Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.1% → 99.8%

Time: 13.9s

Alternatives: 25

Speedup: 2.4×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.1% 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

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{y\_m}{x\_m} \cdot \cosh x\_m\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+209}:\\ \;\;\;\;\frac{t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \cosh x\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \end{array} \]

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 (* (/ y_m x_m) (cosh x_m))))
   (*
    x_s
    (* y_s (if (<= t_0 2e+209) (/ t_0 z) (/ (/ (* y_m (cosh x_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 t_0 = (y_m / x_m) * cosh(x_m);
	double tmp;
	if (t_0 <= 2e+209) {
		tmp = t_0 / z;
	} else {
		tmp = ((y_m * cosh(x_m)) / z) / 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 = (y_m / x_m) * cosh(x_m)
    if (t_0 <= 2d+209) then
        tmp = t_0 / z
    else
        tmp = ((y_m * cosh(x_m)) / z) / 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 = (y_m / x_m) * Math.cosh(x_m);
	double tmp;
	if (t_0 <= 2e+209) {
		tmp = t_0 / z;
	} else {
		tmp = ((y_m * Math.cosh(x_m)) / z) / 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 = (y_m / x_m) * math.cosh(x_m)
	tmp = 0
	if t_0 <= 2e+209:
		tmp = t_0 / z
	else:
		tmp = ((y_m * math.cosh(x_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)
	t_0 = Float64(Float64(y_m / x_m) * cosh(x_m))
	tmp = 0.0
	if (t_0 <= 2e+209)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(Float64(Float64(y_m * cosh(x_m)) / z) / 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 = (y_m / x_m) * cosh(x_m);
	tmp = 0.0;
	if (t_0 <= 2e+209)
		tmp = t_0 / z;
	else
		tmp = ((y_m * cosh(x_m)) / z) / 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[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$0, 2e+209], N[(t$95$0 / z), $MachinePrecision], N[(N[(N[(y$95$m * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.0000000000000001e209

   1. Initial program 96.4%

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

   if 2.0000000000000001e209 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+209}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \cosh x}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\frac{y\_m}{x\_m} \cdot \cosh x\_m}{z}\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+26}:\\ \;\;\;\;\frac{y\_m}{z \cdot x\_m} \cdot \cosh x\_m\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\cosh x\_m}{x\_m} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot 0.001388888888888889\right) \cdot x\_m\right) \cdot \frac{1}{z}\right) \cdot y\_m\\ \end{array}\right) \end{array} \end{array} \]

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 (/ (* (/ y_m x_m) (cosh x_m)) z)))
   (*
    x_s
    (*
     y_s
     (if (<= t_0 5e+26)
       (* (/ y_m (* z x_m)) (cosh x_m))
       (if (<= t_0 INFINITY)
         (* (/ (cosh x_m) x_m) (/ y_m z))
         (*
          (*
           (* (* (* (* (* x_m x_m) x_m) x_m) 0.001388888888888889) x_m)
           (/ 1.0 z))
          y_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 = ((y_m / x_m) * cosh(x_m)) / z;
	double tmp;
	if (t_0 <= 5e+26) {
		tmp = (y_m / (z * x_m)) * cosh(x_m);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (cosh(x_m) / x_m) * (y_m / z);
	} else {
		tmp = ((((((x_m * x_m) * x_m) * x_m) * 0.001388888888888889) * x_m) * (1.0 / z)) * y_m;
	}
	return x_s * (y_s * tmp);
}

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 = ((y_m / x_m) * Math.cosh(x_m)) / z;
	double tmp;
	if (t_0 <= 5e+26) {
		tmp = (y_m / (z * x_m)) * Math.cosh(x_m);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.cosh(x_m) / x_m) * (y_m / z);
	} else {
		tmp = ((((((x_m * x_m) * x_m) * x_m) * 0.001388888888888889) * x_m) * (1.0 / z)) * y_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 = ((y_m / x_m) * math.cosh(x_m)) / z
	tmp = 0
	if t_0 <= 5e+26:
		tmp = (y_m / (z * x_m)) * math.cosh(x_m)
	elif t_0 <= math.inf:
		tmp = (math.cosh(x_m) / x_m) * (y_m / z)
	else:
		tmp = ((((((x_m * x_m) * x_m) * x_m) * 0.001388888888888889) * x_m) * (1.0 / z)) * y_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(Float64(Float64(y_m / x_m) * cosh(x_m)) / z)
	tmp = 0.0
	if (t_0 <= 5e+26)
		tmp = Float64(Float64(y_m / Float64(z * x_m)) * cosh(x_m));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(cosh(x_m) / x_m) * Float64(y_m / z));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(x_m * x_m) * x_m) * x_m) * 0.001388888888888889) * x_m) * Float64(1.0 / z)) * y_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 = ((y_m / x_m) * cosh(x_m)) / z;
	tmp = 0.0;
	if (t_0 <= 5e+26)
		tmp = (y_m / (z * x_m)) * cosh(x_m);
	elseif (t_0 <= Inf)
		tmp = (cosh(x_m) / x_m) * (y_m / z);
	else
		tmp = ((((((x_m * x_m) * x_m) * x_m) * 0.001388888888888889) * x_m) * (1.0 / z)) * y_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[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$0, 5e+26], N[(N[(y$95$m / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.0000000000000001e26

   1. Initial program 96.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 88.6

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

   4. Applied rewrites 88.6%

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

   if 5.0000000000000001e26 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < +inf.0

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. div-inv N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 98.6

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

   4. Applied rewrites 98.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{720} \cdot \color{blue}{\frac{{x}^{5} \cdot y}{z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.6%

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

   10. Applied rewrites 95.9%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 5 \cdot 10^{+26}:\\ \;\;\;\;\frac{y}{z \cdot x} \cdot \cosh x\\ \mathbf{elif}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq \infty:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.001388888888888889\right) \cdot x\right) \cdot \frac{1}{z}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{y\_m}{x\_m} \cdot \cosh x\_m\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+209}:\\ \;\;\;\;\frac{t\_0}{z}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\cosh x\_m}{x\_m} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot 0.001388888888888889\right) \cdot x\_m\right) \cdot \frac{1}{z}\right) \cdot y\_m\\ \end{array}\right) \end{array} \end{array} \]

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 (* (/ y_m x_m) (cosh x_m))))
   (*
    x_s
    (*
     y_s
     (if (<= t_0 2e+209)
       (/ t_0 z)
       (if (<= t_0 INFINITY)
         (* (/ (cosh x_m) x_m) (/ y_m z))
         (*
          (*
           (* (* (* (* (* x_m x_m) x_m) x_m) 0.001388888888888889) x_m)
           (/ 1.0 z))
          y_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 = (y_m / x_m) * cosh(x_m);
	double tmp;
	if (t_0 <= 2e+209) {
		tmp = t_0 / z;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (cosh(x_m) / x_m) * (y_m / z);
	} else {
		tmp = ((((((x_m * x_m) * x_m) * x_m) * 0.001388888888888889) * x_m) * (1.0 / z)) * y_m;
	}
	return x_s * (y_s * tmp);
}

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 = (y_m / x_m) * Math.cosh(x_m);
	double tmp;
	if (t_0 <= 2e+209) {
		tmp = t_0 / z;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.cosh(x_m) / x_m) * (y_m / z);
	} else {
		tmp = ((((((x_m * x_m) * x_m) * x_m) * 0.001388888888888889) * x_m) * (1.0 / z)) * y_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 = (y_m / x_m) * math.cosh(x_m)
	tmp = 0
	if t_0 <= 2e+209:
		tmp = t_0 / z
	elif t_0 <= math.inf:
		tmp = (math.cosh(x_m) / x_m) * (y_m / z)
	else:
		tmp = ((((((x_m * x_m) * x_m) * x_m) * 0.001388888888888889) * x_m) * (1.0 / z)) * y_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(Float64(y_m / x_m) * cosh(x_m))
	tmp = 0.0
	if (t_0 <= 2e+209)
		tmp = Float64(t_0 / z);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(cosh(x_m) / x_m) * Float64(y_m / z));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(x_m * x_m) * x_m) * x_m) * 0.001388888888888889) * x_m) * Float64(1.0 / z)) * y_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 = (y_m / x_m) * cosh(x_m);
	tmp = 0.0;
	if (t_0 <= 2e+209)
		tmp = t_0 / z;
	elseif (t_0 <= Inf)
		tmp = (cosh(x_m) / x_m) * (y_m / z);
	else
		tmp = ((((((x_m * x_m) * x_m) * x_m) * 0.001388888888888889) * x_m) * (1.0 / z)) * y_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[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$0, 2e+209], N[(t$95$0 / z), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.0000000000000001e209

   1. Initial program 96.4%

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

   if 2.0000000000000001e209 < (*.f64 (cosh.f64 x) (/.f64 y x)) < +inf.0

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. div-inv N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

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

   1. Initial program 0.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. Applied rewrites 42.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{720} \cdot \color{blue}{\frac{{x}^{5} \cdot y}{z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.6%

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

   10. Applied rewrites 95.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+209}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \cosh x}{z}\\ \mathbf{elif}\;\frac{y}{x} \cdot \cosh x \leq \infty:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.001388888888888889\right) \cdot x\right) \cdot \frac{1}{z}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{y\_m}{x\_m} \cdot \cosh x\_m}{z} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{t\_0}{x\_m} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \end{array} \]

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
          (fma
           (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
           (* x_m x_m)
           0.5)
          (* x_m x_m)
          1.0)))
   (*
    x_s
    (*
     y_s
     (if (<= (/ (* (/ y_m x_m) (cosh x_m)) z) 5e-46)
       (/ (* (/ t_0 x_m) y_m) z)
       (/ (/ (* t_0 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 t_0 = fma(fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0);
	double tmp;
	if ((((y_m / x_m) * cosh(x_m)) / z) <= 5e-46) {
		tmp = ((t_0 / x_m) * y_m) / z;
	} else {
		tmp = ((t_0 * 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)
	t_0 = fma(fma(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0)
	tmp = 0.0
	if (Float64(Float64(Float64(y_m / x_m) * cosh(x_m)) / z) <= 5e-46)
		tmp = Float64(Float64(Float64(t_0 / x_m) * y_m) / z);
	else
		tmp = Float64(Float64(Float64(t_0 * 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_] := Block[{t$95$0 = N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e-46], N[(N[(N[(t$95$0 / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(t$95$0 * 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) < 4.99999999999999992e-46

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 91.8%

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

   if 4.99999999999999992e-46 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 74.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 92.9

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

   7. Applied rewrites 92.9%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 10^{+229}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right) \cdot x\_m\right) \cdot x\_m, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

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 (<= (* (/ y_m x_m) (cosh x_m)) 1e+229)
     (/
      (*
       (/
        (fma
         (fma
          (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
          (* x_m x_m)
          0.5)
         (* x_m x_m)
         1.0)
        x_m)
       y_m)
      z)
     (/
      (/
       (*
        (fma
         (*
          (* (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664) x_m)
          x_m)
         (* x_m x_m)
         1.0)
        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 (((y_m / x_m) * cosh(x_m)) <= 1e+229) {
		tmp = ((fma(fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y_m) / z;
	} else {
		tmp = ((fma(((fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664) * x_m) * x_m), (x_m * x_m), 1.0) * 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(y_m / x_m) * cosh(x_m)) <= 1e+229)
		tmp = Float64(Float64(Float64(fma(fma(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) * y_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664) * x_m) * x_m), Float64(x_m * x_m), 1.0) * 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[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 1e+229], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 9.9999999999999999e228

   1. Initial program 96.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.6%

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

   if 9.9999999999999999e228 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. Applied rewrites 66.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 66.8%

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

   10. Applied rewrites 94.4%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 10^{+229}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x\right) \cdot x, x \cdot x, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{y\_m}{x\_m} \cdot \cosh x\_m}{z} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{t\_0}{x\_m} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \end{array} \]

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 (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)))
   (*
    x_s
    (*
     y_s
     (if (<= (/ (* (/ y_m x_m) (cosh x_m)) z) 5e-46)
       (/ (* (/ t_0 x_m) y_m) z)
       (/ (/ (* t_0 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 t_0 = fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0);
	double tmp;
	if ((((y_m / x_m) * cosh(x_m)) / z) <= 5e-46) {
		tmp = ((t_0 / x_m) * y_m) / z;
	} else {
		tmp = ((t_0 * 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)
	t_0 = fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0)
	tmp = 0.0
	if (Float64(Float64(Float64(y_m / x_m) * cosh(x_m)) / z) <= 5e-46)
		tmp = Float64(Float64(Float64(t_0 / x_m) * y_m) / z);
	else
		tmp = Float64(Float64(Float64(t_0 * 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_] := Block[{t$95$0 = N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e-46], N[(N[(N[(t$95$0 / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(t$95$0 * 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) < 4.99999999999999992e-46

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 89.0%

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

   if 4.99999999999999992e-46 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 74.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.2

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

   7. Applied rewrites 91.2%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 10^{+229}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right) \cdot x\_m\right) \cdot x\_m, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

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 (<= (* (/ y_m x_m) (cosh x_m)) 1e+229)
     (/
      (*
       (/ (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0) x_m)
       y_m)
      z)
     (/
      (/
       (*
        (fma
         (*
          (* (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664) x_m)
          x_m)
         (* x_m x_m)
         1.0)
        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 (((y_m / x_m) * cosh(x_m)) <= 1e+229) {
		tmp = ((fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y_m) / z;
	} else {
		tmp = ((fma(((fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664) * x_m) * x_m), (x_m * x_m), 1.0) * 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(y_m / x_m) * cosh(x_m)) <= 1e+229)
		tmp = Float64(Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) * y_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664) * x_m) * x_m), Float64(x_m * x_m), 1.0) * 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[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 1e+229], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 9.9999999999999999e228

   1. Initial program 96.4%

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 89.3%

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

   if 9.9999999999999999e228 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. Applied rewrites 66.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 66.8%

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

   10. Applied rewrites 94.4%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 10^{+229}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x\right) \cdot x, x \cdot x, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 4 \cdot 10^{+245}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, 0.5, \frac{1}{x\_m}\right) \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z} \cdot y\_m}{x\_m}\\ \end{array}\right) \end{array} \]

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 (<= (* (/ y_m x_m) (cosh x_m)) 4e+245)
     (/ (* (fma x_m 0.5 (/ 1.0 x_m)) y_m) z)
     (/ (* (/ (fma (* x_m x_m) 0.5 1.0) z) y_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 (((y_m / x_m) * cosh(x_m)) <= 4e+245) {
		tmp = (fma(x_m, 0.5, (1.0 / x_m)) * y_m) / z;
	} else {
		tmp = ((fma((x_m * x_m), 0.5, 1.0) / z) * y_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 (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 4e+245)
		tmp = Float64(Float64(fma(x_m, 0.5, Float64(1.0 / x_m)) * y_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / z) * y_m) / 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[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 4e+245], N[(N[(N[(x$95$m * 0.5 + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 4.00000000000000018e245

   1. Initial program 96.5%

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      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. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-out N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 80.0%

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

   if 4.00000000000000018e245 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 66.9%

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 51.1

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

   5. Applied rewrites 51.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z \cdot x}}\right) \cdot y \]

   7. Applied rewrites 57.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 57.9

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

   9. Applied rewrites 57.9%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. associate-*l/ N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 83.7

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

   11. Applied rewrites 83.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 4 \cdot 10^{+245}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 0.5, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{z} \cdot y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, 0.5, \frac{1}{x\_m}\right) \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)}{z}}{x\_m} \cdot y\_m\\ \end{array}\right) \end{array} \]

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 (<= (* (/ y_m x_m) (cosh x_m)) 5e+291)
     (/ (* (fma x_m 0.5 (/ 1.0 x_m)) y_m) z)
     (* (/ (/ (fma 0.5 (* x_m x_m) 1.0) z) x_m) y_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 (((y_m / x_m) * cosh(x_m)) <= 5e+291) {
		tmp = (fma(x_m, 0.5, (1.0 / x_m)) * y_m) / z;
	} else {
		tmp = ((fma(0.5, (x_m * x_m), 1.0) / z) / x_m) * y_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(y_m / x_m) * cosh(x_m)) <= 5e+291)
		tmp = Float64(Float64(fma(x_m, 0.5, Float64(1.0 / x_m)) * y_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) / z) / x_m) * y_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[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 5e+291], N[(N[(N[(x$95$m * 0.5 + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.0000000000000001e291

   1. Initial program 96.5%

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      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. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-out N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 80.3%

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

   if 5.0000000000000001e291 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 66.1%

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 49.9

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

   5. Applied rewrites 49.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z \cdot x}}\right) \cdot y \]

   7. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 81.1

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

   9. Applied rewrites 81.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 0.5, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z}}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.1% accurate, 0.8× speedup

Math

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

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 (<= (* (/ y_m x_m) (cosh x_m)) INFINITY)
     (* (/ (fma 0.5 (* x_m x_m) 1.0) z) (/ y_m x_m))
     (* (/ (* (* x_m x_m) 0.5) (* z x_m)) y_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 (((y_m / x_m) * cosh(x_m)) <= ((double) INFINITY)) {
		tmp = (fma(0.5, (x_m * x_m), 1.0) / z) * (y_m / x_m);
	} else {
		tmp = (((x_m * x_m) * 0.5) / (z * x_m)) * y_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(y_m / x_m) * cosh(x_m)) <= Inf)
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) / z) * Float64(y_m / x_m));
	else
		tmp = Float64(Float64(Float64(Float64(x_m * x_m) * 0.5) / Float64(z * x_m)) * y_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[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.0

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

   5. Applied rewrites 79.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 80.6

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

   7. Applied rewrites 80.6%

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

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

   1. Initial program 0.0%

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 0.6

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

   5. Applied rewrites 0.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z \cdot x}}\right) \cdot y \]

   7. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 42.3

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

   9. Applied rewrites 42.3%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 42.3%

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

4. Final simplification 77.0%

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

Alternative 11: 72.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, 0.5, \frac{1}{x\_m}\right) \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z \cdot x\_m}\\ \end{array}\right) \end{array} \]

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 (<= (* (/ y_m x_m) (cosh x_m)) 5e+232)
     (/ (* (fma x_m 0.5 (/ 1.0 x_m)) y_m) z)
     (/ (* (fma 0.5 (* x_m x_m) 1.0) 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 (((y_m / x_m) * cosh(x_m)) <= 5e+232) {
		tmp = (fma(x_m, 0.5, (1.0 / x_m)) * y_m) / z;
	} else {
		tmp = (fma(0.5, (x_m * x_m), 1.0) * 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(y_m / x_m) * cosh(x_m)) <= 5e+232)
		tmp = Float64(Float64(fma(x_m, 0.5, Float64(1.0 / x_m)) * y_m) / z);
	else
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / Float64(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[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 5e+232], N[(N[(N[(x$95$m * 0.5 + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 4.99999999999999987e232

   1. Initial program 96.5%

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      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. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-out N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 79.8%

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

   if 4.99999999999999987e232 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 67.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 52.2

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

   5. Applied rewrites 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 0.5, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{y\_m}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z \cdot x\_m}\\ \end{array}\right) \end{array} \]

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 (<= (* (/ y_m x_m) (cosh x_m)) 2e+150)
     (* (/ 1.0 z) (/ y_m x_m))
     (/ (* (fma 0.5 (* x_m x_m) 1.0) 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 (((y_m / x_m) * cosh(x_m)) <= 2e+150) {
		tmp = (1.0 / z) * (y_m / x_m);
	} else {
		tmp = (fma(0.5, (x_m * x_m), 1.0) * 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(y_m / x_m) * cosh(x_m)) <= 2e+150)
		tmp = Float64(Float64(1.0 / z) * Float64(y_m / x_m));
	else
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / Float64(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[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+150], N[(N[(1.0 / z), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.99999999999999996e150

   1. Initial program 96.2%

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

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

      3. lower-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

   7. Applied rewrites 66.7%

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

   if 1.99999999999999996e150 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 71.3%

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 60.8

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

   7. Applied rewrites 60.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{y\_m}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)}{z \cdot x\_m} \cdot y\_m\\ \end{array}\right) \end{array} \]

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 (<= (* (/ y_m x_m) (cosh x_m)) 2e+131)
     (* (/ 1.0 z) (/ y_m x_m))
     (* (/ (fma 0.5 (* x_m x_m) 1.0) (* z x_m)) y_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 (((y_m / x_m) * cosh(x_m)) <= 2e+131) {
		tmp = (1.0 / z) * (y_m / x_m);
	} else {
		tmp = (fma(0.5, (x_m * x_m), 1.0) / (z * x_m)) * y_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(y_m / x_m) * cosh(x_m)) <= 2e+131)
		tmp = Float64(Float64(1.0 / z) * Float64(y_m / x_m));
	else
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) / Float64(z * x_m)) * y_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[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+131], N[(N[(1.0 / z), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.9999999999999998e131

   1. Initial program 96.2%

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

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

      3. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

   7. Applied rewrites 66.5%

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

   if 1.9999999999999998e131 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 71.6%

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z \cdot x}}\right) \cdot y \]

   7. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 63.2

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

   9. Applied rewrites 63.2%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z \cdot x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 10^{+229}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

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 (<= (* (/ y_m x_m) (cosh x_m)) 1e+229)
     (/ (/ y_m x_m) z)
     (/ (/ 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 (((y_m / x_m) * cosh(x_m)) <= 1e+229) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = (y_m / z) / 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 (((y_m / x_m) * cosh(x_m)) <= 1d+229) then
        tmp = (y_m / x_m) / z
    else
        tmp = (y_m / z) / 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 (((y_m / x_m) * Math.cosh(x_m)) <= 1e+229) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = (y_m / z) / 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 ((y_m / x_m) * math.cosh(x_m)) <= 1e+229:
		tmp = (y_m / x_m) / z
	else:
		tmp = (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(y_m / x_m) * cosh(x_m)) <= 1e+229)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(Float64(y_m / z) / 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 (((y_m / x_m) * cosh(x_m)) <= 1e+229)
		tmp = (y_m / x_m) / z;
	else
		tmp = (y_m / z) / 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[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 1e+229], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 9.9999999999999999e228

   1. Initial program 96.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   if 9.9999999999999999e228 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 32.7

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

   7. Applied rewrites 32.7%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 10^{+229}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot x\_m}\\ \end{array}\right) \end{array} \]

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 (<= (* (/ y_m x_m) (cosh x_m)) 5e+232)
     (/ (/ y_m x_m) z)
     (/ 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 (((y_m / x_m) * cosh(x_m)) <= 5e+232) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = y_m / (z * 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 (((y_m / x_m) * cosh(x_m)) <= 5d+232) then
        tmp = (y_m / x_m) / z
    else
        tmp = y_m / (z * 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 (((y_m / x_m) * Math.cosh(x_m)) <= 5e+232) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = y_m / (z * 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 ((y_m / x_m) * math.cosh(x_m)) <= 5e+232:
		tmp = (y_m / x_m) / z
	else:
		tmp = 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(y_m / x_m) * cosh(x_m)) <= 5e+232)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(y_m / Float64(z * 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 (((y_m / x_m) * cosh(x_m)) <= 5e+232)
		tmp = (y_m / x_m) / z;
	else
		tmp = y_m / (z * 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[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 5e+232], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 4.99999999999999987e232

   1. Initial program 96.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 68.4

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

   5. Applied rewrites 68.4%

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

   if 4.99999999999999987e232 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 67.7%

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

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

      3. lower-*.f64 25.3

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

   5. Applied rewrites 25.3%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{x\_m}{y\_m} \cdot z}\\ \mathbf{elif}\;x\_m \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{y\_m \cdot \cosh x\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.001388888888888889 \cdot y\_m\right) \cdot \left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right)}{z}\\ \end{array}\right) \end{array} \]

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 3.6e-195)
     (/ 1.0 (* (/ x_m y_m) z))
     (if (<= x_m 4.5e+61)
       (/ (* y_m (cosh x_m)) (* z x_m))
       (/
        (* (* 0.001388888888888889 y_m) (* (* (* (* x_m x_m) x_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 <= 3.6e-195) {
		tmp = 1.0 / ((x_m / y_m) * z);
	} else if (x_m <= 4.5e+61) {
		tmp = (y_m * cosh(x_m)) / (z * x_m);
	} else {
		tmp = ((0.001388888888888889 * y_m) * ((((x_m * x_m) * x_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 <= 3.6d-195) then
        tmp = 1.0d0 / ((x_m / y_m) * z)
    else if (x_m <= 4.5d+61) then
        tmp = (y_m * cosh(x_m)) / (z * x_m)
    else
        tmp = ((0.001388888888888889d0 * y_m) * ((((x_m * x_m) * x_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 <= 3.6e-195) {
		tmp = 1.0 / ((x_m / y_m) * z);
	} else if (x_m <= 4.5e+61) {
		tmp = (y_m * Math.cosh(x_m)) / (z * x_m);
	} else {
		tmp = ((0.001388888888888889 * y_m) * ((((x_m * x_m) * x_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 <= 3.6e-195:
		tmp = 1.0 / ((x_m / y_m) * z)
	elif x_m <= 4.5e+61:
		tmp = (y_m * math.cosh(x_m)) / (z * x_m)
	else:
		tmp = ((0.001388888888888889 * y_m) * ((((x_m * x_m) * x_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 <= 3.6e-195)
		tmp = Float64(1.0 / Float64(Float64(x_m / y_m) * z));
	elseif (x_m <= 4.5e+61)
		tmp = Float64(Float64(y_m * cosh(x_m)) / Float64(z * x_m));
	else
		tmp = Float64(Float64(Float64(0.001388888888888889 * y_m) * Float64(Float64(Float64(Float64(x_m * x_m) * x_m) * 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 <= 3.6e-195)
		tmp = 1.0 / ((x_m / y_m) * z);
	elseif (x_m <= 4.5e+61)
		tmp = (y_m * cosh(x_m)) / (z * x_m);
	else
		tmp = ((0.001388888888888889 * y_m) * ((((x_m * x_m) * x_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, 3.6e-195], N[(1.0 / N[(N[(x$95$m / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 4.5e+61], N[(N[(y$95$m * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.001388888888888889 * y$95$m), $MachinePrecision] * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 3.6e-195

   1. Initial program 88.5%

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

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

      3. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

   7. Applied rewrites 61.4%

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

   9. Applied rewrites 61.3%

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

   if 3.6e-195 < x < 4.5e61

   1. Initial program 94.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 94.5

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

   4. Applied rewrites 94.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 97.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 97.9

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

   6. Applied rewrites 97.9%

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

   if 4.5e61 < x

   1. Initial program 72.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{720} \cdot \color{blue}{\frac{{x}^{5} \cdot y}{z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.3%

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

   10. Applied rewrites 100.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot z}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{y \cdot \cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.001388888888888889 \cdot y\right) \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{x\_m}{y\_m} \cdot z}\\ \mathbf{elif}\;x\_m \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\cosh x\_m}{z \cdot x\_m} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.001388888888888889 \cdot y\_m\right) \cdot \left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right)}{z}\\ \end{array}\right) \end{array} \]

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 3.6e-195)
     (/ 1.0 (* (/ x_m y_m) z))
     (if (<= x_m 4.5e+61)
       (* (/ (cosh x_m) (* z x_m)) y_m)
       (/
        (* (* 0.001388888888888889 y_m) (* (* (* (* x_m x_m) x_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 <= 3.6e-195) {
		tmp = 1.0 / ((x_m / y_m) * z);
	} else if (x_m <= 4.5e+61) {
		tmp = (cosh(x_m) / (z * x_m)) * y_m;
	} else {
		tmp = ((0.001388888888888889 * y_m) * ((((x_m * x_m) * x_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 <= 3.6d-195) then
        tmp = 1.0d0 / ((x_m / y_m) * z)
    else if (x_m <= 4.5d+61) then
        tmp = (cosh(x_m) / (z * x_m)) * y_m
    else
        tmp = ((0.001388888888888889d0 * y_m) * ((((x_m * x_m) * x_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 <= 3.6e-195) {
		tmp = 1.0 / ((x_m / y_m) * z);
	} else if (x_m <= 4.5e+61) {
		tmp = (Math.cosh(x_m) / (z * x_m)) * y_m;
	} else {
		tmp = ((0.001388888888888889 * y_m) * ((((x_m * x_m) * x_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 <= 3.6e-195:
		tmp = 1.0 / ((x_m / y_m) * z)
	elif x_m <= 4.5e+61:
		tmp = (math.cosh(x_m) / (z * x_m)) * y_m
	else:
		tmp = ((0.001388888888888889 * y_m) * ((((x_m * x_m) * x_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 <= 3.6e-195)
		tmp = Float64(1.0 / Float64(Float64(x_m / y_m) * z));
	elseif (x_m <= 4.5e+61)
		tmp = Float64(Float64(cosh(x_m) / Float64(z * x_m)) * y_m);
	else
		tmp = Float64(Float64(Float64(0.001388888888888889 * y_m) * Float64(Float64(Float64(Float64(x_m * x_m) * x_m) * 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 <= 3.6e-195)
		tmp = 1.0 / ((x_m / y_m) * z);
	elseif (x_m <= 4.5e+61)
		tmp = (cosh(x_m) / (z * x_m)) * y_m;
	else
		tmp = ((0.001388888888888889 * y_m) * ((((x_m * x_m) * x_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, 3.6e-195], N[(1.0 / N[(N[(x$95$m / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 4.5e+61], N[(N[(N[Cosh[x$95$m], $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(0.001388888888888889 * y$95$m), $MachinePrecision] * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 3.6e-195

   1. Initial program 88.5%

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

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

      3. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

   7. Applied rewrites 61.4%

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

   9. Applied rewrites 61.3%

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

   if 3.6e-195 < x < 4.5e61

   1. Initial program 94.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 96.0

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

   4. Applied rewrites 96.0%

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

   if 4.5e61 < x

   1. Initial program 72.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{720} \cdot \color{blue}{\frac{{x}^{5} \cdot y}{z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.3%

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

   10. Applied rewrites 100.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot z}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\cosh x}{z \cdot x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.001388888888888889 \cdot y\right) \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 87.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{x\_m}{y\_m} \cdot z}\\ \mathbf{elif}\;x\_m \leq 1.1 \cdot 10^{+33}:\\ \;\;\;\;\frac{t\_0 \cdot y\_m}{z \cdot x\_m}\\ \mathbf{elif}\;x\_m \leq 5.4 \cdot 10^{+149}:\\ \;\;\;\;\left(\frac{0.001388888888888889}{z} \cdot y\_m\right) \cdot \left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{z}}{x\_m} \cdot y\_m\\ \end{array}\right) \end{array} \end{array} \]

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 0.5 (* x_m x_m) 1.0)))
   (*
    x_s
    (*
     y_s
     (if (<= x_m 3.6e-195)
       (/ 1.0 (* (/ x_m y_m) z))
       (if (<= x_m 1.1e+33)
         (/ (* t_0 y_m) (* z x_m))
         (if (<= x_m 5.4e+149)
           (*
            (* (/ 0.001388888888888889 z) y_m)
            (* (* (* (* x_m x_m) x_m) x_m) x_m))
           (* (/ (/ t_0 z) x_m) y_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(0.5, (x_m * x_m), 1.0);
	double tmp;
	if (x_m <= 3.6e-195) {
		tmp = 1.0 / ((x_m / y_m) * z);
	} else if (x_m <= 1.1e+33) {
		tmp = (t_0 * y_m) / (z * x_m);
	} else if (x_m <= 5.4e+149) {
		tmp = ((0.001388888888888889 / z) * y_m) * ((((x_m * x_m) * x_m) * x_m) * x_m);
	} else {
		tmp = ((t_0 / z) / x_m) * y_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(0.5, Float64(x_m * x_m), 1.0)
	tmp = 0.0
	if (x_m <= 3.6e-195)
		tmp = Float64(1.0 / Float64(Float64(x_m / y_m) * z));
	elseif (x_m <= 1.1e+33)
		tmp = Float64(Float64(t_0 * y_m) / Float64(z * x_m));
	elseif (x_m <= 5.4e+149)
		tmp = Float64(Float64(Float64(0.001388888888888889 / z) * y_m) * Float64(Float64(Float64(Float64(x_m * x_m) * x_m) * x_m) * x_m));
	else
		tmp = Float64(Float64(Float64(t_0 / z) / x_m) * y_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[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 3.6e-195], N[(1.0 / N[(N[(x$95$m / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 1.1e+33], N[(N[(t$95$0 * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 5.4e+149], N[(N[(N[(0.001388888888888889 / z), $MachinePrecision] * y$95$m), $MachinePrecision] * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if x < 3.6e-195

   1. Initial program 88.5%

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

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

      3. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

   7. Applied rewrites 61.4%

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

   9. Applied rewrites 61.3%

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

   if 3.6e-195 < x < 1.09999999999999997e33

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.9

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

   5. Applied rewrites 77.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 79.8

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

   7. Applied rewrites 79.8%

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

   if 1.09999999999999997e33 < x < 5.4000000000000002e149

   1. Initial program 82.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{720} \cdot \color{blue}{\frac{{x}^{5} \cdot y}{z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.0%

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

   10. Applied rewrites 83.0%

       \[\leadsto \left(y \cdot \frac{0.001388888888888889}{z}\right) \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \]

   if 5.4000000000000002e149 < x

   1. Initial program 67.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z \cdot x}}\right) \cdot y \]

   7. Applied rewrites 45.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.8

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

   9. Applied rewrites 96.8%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot z}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+149}:\\ \;\;\;\;\left(\frac{0.001388888888888889}{z} \cdot y\right) \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z}}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 90.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{x\_m}{y\_m} \cdot z}\\ \mathbf{elif}\;x\_m \leq 6.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot \frac{y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.001388888888888889 \cdot y\_m\right) \cdot \left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right)}{z}\\ \end{array}\right) \end{array} \]

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 3.6e-195)
     (/ 1.0 (* (/ x_m y_m) z))
     (if (<= x_m 6.4)
       (*
        (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)
        (/ y_m (* z x_m)))
       (/
        (* (* 0.001388888888888889 y_m) (* (* (* (* x_m x_m) x_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 <= 3.6e-195) {
		tmp = 1.0 / ((x_m / y_m) * z);
	} else if (x_m <= 6.4) {
		tmp = fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) * (y_m / (z * x_m));
	} else {
		tmp = ((0.001388888888888889 * y_m) * ((((x_m * x_m) * x_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 <= 3.6e-195)
		tmp = Float64(1.0 / Float64(Float64(x_m / y_m) * z));
	elseif (x_m <= 6.4)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * Float64(y_m / Float64(z * x_m)));
	else
		tmp = Float64(Float64(Float64(0.001388888888888889 * y_m) * Float64(Float64(Float64(Float64(x_m * x_m) * x_m) * 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, 3.6e-195], N[(1.0 / N[(N[(x$95$m / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 6.4], N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.001388888888888889 * y$95$m), $MachinePrecision] * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 3.6e-195

   1. Initial program 88.5%

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

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

      3. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

   7. Applied rewrites 61.4%

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

   9. Applied rewrites 61.3%

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

   if 3.6e-195 < x < 6.4000000000000004

   1. Initial program 94.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 95.7%

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

   if 6.4000000000000004 < x

   1. Initial program 78.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. Applied rewrites 52.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{720} \cdot \color{blue}{\frac{{x}^{5} \cdot y}{z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.4%

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

   10. Applied rewrites 84.9%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot z}\\ \mathbf{elif}\;x \leq 6.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.001388888888888889 \cdot y\right) \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 90.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5.2 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right) \cdot y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

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 (<= y_m 5.2e+66)
     (/
      (*
       (/ (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0) x_m)
       y_m)
      z)
     (/ (/ (* (fma (* x_m x_m) 0.5 1.0) 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 (y_m <= 5.2e+66) {
		tmp = ((fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y_m) / z;
	} else {
		tmp = ((fma((x_m * x_m), 0.5, 1.0) * 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 (y_m <= 5.2e+66)
		tmp = Float64(Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) * y_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) * 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[y$95$m, 5.2e+66], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.20000000000000024e66

   1. Initial program 85.3%

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 87.8%

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

   if 5.20000000000000024e66 < y

   1. Initial program 91.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z \cdot x}}\right) \cdot y \]

   7. Applied rewrites 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 78.7

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

   9. Applied rewrites 78.7%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/r* N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 93.9

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

   11. Applied rewrites 93.9%

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

Alternative 21: 90.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{x\_m}{y\_m} \cdot z}\\ \mathbf{elif}\;x\_m \leq 4.5:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right) \cdot \frac{y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.001388888888888889 \cdot y\_m\right) \cdot \left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right)}{z}\\ \end{array}\right) \end{array} \]

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 3.6e-195)
     (/ 1.0 (* (/ x_m y_m) z))
     (if (<= x_m 4.5)
       (* (fma (* x_m x_m) 0.5 1.0) (/ y_m (* z x_m)))
       (/
        (* (* 0.001388888888888889 y_m) (* (* (* (* x_m x_m) x_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 <= 3.6e-195) {
		tmp = 1.0 / ((x_m / y_m) * z);
	} else if (x_m <= 4.5) {
		tmp = fma((x_m * x_m), 0.5, 1.0) * (y_m / (z * x_m));
	} else {
		tmp = ((0.001388888888888889 * y_m) * ((((x_m * x_m) * x_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 <= 3.6e-195)
		tmp = Float64(1.0 / Float64(Float64(x_m / y_m) * z));
	elseif (x_m <= 4.5)
		tmp = Float64(fma(Float64(x_m * x_m), 0.5, 1.0) * Float64(y_m / Float64(z * x_m)));
	else
		tmp = Float64(Float64(Float64(0.001388888888888889 * y_m) * Float64(Float64(Float64(Float64(x_m * x_m) * x_m) * 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, 3.6e-195], N[(1.0 / N[(N[(x$95$m / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 4.5], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(y$95$m / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.001388888888888889 * y$95$m), $MachinePrecision] * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 3.6e-195

   1. Initial program 88.5%

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

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

      3. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

   7. Applied rewrites 61.4%

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

   9. Applied rewrites 61.3%

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

   if 3.6e-195 < x < 4.5

   1. Initial program 94.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-/l/ N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if 4.5 < x

   1. Initial program 78.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. Applied rewrites 52.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{720} \cdot \color{blue}{\frac{{x}^{5} \cdot y}{z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.4%

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

   10. Applied rewrites 84.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot z}\\ \mathbf{elif}\;x \leq 4.5:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.001388888888888889 \cdot y\right) \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 69.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{x\_m}{y\_m} \cdot z}\\ \mathbf{elif}\;x\_m \leq 5 \cdot 10^{+179}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x\_m \cdot x\_m\right) \cdot 0.5\right) \cdot \frac{y\_m}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

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 (<= x_m 3.6e-195)
     (/ 1.0 (* (/ x_m y_m) z))
     (if (<= x_m 5e+179)
       (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) (* z x_m))
       (/ (* (* (* x_m x_m) 0.5) (/ 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) {
	double tmp;
	if (x_m <= 3.6e-195) {
		tmp = 1.0 / ((x_m / y_m) * z);
	} else if (x_m <= 5e+179) {
		tmp = (fma(0.5, (x_m * x_m), 1.0) * y_m) / (z * x_m);
	} else {
		tmp = (((x_m * x_m) * 0.5) * (y_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 <= 3.6e-195)
		tmp = Float64(1.0 / Float64(Float64(x_m / y_m) * z));
	elseif (x_m <= 5e+179)
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / Float64(z * x_m));
	else
		tmp = Float64(Float64(Float64(Float64(x_m * x_m) * 0.5) * Float64(y_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, 3.6e-195], N[(1.0 / N[(N[(x$95$m / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 5e+179], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 3.6e-195

   1. Initial program 88.5%

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

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

      3. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

   7. Applied rewrites 61.4%

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

   9. Applied rewrites 61.3%

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

   if 3.6e-195 < x < 5e179

   1. Initial program 86.1%

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 58.6

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

   5. Applied rewrites 58.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 68.3

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

   7. Applied rewrites 68.3%

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

   if 5e179 < x

   1. Initial program 77.3%

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 77.3%

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

Alternative 23: 68.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{x\_m}{y\_m} \cdot z}\\ \mathbf{elif}\;x\_m \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right) \cdot \frac{y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x\_m \cdot x\_m\right) \cdot 0.5}{z \cdot x\_m} \cdot y\_m\\ \end{array}\right) \end{array} \]

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 (<= x_m 3.6e-195)
     (/ 1.0 (* (/ x_m y_m) z))
     (if (<= x_m 1e+14)
       (* (fma (* x_m x_m) 0.5 1.0) (/ y_m (* z x_m)))
       (* (/ (* (* x_m x_m) 0.5) (* z x_m)) y_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 <= 3.6e-195) {
		tmp = 1.0 / ((x_m / y_m) * z);
	} else if (x_m <= 1e+14) {
		tmp = fma((x_m * x_m), 0.5, 1.0) * (y_m / (z * x_m));
	} else {
		tmp = (((x_m * x_m) * 0.5) / (z * x_m)) * y_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 <= 3.6e-195)
		tmp = Float64(1.0 / Float64(Float64(x_m / y_m) * z));
	elseif (x_m <= 1e+14)
		tmp = Float64(fma(Float64(x_m * x_m), 0.5, 1.0) * Float64(y_m / Float64(z * x_m)));
	else
		tmp = Float64(Float64(Float64(Float64(x_m * x_m) * 0.5) / Float64(z * x_m)) * y_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[x$95$m, 3.6e-195], N[(1.0 / N[(N[(x$95$m / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 1e+14], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(y$95$m / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 3.6e-195

   1. Initial program 88.5%

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

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

      3. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

   7. Applied rewrites 61.4%

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

   9. Applied rewrites 61.3%

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

   if 3.6e-195 < x < 1e14

   1. Initial program 95.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-/l/ N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if 1e14 < x

   1. Initial program 75.9%

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.9

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

   5. Applied rewrites 47.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z \cdot x}}\right) \cdot y \]

   7. Applied rewrites 39.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 39.2

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

   9. Applied rewrites 39.2%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 39.2%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot z}\\ \mathbf{elif}\;x \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.5}{z \cdot x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 68.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{x\_m}{y\_m} \cdot z}\\ \mathbf{elif}\;x\_m \leq 1.4:\\ \;\;\;\;\frac{y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x\_m \cdot x\_m\right) \cdot 0.5}{z \cdot x\_m} \cdot y\_m\\ \end{array}\right) \end{array} \]

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 (<= x_m 3.6e-195)
     (/ 1.0 (* (/ x_m y_m) z))
     (if (<= x_m 1.4)
       (/ y_m (* z x_m))
       (* (/ (* (* x_m x_m) 0.5) (* z x_m)) y_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 <= 3.6e-195) {
		tmp = 1.0 / ((x_m / y_m) * z);
	} else if (x_m <= 1.4) {
		tmp = y_m / (z * x_m);
	} else {
		tmp = (((x_m * x_m) * 0.5) / (z * x_m)) * y_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 <= 3.6d-195) then
        tmp = 1.0d0 / ((x_m / y_m) * z)
    else if (x_m <= 1.4d0) then
        tmp = y_m / (z * x_m)
    else
        tmp = (((x_m * x_m) * 0.5d0) / (z * x_m)) * y_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 <= 3.6e-195) {
		tmp = 1.0 / ((x_m / y_m) * z);
	} else if (x_m <= 1.4) {
		tmp = y_m / (z * x_m);
	} else {
		tmp = (((x_m * x_m) * 0.5) / (z * x_m)) * y_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 <= 3.6e-195:
		tmp = 1.0 / ((x_m / y_m) * z)
	elif x_m <= 1.4:
		tmp = y_m / (z * x_m)
	else:
		tmp = (((x_m * x_m) * 0.5) / (z * x_m)) * y_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 <= 3.6e-195)
		tmp = Float64(1.0 / Float64(Float64(x_m / y_m) * z));
	elseif (x_m <= 1.4)
		tmp = Float64(y_m / Float64(z * x_m));
	else
		tmp = Float64(Float64(Float64(Float64(x_m * x_m) * 0.5) / Float64(z * x_m)) * y_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 <= 3.6e-195)
		tmp = 1.0 / ((x_m / y_m) * z);
	elseif (x_m <= 1.4)
		tmp = y_m / (z * x_m);
	else
		tmp = (((x_m * x_m) * 0.5) / (z * x_m)) * y_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, 3.6e-195], N[(1.0 / N[(N[(x$95$m / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 1.4], N[(y$95$m / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 3.6e-195

   1. Initial program 88.5%

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

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

      3. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

   7. Applied rewrites 61.4%

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

   9. Applied rewrites 61.3%

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

   if 3.6e-195 < x < 1.3999999999999999

   1. Initial program 94.6%

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

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

      3. lower-*.f64 95.0

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

   5. Applied rewrites 95.0%

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

   if 1.3999999999999999 < x

   1. Initial program 78.1%

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 45.2

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

   5. Applied rewrites 45.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z \cdot x}}\right) \cdot y \]

   7. Applied rewrites 37.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 37.4

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

   9. Applied rewrites 37.4%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 37.4%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.5}{z \cdot x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 48.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \frac{y\_m}{z \cdot x\_m}\right) \end{array} \]

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 (* 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) {
	return x_s * (y_s * (y_m / (z * x_m)));
}

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 / (z * x_m)))
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 / (z * x_m)));
}

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 / (z * x_m)))

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(z * x_m))))
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 / (z * x_m)));
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[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 52.7

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

5. Applied rewrites 52.7%

   \[\leadsto \color{blue}{\frac{y}{z \cdot x}} \]
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 2024235 
(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))