmp = (fma((x_m * (x_m * y_m)), fma((x_m * x_m), fma(x_m, (x_m * 0.001388888888888889), 0.041666666666666664), 0.5), y_m) / x_m) / z;
	} else {
		tmp = 1.0 / (x_m / (y_m * (fma((x_m * x_m), fma(x_m, (x_m * fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0) / z)));
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 5e+135)
		tmp = Float64(Float64(fma(Float64(x_m * Float64(x_m * y_m)), fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.001388888888888889), 0.041666666666666664), 0.5), y_m) / x_m) / z);
	else
		tmp = Float64(1.0 / Float64(x_m / 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) / z))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e+135], N[(N[(N[(N[(x$95$m * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(1.0 / N[(x$95$m / 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] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.00000000000000029e135

   1. Initial program 95.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 82.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), y\right)}{x \cdot z}} \]
   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. 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 egg-rr 90.7%

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

   if 5.00000000000000029e135 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 68.6%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 95.0

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

   4. Applied egg-rr 95.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Simplified 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   9. Applied egg-rr 87.9%

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

   11. Applied egg-rr 92.8%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 5 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot y\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), y\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, 0.041666666666666664\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 10^{+279}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot \left(x\_m \cdot y\_m\right), \mathsf{fma}\left(x\_m \cdot x\_m, t\_0, 0.5\right), y\_m\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot t\_0, 0.5\right), 1\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (let* ((t_0 (fma x_m (* x_m 0.001388888888888889) 0.041666666666666664)))
   (*
    x_s
    (*
     y_s
     (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 1e+279)
       (/ (/ (fma (* x_m (* x_m y_m)) (fma (* x_m x_m) t_0 0.5) y_m) x_m) z)
       (*
        y_m
        (/ (/ (fma (* x_m x_m) (fma x_m (* x_m t_0) 0.5) 1.0) z) x_m)))))))

C

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 x_m, double y_m, double z) {
	double t_0 = fma(x_m, (x_m * 0.001388888888888889), 0.041666666666666664);
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 1e+279) {
		tmp = (fma((x_m * (x_m * y_m)), fma((x_m * x_m), t_0, 0.5), y_m) / x_m) / z;
	} else {
		tmp = y_m * ((fma((x_m * x_m), fma(x_m, (x_m * t_0), 0.5), 1.0) / z) / x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	t_0 = fma(x_m, Float64(x_m * 0.001388888888888889), 0.041666666666666664)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 1e+279)
		tmp = Float64(Float64(fma(Float64(x_m * Float64(x_m * y_m)), fma(Float64(x_m * x_m), t_0, 0.5), y_m) / x_m) / z);
	else
		tmp = Float64(y_m * Float64(Float64(fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * t_0), 0.5), 1.0) / z) / x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+279], N[(N[(N[(N[(x$95$m * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0 + 0.5), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * t$95$0), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.00000000000000006e279

   1. Initial program 95.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 80.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), y\right)}{x \cdot z}} \]
   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. 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 egg-rr 91.1%

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

   if 1.00000000000000006e279 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 65.8%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Simplified 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   9. Applied egg-rr 92.3%

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

Alternative 7: 94.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m} \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (cosh x_m) (/ y_m x_m)) 5e+123)
     (/
      (*
       (/ 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)
     (*
      y_m
      (/
       (/
        (fma
         (* x_m x_m)
         (fma
          x_m
          (* x_m (fma x_m (* x_m 0.001388888888888889) 0.041666666666666664))
          0.5)
         1.0)
        z)
       x_m))))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 5e+123) {
		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;
	} else {
		tmp = y_m * ((fma((x_m * x_m), fma(x_m, (x_m * fma(x_m, (x_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) / z) / x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 5e+123)
		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);
	else
		tmp = Float64(y_m * Float64(Float64(fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) / z) / x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 5e+123], 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), $MachinePrecision], N[(y$95$m * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 4.99999999999999974e123

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{y}{x}}{z} \]
   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 91.1

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

   5. Simplified 91.1%

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

   if 4.99999999999999974e123 < (*.f64 (cosh.f64 x) (/.f64 y x))

   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. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Simplified 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   9. Applied egg-rr 93.7%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right)\right), y\_m\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (cosh x_m) (/ y_m x_m)) 5e+293)
     (/
      (/
       (fma x_m (* x_m (* y_m (fma x_m (* x_m 0.041666666666666664) 0.5))) y_m)
       x_m)
      z)
     (*
      y_m
      (/
       (/
        (fma
         (* x_m x_m)
         (fma
          x_m
          (* x_m (fma x_m (* x_m 0.001388888888888889) 0.041666666666666664))
          0.5)
         1.0)
        z)
       x_m))))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 5e+293) {
		tmp = (fma(x_m, (x_m * (y_m * fma(x_m, (x_m * 0.041666666666666664), 0.5))), y_m) / x_m) / z;
	} else {
		tmp = y_m * ((fma((x_m * x_m), fma(x_m, (x_m * fma(x_m, (x_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) / z) / x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 5e+293)
		tmp = Float64(Float64(fma(x_m, Float64(x_m * Float64(y_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5))), y_m) / x_m) / z);
	else
		tmp = Float64(y_m * Float64(Float64(fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) / z) / x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 5e+293], N[(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] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.00000000000000033e293

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Simplified 89.4%

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

   if 5.00000000000000033e293 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 62.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Simplified 91.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   9. Applied egg-rr 92.5%

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

Alternative 9: 94.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right)\right), y\_m\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889\right), 0.5\right), 1\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (cosh x_m) (/ y_m x_m)) 5e+293)
     (/
      (/
       (fma x_m (* x_m (* y_m (fma x_m (* x_m 0.041666666666666664) 0.5))) y_m)
       x_m)
      z)
     (*
      y_m
      (/
       (/
        (fma
         (* x_m x_m)
         (fma x_m (* x_m (* (* x_m x_m) 0.001388888888888889)) 0.5)
         1.0)
        z)
       x_m))))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 5e+293) {
		tmp = (fma(x_m, (x_m * (y_m * fma(x_m, (x_m * 0.041666666666666664), 0.5))), y_m) / x_m) / z;
	} 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) / x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 5e+293)
		tmp = Float64(Float64(fma(x_m, Float64(x_m * Float64(y_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5))), y_m) / x_m) / z);
	else
		tmp = Float64(y_m * Float64(Float64(fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * Float64(Float64(x_m * x_m) * 0.001388888888888889)), 0.5), 1.0) / z) / x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 5e+293], N[(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] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * 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), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.00000000000000033e293

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Simplified 89.4%

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

   if 5.00000000000000033e293 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 62.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Simplified 91.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   9. Applied egg-rr 92.5%

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

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 92.5

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

   12. Simplified 92.5%

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

Alternative 10: 94.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right)\right), y\_m\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot 0.001388888888888889\right)\right), 0.5\right), 1\right)}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (cosh x_m) (/ y_m x_m)) 2e+306)
     (/
      (/
       (fma x_m (* x_m (* y_m (fma x_m (* x_m 0.041666666666666664) 0.5))) y_m)
       x_m)
      z)
     (*
      y_m
      (/
       (/
        (fma
         x_m
         (* x_m (fma x_m (* x_m (* x_m (* x_m 0.001388888888888889))) 0.5))
         1.0)
        x_m)
       z))))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 2e+306) {
		tmp = (fma(x_m, (x_m * (y_m * fma(x_m, (x_m * 0.041666666666666664), 0.5))), y_m) / x_m) / z;
	} else {
		tmp = y_m * ((fma(x_m, (x_m * fma(x_m, (x_m * (x_m * (x_m * 0.001388888888888889))), 0.5)), 1.0) / x_m) / z);
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 2e+306)
		tmp = Float64(Float64(fma(x_m, Float64(x_m * Float64(y_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5))), y_m) / x_m) / z);
	else
		tmp = Float64(y_m * Float64(Float64(fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * Float64(x_m * Float64(x_m * 0.001388888888888889))), 0.5)), 1.0) / x_m) / z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 2e+306], N[(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] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000003e306

   1. Initial program 97.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Simplified 89.5%

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

   if 2.00000000000000003e306 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 62.1%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Simplified 91.2%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 91.2

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

   10. Simplified 91.2%

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

Alternative 11: 91.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot t\_0\right), y\_m\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, t\_0, 1\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (let* ((t_0 (fma x_m (* x_m 0.041666666666666664) 0.5)))
   (*
    x_s
    (*
     y_s
     (if (<= (* (cosh x_m) (/ y_m x_m)) 5e+293)
       (/ (/ (fma x_m (* x_m (* y_m t_0)) y_m) x_m) z)
       (* y_m (/ (/ (fma (* x_m x_m) t_0 1.0) z) x_m)))))))

C

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 x_m, double y_m, double z) {
	double t_0 = fma(x_m, (x_m * 0.041666666666666664), 0.5);
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 5e+293) {
		tmp = (fma(x_m, (x_m * (y_m * t_0)), y_m) / x_m) / z;
	} else {
		tmp = y_m * ((fma((x_m * x_m), t_0, 1.0) / z) / x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	t_0 = fma(x_m, Float64(x_m * 0.041666666666666664), 0.5)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 5e+293)
		tmp = Float64(Float64(fma(x_m, Float64(x_m * Float64(y_m * t_0)), y_m) / x_m) / z);
	else
		tmp = Float64(y_m * Float64(Float64(fma(Float64(x_m * x_m), t_0, 1.0) / z) / x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 5e+293], N[(N[(N[(x$95$m * N[(x$95$m * N[(y$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.00000000000000033e293

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Simplified 89.4%

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

   if 5.00000000000000033e293 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 62.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Simplified 91.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   9. Applied egg-rr 92.5%

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

   10. Taylor expanded in x around 0

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

   12. Simplified 88.0%

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

Alternative 12: 91.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m} \cdot t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{t\_0}{z}}{x\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (let* ((t_0 (fma (* x_m x_m) (fma x_m (* x_m 0.041666666666666664) 0.5) 1.0)))
   (*
    x_s
    (*
     y_s
     (if (<= (* (cosh x_m) (/ y_m x_m)) 5e+123)
       (/ (* (/ y_m x_m) t_0) z)
       (* y_m (/ (/ t_0 z) x_m)))))))

C

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 x_m, double y_m, double z) {
	double t_0 = fma((x_m * x_m), fma(x_m, (x_m * 0.041666666666666664), 0.5), 1.0);
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 5e+123) {
		tmp = ((y_m / x_m) * t_0) / z;
	} else {
		tmp = y_m * ((t_0 / z) / x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	t_0 = fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.041666666666666664), 0.5), 1.0)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 5e+123)
		tmp = Float64(Float64(Float64(y_m / x_m) * t_0) / z);
	else
		tmp = Float64(y_m * Float64(Float64(t_0 / z) / x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = 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]}, N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 5e+123], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(t$95$0 / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 4.99999999999999974e123

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 89.4

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

   5. Simplified 89.4%

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

   if 4.99999999999999974e123 < (*.f64 (cosh.f64 x) (/.f64 y x))

   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. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Simplified 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   9. Applied egg-rr 93.7%

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

   10. Taylor expanded in x around 0

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

   12. Simplified 90.0%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\frac{y\_m}{x\_m} \cdot \frac{\mathsf{fma}\left(x\_m, x\_m \cdot t\_0, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, t\_0, 1\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (let* ((t_0 (fma x_m (* x_m 0.041666666666666664) 0.5)))
   (*
    x_s
    (*
     y_s
     (if (<= (* (cosh x_m) (/ y_m x_m)) 5e+123)
       (* (/ y_m x_m) (/ (fma x_m (* x_m t_0) 1.0) z))
       (* y_m (/ (/ (fma (* x_m x_m) t_0 1.0) z) x_m)))))))

C

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 x_m, double y_m, double z) {
	double t_0 = fma(x_m, (x_m * 0.041666666666666664), 0.5);
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 5e+123) {
		tmp = (y_m / x_m) * (fma(x_m, (x_m * t_0), 1.0) / z);
	} else {
		tmp = y_m * ((fma((x_m * x_m), t_0, 1.0) / z) / x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	t_0 = fma(x_m, Float64(x_m * 0.041666666666666664), 0.5)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 5e+123)
		tmp = Float64(Float64(y_m / x_m) * Float64(fma(x_m, Float64(x_m * t_0), 1.0) / z));
	else
		tmp = Float64(y_m * Float64(Float64(fma(Float64(x_m * x_m), t_0, 1.0) / z) / x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 5e+123], N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(N[(x$95$m * N[(x$95$m * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 4.99999999999999974e123

   1. Initial program 96.6%

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

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. div-inv N/A

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

      16. lower-/.f64 88.7

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

   4. Applied egg-rr 88.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 82.3

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

   7. Simplified 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   9. Applied egg-rr 91.3%

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

   if 4.99999999999999974e123 < (*.f64 (cosh.f64 x) (/.f64 y x))

   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. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Simplified 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   9. Applied egg-rr 93.7%

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

   10. Taylor expanded in x around 0

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

   12. Simplified 90.0%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot 0.5\right), y\_m\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (cosh x_m) (/ y_m x_m)) 5e+293)
     (/ (/ (fma x_m (* x_m (* y_m 0.5)) y_m) x_m) z)
     (*
      y_m
      (/
       (/ (fma (* x_m x_m) (fma x_m (* x_m 0.041666666666666664) 0.5) 1.0) z)
       x_m))))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 5e+293) {
		tmp = (fma(x_m, (x_m * (y_m * 0.5)), y_m) / x_m) / z;
	} else {
		tmp = y_m * ((fma((x_m * x_m), fma(x_m, (x_m * 0.041666666666666664), 0.5), 1.0) / z) / x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 5e+293)
		tmp = Float64(Float64(fma(x_m, Float64(x_m * Float64(y_m * 0.5)), y_m) / x_m) / z);
	else
		tmp = Float64(y_m * Float64(Float64(fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.041666666666666664), 0.5), 1.0) / z) / x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 5e+293], N[(N[(N[(x$95$m * N[(x$95$m * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(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] / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.00000000000000033e293

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   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 85.8

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

   5. Simplified 85.8%

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

   6. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 84.6

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

   8. Simplified 84.6%

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

   if 5.00000000000000033e293 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 62.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Simplified 91.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   9. Applied egg-rr 92.5%

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

   10. Taylor expanded in x around 0

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

   12. Simplified 88.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot 0.5\right), y\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 91.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot 0.5\right), y\_m\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (cosh x_m) (/ y_m x_m)) 2e+306)
     (/ (/ (fma x_m (* x_m (* y_m 0.5)) y_m) x_m) z)
     (*
      y_m
      (/
       (/ (fma (* x_m x_m) (fma x_m (* x_m 0.041666666666666664) 0.5) 1.0) x_m)
       z))))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 2e+306) {
		tmp = (fma(x_m, (x_m * (y_m * 0.5)), y_m) / x_m) / z;
	} else {
		tmp = y_m * ((fma((x_m * x_m), fma(x_m, (x_m * 0.041666666666666664), 0.5), 1.0) / x_m) / z);
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 2e+306)
		tmp = Float64(Float64(fma(x_m, Float64(x_m * Float64(y_m * 0.5)), y_m) / x_m) / z);
	else
		tmp = Float64(y_m * Float64(Float64(fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.041666666666666664), 0.5), 1.0) / x_m) / z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 2e+306], N[(N[(N[(x$95$m * N[(x$95$m * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(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] / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000003e306

   1. Initial program 97.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 86.0

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

   5. Simplified 86.0%

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

   6. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 84.8

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

   8. Simplified 84.8%

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

   if 2.00000000000000003e306 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 62.1%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 87.7

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

   7. Simplified 87.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot 0.5\right), y\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 80.9% accurate, 0.7× speedup

Math

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

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 2e-10)
     (/ (/ (fma x_m (* x_m (* y_m 0.5)) y_m) x_m) z)
     (/ (/ (fma y_m (* x_m (* x_m 0.5)) y_m) z) x_m)))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 2e-10) {
		tmp = (fma(x_m, (x_m * (y_m * 0.5)), y_m) / x_m) / z;
	} else {
		tmp = (fma(y_m, (x_m * (x_m * 0.5)), y_m) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 2e-10)
		tmp = Float64(Float64(fma(x_m, Float64(x_m * Float64(y_m * 0.5)), y_m) / x_m) / z);
	else
		tmp = Float64(Float64(fma(y_m, Float64(x_m * Float64(x_m * 0.5)), y_m) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e-10], N[(N[(N[(x$95$m * N[(x$95$m * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y$95$m * N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 2.00000000000000007e-10

   1. Initial program 94.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   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 83.3

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

   5. Simplified 83.3%

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

   6. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 80.6

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

   8. Simplified 80.6%

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

   if 2.00000000000000007e-10 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 72.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   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 53.2

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

   5. Simplified 53.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   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. div-inv N/A

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

      6. associate-*l* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   7. Applied egg-rr 78.6%

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

4. Final simplification 79.8%

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

Alternative 17: 80.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 10^{+279}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot 0.5\right), y\_m\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot 0.5, 1\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 1e+279)
     (/ (/ (fma x_m (* x_m (* y_m 0.5)) y_m) x_m) z)
     (* y_m (/ (/ (fma x_m (* x_m 0.5) 1.0) z) x_m))))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 1e+279) {
		tmp = (fma(x_m, (x_m * (y_m * 0.5)), y_m) / x_m) / z;
	} else {
		tmp = y_m * ((fma(x_m, (x_m * 0.5), 1.0) / z) / x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 1e+279)
		tmp = Float64(Float64(fma(x_m, Float64(x_m * Float64(y_m * 0.5)), y_m) / x_m) / z);
	else
		tmp = Float64(y_m * Float64(Float64(fma(x_m, Float64(x_m * 0.5), 1.0) / z) / x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+279], N[(N[(N[(x$95$m * N[(x$95$m * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.00000000000000006e279

   1. Initial program 95.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   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 84.8

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

   5. Simplified 84.8%

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

   6. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 82.5

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

   8. Simplified 82.5%

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

   if 1.00000000000000006e279 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 65.8%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Simplified 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   9. Applied egg-rr 92.3%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 75.5

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

   12. Simplified 75.5%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 10^{+279}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot 0.5\right), y\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(x, x \cdot 0.5, 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 82.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m} \cdot \mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot 0.5, 1\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (cosh x_m) (/ y_m x_m)) 5e+123)
     (/ (* (/ y_m x_m) (fma 0.5 (* x_m x_m) 1.0)) z)
     (* y_m (/ (/ (fma x_m (* x_m 0.5) 1.0) z) x_m))))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 5e+123) {
		tmp = ((y_m / x_m) * fma(0.5, (x_m * x_m), 1.0)) / z;
	} else {
		tmp = y_m * ((fma(x_m, (x_m * 0.5), 1.0) / z) / x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 5e+123)
		tmp = Float64(Float64(Float64(y_m / x_m) * fma(0.5, Float64(x_m * x_m), 1.0)) / z);
	else
		tmp = Float64(y_m * Float64(Float64(fma(x_m, Float64(x_m * 0.5), 1.0) / z) / x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 5e+123], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 4.99999999999999974e123

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   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 84.1

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

   5. Simplified 84.1%

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

   if 4.99999999999999974e123 < (*.f64 (cosh.f64 x) (/.f64 y x))

   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. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Simplified 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   9. Applied egg-rr 93.7%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 79.9

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

   12. Simplified 79.9%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(x, x \cdot 0.5, 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 82.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, x\_m \cdot 0.5, \frac{y\_m}{x\_m}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot 0.5, 1\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (cosh x_m) (/ y_m x_m)) 5e+123)
     (/ (fma y_m (* x_m 0.5) (/ y_m x_m)) z)
     (* y_m (/ (/ (fma x_m (* x_m 0.5) 1.0) z) x_m))))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 5e+123) {
		tmp = fma(y_m, (x_m * 0.5), (y_m / x_m)) / z;
	} else {
		tmp = y_m * ((fma(x_m, (x_m * 0.5), 1.0) / z) / x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 5e+123)
		tmp = Float64(fma(y_m, Float64(x_m * 0.5), Float64(y_m / x_m)) / z);
	else
		tmp = Float64(y_m * Float64(Float64(fma(x_m, Float64(x_m * 0.5), 1.0) / z) / x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 5e+123], N[(N[(y$95$m * N[(x$95$m * 0.5), $MachinePrecision] + N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 4.99999999999999974e123

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. associate-/l* N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 75.8

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

   5. Simplified 75.8%

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

   if 4.99999999999999974e123 < (*.f64 (cosh.f64 x) (/.f64 y x))

   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. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Simplified 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   9. Applied egg-rr 93.7%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 79.9

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

   12. Simplified 79.9%

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

Alternative 20: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\cosh x\_m}{x\_m}\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+15}:\\ \;\;\;\;t\_0 \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{t\_0}{z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (cosh x_m) x_m)))
   (* x_s (* y_s (if (<= z 3e+15) (* t_0 (/ y_m z)) (* y_m (/ t_0 z)))))))

C

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 x_m, double y_m, double z) {
	double t_0 = cosh(x_m) / x_m;
	double tmp;
	if (z <= 3e+15) {
		tmp = t_0 * (y_m / z);
	} else {
		tmp = y_m * (t_0 / z);
	}
	return x_s * (y_s * tmp);
}

Fortran

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, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x_m) / x_m
    if (z <= 3d+15) then
        tmp = t_0 * (y_m / z)
    else
        tmp = y_m * (t_0 / z)
    end if
    code = x_s * (y_s * tmp)
end function

Java

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 x_m, double y_m, double z) {
	double t_0 = Math.cosh(x_m) / x_m;
	double tmp;
	if (z <= 3e+15) {
		tmp = t_0 * (y_m / z);
	} else {
		tmp = y_m * (t_0 / z);
	}
	return x_s * (y_s * tmp);
}

Python

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, x_m, y_m, z):
	t_0 = math.cosh(x_m) / x_m
	tmp = 0
	if z <= 3e+15:
		tmp = t_0 * (y_m / z)
	else:
		tmp = y_m * (t_0 / z)
	return x_s * (y_s * tmp)

Julia

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, x_m, y_m, z)
	t_0 = Float64(cosh(x_m) / x_m)
	tmp = 0.0
	if (z <= 3e+15)
		tmp = Float64(t_0 * Float64(y_m / z));
	else
		tmp = Float64(y_m * Float64(t_0 / z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

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, x_m, y_m, z)
	t_0 = cosh(x_m) / x_m;
	tmp = 0.0;
	if (z <= 3e+15)
		tmp = t_0 * (y_m / z);
	else
		tmp = y_m * (t_0 / z);
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[z, 3e+15], N[(t$95$0 * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3e15

   1. Initial program 86.6%

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

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. div-inv N/A

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

      16. lower-/.f64 92.0

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

   4. Applied egg-rr 92.0%

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

   if 3e15 < z

   1. Initial program 81.6%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 99.7

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

   4. Applied egg-rr 99.7%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.6 \cdot 10^{+38}:\\ \;\;\;\;\frac{\cosh x\_m \cdot y\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x\_m}{y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}}}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 1.6e+38)
     (/ (* (cosh x_m) y_m) (* x_m z))
     (/
      1.0
      (/
       x_m
       (*
        y_m
        (/
         (fma
          (* x_m x_m)
          (fma
           x_m
           (* x_m (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664))
           0.5)
          1.0)
         z))))))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.6e+38) {
		tmp = (cosh(x_m) * y_m) / (x_m * z);
	} else {
		tmp = 1.0 / (x_m / (y_m * (fma((x_m * x_m), fma(x_m, (x_m * fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0) / z)));
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.6e+38)
		tmp = Float64(Float64(cosh(x_m) * y_m) / Float64(x_m * z));
	else
		tmp = Float64(1.0 / Float64(x_m / 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) / z))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.6e+38], N[(N[(N[Cosh[x$95$m], $MachinePrecision] * y$95$m), $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x$95$m / 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] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.59999999999999993e38

   1. Initial program 87.5%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 88.0

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

   4. Applied egg-rr 88.0%

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

   if 1.59999999999999993e38 < x

   1. Initial program 75.5%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Simplified 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   9. Applied egg-rr 98.0%

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

   11. Applied egg-rr 98.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+38}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.6 \cdot 10^{+38}:\\ \;\;\;\;y\_m \cdot \frac{\cosh x\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x\_m}{y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}}}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 1.6e+38)
     (* y_m (/ (cosh x_m) (* x_m z)))
     (/
      1.0
      (/
       x_m
       (*
        y_m
        (/
         (fma
          (* x_m x_m)
          (fma
           x_m
           (* x_m (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664))
           0.5)
          1.0)
         z))))))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.6e+38) {
		tmp = y_m * (cosh(x_m) / (x_m * z));
	} else {
		tmp = 1.0 / (x_m / (y_m * (fma((x_m * x_m), fma(x_m, (x_m * fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0) / z)));
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.6e+38)
		tmp = Float64(y_m * Float64(cosh(x_m) / Float64(x_m * z)));
	else
		tmp = Float64(1.0 / Float64(x_m / 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) / z))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.6e+38], N[(y$95$m * N[(N[Cosh[x$95$m], $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x$95$m / 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] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.59999999999999993e38

   1. Initial program 87.5%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. *-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 87.9

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

   4. Applied egg-rr 87.9%

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

   if 1.59999999999999993e38 < x

   1. Initial program 75.5%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Simplified 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   9. Applied egg-rr 98.0%

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

   11. Applied egg-rr 98.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+38}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 85.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.36 \cdot 10^{+122}:\\ \;\;\;\;y\_m \cdot \frac{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x\_m \cdot \left(y\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 1.36e+122)
     (*
      y_m
      (/
       (fma x_m (* x_m (fma x_m (* x_m 0.041666666666666664) 0.5)) 1.0)
       (* x_m z)))
     (/ (* 0.041666666666666664 (* x_m (* y_m (* x_m x_m)))) z)))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.36e+122) {
		tmp = y_m * (fma(x_m, (x_m * fma(x_m, (x_m * 0.041666666666666664), 0.5)), 1.0) / (x_m * z));
	} else {
		tmp = (0.041666666666666664 * (x_m * (y_m * (x_m * x_m)))) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.36e+122)
		tmp = Float64(y_m * Float64(fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5)), 1.0) / Float64(x_m * z)));
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(x_m * Float64(y_m * Float64(x_m * x_m)))) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.36e+122], N[(y$95$m * N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.041666666666666664 * N[(x$95$m * N[(y$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.36000000000000004e122

   1. Initial program 87.3%

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

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

      9. times-frac N/A

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

      10. div-inv N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(\frac{1}{x} \cdot \cosh x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{z}} \cdot \left(\frac{1}{x} \cdot \cosh x\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \]

      15. div-inv N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\cosh x}{x}} \]

      16. lower-/.f64 90.9

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\cosh x}{x}} \]

   4. Applied egg-rr 90.9%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\cosh x}{x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right)}{x} \]

      9. associate-*l* N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right)}{x} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \]

      11. lower-*.f64 81.7

          \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right)}{x} \]

   7. Simplified 81.7%

      \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right) + \frac{1}{2}\right) + 1}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}\right) + 1}{x} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)} + 1}{x} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}}{x} \]

      5. frac-times N/A

         \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{z \cdot x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{\color{blue}{x \cdot z}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{\color{blue}{x \cdot z}} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x \cdot z}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x \cdot z}} \]

      10. lower-/.f64 79.3

          \[\leadsto y \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z}} \]

   9. Applied egg-rr 79.3%

      \[\leadsto \color{blue}{y \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z}} \]

   if 1.36000000000000004e122 < x

   1. Initial program 73.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\cosh x}{x}} \]

      10. div-inv N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(\frac{1}{x} \cdot \cosh x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{z}} \cdot \left(\frac{1}{x} \cdot \cosh x\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \]

      15. div-inv N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\cosh x}{x}} \]

      16. lower-/.f64 81.1

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\cosh x}{x}} \]

   4. Applied egg-rr 81.1%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\cosh x}{x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right)}{x} \]

      9. associate-*l* N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right)}{x} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \]

      11. lower-*.f64 81.1

          \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right)}{x} \]

   7. Simplified 81.1%

      \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right) + \frac{1}{2}\right) + 1}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}\right) + 1}{x} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)} + 1}{x} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}}{x} \]

      5. frac-times N/A

         \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{z \cdot x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{\color{blue}{x \cdot z}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{\color{blue}{x \cdot z}} \]

      8. div-inv N/A

         \[\leadsto \color{blue}{\left(y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)\right) \cdot \frac{1}{x \cdot z}} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)\right) \cdot \color{blue}{\frac{1}{x \cdot z}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)\right) \cdot \frac{1}{x \cdot z}} \]

   9. Applied egg-rr 45.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\right) \cdot \frac{1}{x \cdot z}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}{z}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}{z}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]

       4. cube-mult N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot y\right)}{z} \]

       5. unpow2 N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot y\right)}{z} \]

       6. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot y\right)}\right)}{z} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

       10. lower-*.f64 97.4

           \[\leadsto \frac{0.041666666666666664 \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

   12. Simplified 97.4%

       \[\leadsto \color{blue}{\frac{0.041666666666666664 \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot y\right)\right)}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.36 \cdot 10^{+122}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x \cdot \left(y \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 84.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.25:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot 0.5\right), y\_m\right)}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x\_m \cdot \left(y\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 2.25)
     (/ (fma x_m (* x_m (* y_m 0.5)) y_m) (* x_m z))
     (/ (* 0.041666666666666664 (* x_m (* y_m (* x_m x_m)))) z)))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 2.25) {
		tmp = fma(x_m, (x_m * (y_m * 0.5)), y_m) / (x_m * z);
	} else {
		tmp = (0.041666666666666664 * (x_m * (y_m * (x_m * x_m)))) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 2.25)
		tmp = Float64(fma(x_m, Float64(x_m * Float64(y_m * 0.5)), y_m) / Float64(x_m * z));
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(x_m * Float64(y_m * Float64(x_m * x_m)))) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 2.25], N[(N[(x$95$m * N[(x$95$m * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(0.041666666666666664 * N[(x$95$m * N[(y$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   1. Initial program 87.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{y}{x}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \cosh x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x}} \cdot \cosh x}{z} \]

      5. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{x}\right)} \cdot \cosh x}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{x} \cdot \cosh x\right)}}{z} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      9. lower-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      10. *-commutative N/A

          \[\leadsto y \cdot \frac{\color{blue}{\cosh x \cdot \frac{1}{x}}}{z} \]

      11. div-inv N/A

          \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{x}}}{z} \]

      12. lower-/.f64 95.5

          \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{x}}}{z} \]

   4. Applied egg-rr 95.5%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{x}}{z}} \]

   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

         \[\leadsto \frac{\color{blue}{\frac{{x}^{2} \cdot y}{z} \cdot \frac{1}{2}} + \frac{y}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z}} + \frac{y}{z}}{x} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{y}{z}\right)} + \frac{y}{z}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{z}} + \frac{y}{z}}{x} \]

      5. *-lft-identity N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{z} + \color{blue}{1 \cdot \frac{y}{z}}}{x} \]

      6. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{z} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)}}{x} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{y}{z} \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{x} \]

      8. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{1 + \frac{1}{2} \cdot {x}^{2}}{x}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}{z \cdot x}} \]

      10. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}{\color{blue}{x \cdot z}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}{x \cdot z}} \]

   7. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \left(0.5 \cdot y\right), y\right)}{x \cdot z}} \]

   if 2.25 < x

   1. Initial program 78.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\cosh x}{x}} \]

      10. div-inv N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(\frac{1}{x} \cdot \cosh x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{z}} \cdot \left(\frac{1}{x} \cdot \cosh x\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \]

      15. div-inv N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\cosh x}{x}} \]

      16. lower-/.f64 80.4

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\cosh x}{x}} \]

   4. Applied egg-rr 80.4%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\cosh x}{x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right)}{x} \]

      9. associate-*l* N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right)}{x} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \]

      11. lower-*.f64 65.2

          \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right)}{x} \]

   7. Simplified 65.2%

      \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right) + \frac{1}{2}\right) + 1}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}\right) + 1}{x} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)} + 1}{x} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}}{x} \]

      5. frac-times N/A

         \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{z \cdot x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{\color{blue}{x \cdot z}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{\color{blue}{x \cdot z}} \]

      8. div-inv N/A

         \[\leadsto \color{blue}{\left(y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)\right) \cdot \frac{1}{x \cdot z}} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)\right) \cdot \color{blue}{\frac{1}{x \cdot z}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)\right) \cdot \frac{1}{x \cdot z}} \]

   9. Applied egg-rr 42.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\right) \cdot \frac{1}{x \cdot z}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}{z}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}{z}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]

       4. cube-mult N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot y\right)}{z} \]

       5. unpow2 N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot y\right)}{z} \]

       6. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot y\right)}\right)}{z} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

       10. lower-*.f64 70.9

           \[\leadsto \frac{0.041666666666666664 \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

   12. Simplified 70.9%

       \[\leadsto \color{blue}{\frac{0.041666666666666664 \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot y\right)\right)}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot 0.5\right), y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x \cdot \left(y \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 84.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.25:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m \cdot 0.5, 1\right) \cdot \frac{y\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x\_m \cdot \left(y\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 2.25)
     (* (fma x_m (* x_m 0.5) 1.0) (/ y_m (* x_m z)))
     (/ (* 0.041666666666666664 (* x_m (* y_m (* x_m x_m)))) z)))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 2.25) {
		tmp = fma(x_m, (x_m * 0.5), 1.0) * (y_m / (x_m * z));
	} else {
		tmp = (0.041666666666666664 * (x_m * (y_m * (x_m * x_m)))) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 2.25)
		tmp = Float64(fma(x_m, Float64(x_m * 0.5), 1.0) * Float64(y_m / Float64(x_m * z)));
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(x_m * Float64(y_m * Float64(x_m * x_m)))) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 2.25], N[(N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.041666666666666664 * N[(x$95$m * N[(y$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   1. Initial program 87.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   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 76.3

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 76.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   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 73.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \frac{y}{x \cdot z}} \]

      10. lift-fma.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot x\right) + 1\right)} \cdot \frac{y}{x \cdot z} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)} + 1\right) \cdot \frac{y}{x \cdot z} \]

      12. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot x\right) \cdot x} + 1\right) \cdot \frac{y}{x \cdot z} \]

      13. *-commutative N/A

          \[\leadsto \left(\color{blue}{x \cdot \left(\frac{1}{2} \cdot x\right)} + 1\right) \cdot \frac{y}{x \cdot z} \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} \cdot x, 1\right)} \cdot \frac{y}{x \cdot z} \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, 1\right) \cdot \frac{y}{x \cdot z} \]

      16. lower-*.f64 73.8

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot 0.5}, 1\right) \cdot \frac{y}{x \cdot z} \]

   7. Applied egg-rr 73.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.5, 1\right) \cdot \frac{y}{x \cdot z}} \]

   if 2.25 < x

   1. Initial program 78.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\cosh x}{x}} \]

      10. div-inv N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(\frac{1}{x} \cdot \cosh x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{z}} \cdot \left(\frac{1}{x} \cdot \cosh x\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \]

      15. div-inv N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\cosh x}{x}} \]

      16. lower-/.f64 80.4

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\cosh x}{x}} \]

   4. Applied egg-rr 80.4%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\cosh x}{x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right)}{x} \]

      9. associate-*l* N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right)}{x} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \]

      11. lower-*.f64 65.2

          \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right)}{x} \]

   7. Simplified 65.2%

      \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right) + \frac{1}{2}\right) + 1}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}\right) + 1}{x} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)} + 1}{x} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}}{x} \]

      5. frac-times N/A

         \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{z \cdot x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{\color{blue}{x \cdot z}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{\color{blue}{x \cdot z}} \]

      8. div-inv N/A

         \[\leadsto \color{blue}{\left(y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)\right) \cdot \frac{1}{x \cdot z}} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)\right) \cdot \color{blue}{\frac{1}{x \cdot z}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)\right) \cdot \frac{1}{x \cdot z}} \]

   9. Applied egg-rr 42.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\right) \cdot \frac{1}{x \cdot z}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}{z}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}{z}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]

       4. cube-mult N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot y\right)}{z} \]

       5. unpow2 N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot y\right)}{z} \]

       6. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot y\right)}\right)}{z} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

       10. lower-*.f64 70.9

           \[\leadsto \frac{0.041666666666666664 \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

   12. Simplified 70.9%

       \[\leadsto \color{blue}{\frac{0.041666666666666664 \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot y\right)\right)}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.5, 1\right) \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x \cdot \left(y \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 83.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x\_m \cdot \left(y\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 2.2)
     (/ y_m (* x_m z))
     (/ (* 0.041666666666666664 (* x_m (* y_m (* x_m x_m)))) z)))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = (0.041666666666666664 * (x_m * (y_m * (x_m * x_m)))) / z;
	}
	return x_s * (y_s * tmp);
}

Fortran

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, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 2.2d0) then
        tmp = y_m / (x_m * z)
    else
        tmp = (0.041666666666666664d0 * (x_m * (y_m * (x_m * x_m)))) / z
    end if
    code = x_s * (y_s * tmp)
end function

Java

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 x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = (0.041666666666666664 * (x_m * (y_m * (x_m * x_m)))) / z;
	}
	return x_s * (y_s * tmp);
}

Python

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, x_m, y_m, z):
	tmp = 0
	if x_m <= 2.2:
		tmp = y_m / (x_m * z)
	else:
		tmp = (0.041666666666666664 * (x_m * (y_m * (x_m * x_m)))) / z
	return x_s * (y_s * tmp)

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 2.2)
		tmp = Float64(y_m / Float64(x_m * z));
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(x_m * Float64(y_m * Float64(x_m * x_m)))) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

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, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 2.2)
		tmp = y_m / (x_m * z);
	else
		tmp = (0.041666666666666664 * (x_m * (y_m * (x_m * x_m)))) / z;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 2.2], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(0.041666666666666664 * N[(x$95$m * N[(y$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 87.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. lower-*.f64 66.1

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

   5. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 2.2000000000000002 < x

   1. Initial program 78.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\cosh x}{x}} \]

      10. div-inv N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(\frac{1}{x} \cdot \cosh x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{z}} \cdot \left(\frac{1}{x} \cdot \cosh x\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \]

      15. div-inv N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\cosh x}{x}} \]

      16. lower-/.f64 80.4

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\cosh x}{x}} \]

   4. Applied egg-rr 80.4%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\cosh x}{x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right)}{x} \]

      9. associate-*l* N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right)}{x} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \]

      11. lower-*.f64 65.2

          \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right)}{x} \]

   7. Simplified 65.2%

      \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right) + \frac{1}{2}\right) + 1}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}\right) + 1}{x} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)} + 1}{x} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}}{x} \]

      5. frac-times N/A

         \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{z \cdot x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{\color{blue}{x \cdot z}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{\color{blue}{x \cdot z}} \]

      8. div-inv N/A

         \[\leadsto \color{blue}{\left(y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)\right) \cdot \frac{1}{x \cdot z}} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)\right) \cdot \color{blue}{\frac{1}{x \cdot z}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)\right) \cdot \frac{1}{x \cdot z}} \]

   9. Applied egg-rr 42.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\right) \cdot \frac{1}{x \cdot z}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}{z}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}{z}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]

       4. cube-mult N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot y\right)}{z} \]

       5. unpow2 N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot y\right)}{z} \]

       6. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot y\right)}\right)}{z} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{24} \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

       10. lower-*.f64 70.9

           \[\leadsto \frac{0.041666666666666664 \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

   12. Simplified 70.9%

       \[\leadsto \color{blue}{\frac{0.041666666666666664 \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot y\right)\right)}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x \cdot \left(y \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 81.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \left(0.041666666666666664 \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 1.5)
     (/ y_m (* x_m z))
     (* (/ y_m z) (* 0.041666666666666664 (* x_m (* x_m x_m))))))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.5) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = (y_m / z) * (0.041666666666666664 * (x_m * (x_m * x_m)));
	}
	return x_s * (y_s * tmp);
}

Fortran

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, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 1.5d0) then
        tmp = y_m / (x_m * z)
    else
        tmp = (y_m / z) * (0.041666666666666664d0 * (x_m * (x_m * x_m)))
    end if
    code = x_s * (y_s * tmp)
end function

Java

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 x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.5) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = (y_m / z) * (0.041666666666666664 * (x_m * (x_m * x_m)));
	}
	return x_s * (y_s * tmp);
}

Python

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, x_m, y_m, z):
	tmp = 0
	if x_m <= 1.5:
		tmp = y_m / (x_m * z)
	else:
		tmp = (y_m / z) * (0.041666666666666664 * (x_m * (x_m * x_m)))
	return x_s * (y_s * tmp)

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.5)
		tmp = Float64(y_m / Float64(x_m * z));
	else
		tmp = Float64(Float64(y_m / z) * Float64(0.041666666666666664 * Float64(x_m * Float64(x_m * x_m))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

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, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 1.5)
		tmp = y_m / (x_m * z);
	else
		tmp = (y_m / z) * (0.041666666666666664 * (x_m * (x_m * x_m)));
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.5], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] * N[(0.041666666666666664 * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 87.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. lower-*.f64 66.1

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

   5. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 1.5 < x

   1. Initial program 78.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\cosh x}{x}} \]

      10. div-inv N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\frac{1}{x} \cdot \cosh x\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(\frac{1}{x} \cdot \cosh x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{z}} \cdot \left(\frac{1}{x} \cdot \cosh x\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\cosh x \cdot \frac{1}{x}\right)} \]

      15. div-inv N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\cosh x}{x}} \]

      16. lower-/.f64 80.4

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\cosh x}{x}} \]

   4. Applied egg-rr 80.4%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\cosh x}{x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right)}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right)}{x} \]

      9. associate-*l* N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right)}{x} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x} \]

      11. lower-*.f64 65.2

          \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right)}{x} \]

   7. Simplified 65.2%

      \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)} \]

      2. cube-mult N/A

         \[\leadsto \frac{y}{z} \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \frac{y}{z} \cdot \left(\frac{1}{24} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{y}{z} \cdot \left(\frac{1}{24} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      6. lower-*.f64 63.5

         \[\leadsto \frac{y}{z} \cdot \left(0.041666666666666664 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

   10. Simplified 63.5%

       \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 66.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.45:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(y\_m \cdot 0.5\right)}{z}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (* y_s (if (<= x_m 1.45) (/ y_m (* x_m z)) (/ (* x_m (* y_m 0.5)) z)))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.45) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = (x_m * (y_m * 0.5)) / z;
	}
	return x_s * (y_s * tmp);
}

Fortran

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, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 1.45d0) then
        tmp = y_m / (x_m * z)
    else
        tmp = (x_m * (y_m * 0.5d0)) / z
    end if
    code = x_s * (y_s * tmp)
end function

Java

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 x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.45) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = (x_m * (y_m * 0.5)) / z;
	}
	return x_s * (y_s * tmp);
}

Python

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, x_m, y_m, z):
	tmp = 0
	if x_m <= 1.45:
		tmp = y_m / (x_m * z)
	else:
		tmp = (x_m * (y_m * 0.5)) / z
	return x_s * (y_s * tmp)

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.45)
		tmp = Float64(y_m / Float64(x_m * z));
	else
		tmp = Float64(Float64(x_m * Float64(y_m * 0.5)) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

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, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 1.45)
		tmp = y_m / (x_m * z);
	else
		tmp = (x_m * (y_m * 0.5)) / z;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.45], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 87.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. lower-*.f64 66.1

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

   5. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 1.44999999999999996 < x

   1. Initial program 78.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   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 49.4

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 49.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \left(x \cdot y\right)}}{z} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \frac{1}{2}}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \frac{1}{2}\right)}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{2} \cdot y\right)}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{2} \cdot y\right)}}{z} \]

      5. lower-*.f64 30.9

         \[\leadsto \frac{x \cdot \color{blue}{\left(0.5 \cdot y\right)}}{z} \]

   8. Simplified 30.9%

      \[\leadsto \frac{\color{blue}{x \cdot \left(0.5 \cdot y\right)}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot 0.5\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 66.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.45:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m \cdot 0.5}{z}\\ \end{array}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (*
  x_s
  (* y_s (if (<= x_m 1.45) (/ y_m (* x_m z)) (* y_m (/ (* x_m 0.5) z))))))

C

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 x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.45) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = y_m * ((x_m * 0.5) / z);
	}
	return x_s * (y_s * tmp);
}

Fortran

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, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 1.45d0) then
        tmp = y_m / (x_m * z)
    else
        tmp = y_m * ((x_m * 0.5d0) / z)
    end if
    code = x_s * (y_s * tmp)
end function

Java

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 x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.45) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = y_m * ((x_m * 0.5) / z);
	}
	return x_s * (y_s * tmp);
}

Python

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, x_m, y_m, z):
	tmp = 0
	if x_m <= 1.45:
		tmp = y_m / (x_m * z)
	else:
		tmp = y_m * ((x_m * 0.5) / z)
	return x_s * (y_s * tmp)

Julia

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, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.45)
		tmp = Float64(y_m / Float64(x_m * z));
	else
		tmp = Float64(y_m * Float64(Float64(x_m * 0.5) / z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

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, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 1.45)
		tmp = y_m / (x_m * z);
	else
		tmp = y_m * ((x_m * 0.5) / z);
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.45], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(x$95$m * 0.5), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 87.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. lower-*.f64 66.1

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

   5. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 1.44999999999999996 < x

   1. Initial program 78.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   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 49.4

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 49.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \frac{y}{z} + \frac{y}{{x}^{2} \cdot z}\right)} \]
   7. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{y}{z}\right) \cdot x + \frac{y}{{x}^{2} \cdot z} \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{y}{z} \cdot \frac{1}{2}\right)} \cdot x + \frac{y}{{x}^{2} \cdot z} \cdot x \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(\frac{1}{2} \cdot x\right)} + \frac{y}{{x}^{2} \cdot z} \cdot x \]

      4. associate-*l/ N/A

         \[\leadsto \frac{y}{z} \cdot \left(\frac{1}{2} \cdot x\right) + \color{blue}{\frac{y \cdot x}{{x}^{2} \cdot z}} \]

      5. *-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \left(\frac{1}{2} \cdot x\right) + \frac{y \cdot x}{\color{blue}{z \cdot {x}^{2}}} \]

      6. times-frac N/A

         \[\leadsto \frac{y}{z} \cdot \left(\frac{1}{2} \cdot x\right) + \color{blue}{\frac{y}{z} \cdot \frac{x}{{x}^{2}}} \]

      7. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(\frac{1}{2} \cdot x + \frac{x}{{x}^{2}}\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} + \frac{x}{{x}^{2}}\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \frac{y}{z} \cdot \left(x \cdot \frac{1}{2} + \frac{\color{blue}{x \cdot 1}}{{x}^{2}}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \frac{y}{z} \cdot \left(x \cdot \frac{1}{2} + \color{blue}{x \cdot \frac{1}{{x}^{2}}}\right) \]

      11. distribute-lft-in N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{{x}^{2}}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{{x}^{2}}\right)\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{z}} \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{{x}^{2}}\right)\right) \]

      14. distribute-rgt-in N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{{x}^{2}} \cdot x\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{y}{z} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} + \frac{1}{{x}^{2}} \cdot x\right) \]

      16. unpow2 N/A

          \[\leadsto \frac{y}{z} \cdot \left(x \cdot \frac{1}{2} + \frac{1}{\color{blue}{x \cdot x}} \cdot x\right) \]

      17. associate-/r* N/A

          \[\leadsto \frac{y}{z} \cdot \left(x \cdot \frac{1}{2} + \color{blue}{\frac{\frac{1}{x}}{x}} \cdot x\right) \]

      18. associate-*l/ N/A

          \[\leadsto \frac{y}{z} \cdot \left(x \cdot \frac{1}{2} + \color{blue}{\frac{\frac{1}{x} \cdot x}{x}}\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \frac{y}{z} \cdot \left(x \cdot \frac{1}{2} + \frac{\color{blue}{1}}{x}\right) \]

      20. lower-fma.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \frac{1}{x}\right)} \]

      21. lower-/.f64 24.2

          \[\leadsto \frac{y}{z} \cdot \mathsf{fma}\left(x, 0.5, \color{blue}{\frac{1}{x}}\right) \]

   8. Simplified 24.2%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \mathsf{fma}\left(x, 0.5, \frac{1}{x}\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\frac{1}{2} \cdot x\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(x \cdot \frac{1}{2}\right)} \]

       2. lower-*.f64 24.2

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(x \cdot 0.5\right)} \]

   11. Simplified 24.2%

       \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(x \cdot 0.5\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(x \cdot \frac{1}{2}\right)} \]

       2. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{y \cdot \left(x \cdot \frac{1}{2}\right)}{z}} \]

       3. associate-/l* N/A

          \[\leadsto \color{blue}{y \cdot \frac{x \cdot \frac{1}{2}}{z}} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \frac{x \cdot \frac{1}{2}}{z}} \]

       5. lower-/.f64 25.9

          \[\leadsto y \cdot \color{blue}{\frac{x \cdot 0.5}{z}} \]

   13. Applied egg-rr 25.9%

       \[\leadsto \color{blue}{y \cdot \frac{x \cdot 0.5}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 30: 49.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \frac{y\_m}{x\_m \cdot z}\right) \end{array} \]

FPCore

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 x_m y_m z)
 :precision binary64
 (* x_s (* y_s (/ y_m (* x_m z)))))

C

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 x_m, double y_m, double z) {
	return x_s * (y_s * (y_m / (x_m * z)));
}

Fortran

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, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x_s * (y_s * (y_m / (x_m * z)))
end function

Java

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 x_m, double y_m, double z) {
	return x_s * (y_s * (y_m / (x_m * z)));
}

Python

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, x_m, y_m, z):
	return x_s * (y_s * (y_m / (x_m * z)))

Julia

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, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(y_m / Float64(x_m * z))))
end

MATLAB

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, x_m, y_m, z)
	tmp = x_s * (y_s * (y_m / (x_m * z)));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.2%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
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 52.6

      \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

5. Simplified 52.6%

   \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
6. Add Preprocessing

Developer Target 1: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y z) x) (cosh x))))
   (if (< y -4.618902267687042e-52)
     t_0
     (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y / z) / x) * cosh(x)
    if (y < (-4.618902267687042d-52)) then
        tmp = t_0
    else if (y < 1.038530535935153d-39) then
        tmp = ((cosh(x) * y) / x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * Math.cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((Math.cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / z) / x) * math.cosh(x)
	tmp = 0
	if y < -4.618902267687042e-52:
		tmp = t_0
	elif y < 1.038530535935153e-39:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) / x) * cosh(x))
	tmp = 0.0
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / z) / x) * cosh(x);
	tmp = 0.0;
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -4.618902267687042e-52], t$95$0, If[Less[y, 1.038530535935153e-39], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024208 
(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))