Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.7% → 99.8%

Time: 14.7s

Alternatives: 25

Speedup: 2.6×

• 63.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \cosh x\_m \cdot \frac{y\_m}{x\_m}\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\frac{t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\cosh x\_m}{x\_m}}{z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (cosh x_m) (/ y_m x_m))))
   (*
    x_s
    (* y_s (if (<= t_0 2e+230) (/ t_0 z) (* y_m (/ (/ (cosh x_m) x_m) z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 2e+230) {
		tmp = t_0 / z;
	} else {
		tmp = y_m * ((cosh(x_m) / x_m) / z);
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x_m) * (y_m / x_m)
    if (t_0 <= 2d+230) then
        tmp = t_0 / z
    else
        tmp = y_m * ((cosh(x_m) / x_m) / z)
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = Math.cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 2e+230) {
		tmp = t_0 / z;
	} else {
		tmp = y_m * ((Math.cosh(x_m) / x_m) / z);
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	t_0 = math.cosh(x_m) * (y_m / x_m)
	tmp = 0
	if t_0 <= 2e+230:
		tmp = t_0 / z
	else:
		tmp = y_m * ((math.cosh(x_m) / x_m) / z)
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(cosh(x_m) * Float64(y_m / x_m))
	tmp = 0.0
	if (t_0 <= 2e+230)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(y_m * Float64(Float64(cosh(x_m) / x_m) / z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = cosh(x_m) * (y_m / x_m);
	tmp = 0.0;
	if (t_0 <= 2e+230)
		tmp = t_0 / z;
	else
		tmp = y_m * ((cosh(x_m) / x_m) / z);
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$0, 2e+230], N[(t$95$0 / z), $MachinePrecision], N[(y$95$m * N[(N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.0000000000000002e230

   1. Initial program 94.5%

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

   if 2.0000000000000002e230 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 66.1%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

Alternative 2: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)\\ t_1 := \frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z}\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-69}:\\ \;\;\;\;t\_0 \cdot \frac{y\_m}{x\_m \cdot z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{t\_0}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot \left(y\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)\right)\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (fma (* x_m x_m) (fma x_m (* x_m 0.041666666666666664) 0.5) 1.0))
        (t_1 (/ (* (cosh x_m) (/ y_m x_m)) z)))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 2e-69)
       (* t_0 (/ y_m (* x_m z)))
       (if (<= t_1 INFINITY)
         (* (/ y_m z) (/ t_0 x_m))
         (/
          (/ (* x_m (* y_m (* x_m (* (* x_m x_m) 0.041666666666666664)))) z)
          x_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = fma((x_m * x_m), fma(x_m, (x_m * 0.041666666666666664), 0.5), 1.0);
	double t_1 = (cosh(x_m) * (y_m / x_m)) / z;
	double tmp;
	if (t_1 <= 2e-69) {
		tmp = t_0 * (y_m / (x_m * z));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (y_m / z) * (t_0 / x_m);
	} else {
		tmp = ((x_m * (y_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	t_0 = fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.041666666666666664), 0.5), 1.0)
	t_1 = Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z)
	tmp = 0.0
	if (t_1 <= 2e-69)
		tmp = Float64(t_0 * Float64(y_m / Float64(x_m * z)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(y_m / z) * Float64(t_0 / x_m));
	else
		tmp = Float64(Float64(Float64(x_m * Float64(y_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664)))) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, 2e-69], N[(t$95$0 * N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(y$95$m / z), $MachinePrecision] * N[(t$95$0 / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * N[(y$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.9999999999999999e-69

   1. Initial program 94.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-*.f64 85.5

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

   7. Applied rewrites 85.5%

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

   if 1.9999999999999999e-69 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < +inf.0

   1. Initial program 96.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 87.1

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

   7. Applied rewrites 87.1%

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

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 0.1

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

   5. Applied rewrites 0.1%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 0.1

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

   8. Applied rewrites 0.1%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. div-inv N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*r/ N/A

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

   10. Applied rewrites 93.7%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq \infty:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(y \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, y\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), y\_m\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\cosh x\_m}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (cosh x_m) (/ y_m x_m)) 2e+230)
     (/
      (/
       (fma
        (* x_m x_m)
        (*
         y_m
         (fma
          (* x_m x_m)
          (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
          0.5))
        y_m)
       x_m)
      z)
     (* y_m (/ (/ (cosh x_m) x_m) z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 2e+230) {
		tmp = (fma((x_m * x_m), (y_m * fma((x_m * x_m), fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5)), y_m) / x_m) / z;
	} else {
		tmp = y_m * ((cosh(x_m) / x_m) / z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 2e+230)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), Float64(y_m * fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5)), y_m) / x_m) / z);
	else
		tmp = Float64(y_m * Float64(Float64(cosh(x_m) / x_m) / z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 2e+230], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.0000000000000002e230

   1. Initial program 94.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 88.9%

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

   if 2.0000000000000002e230 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 66.1%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

Alternative 4: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{y\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 2e-69)
     (/
      (*
       y_m
       (fma
        (* x_m x_m)
        (fma
         x_m
         (* x_m (fma x_m (* x_m 0.001388888888888889) 0.041666666666666664))
         0.5)
        1.0))
      (* x_m z))
     (/
      (/
       (* y_m (fma (* x_m x_m) (fma x_m (* x_m 0.041666666666666664) 0.5) 1.0))
       z)
      x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 2e-69) {
		tmp = (y_m * fma((x_m * x_m), fma(x_m, (x_m * fma(x_m, (x_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0)) / (x_m * z);
	} else {
		tmp = ((y_m * fma((x_m * x_m), fma(x_m, (x_m * 0.041666666666666664), 0.5), 1.0)) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 2e-69)
		tmp = Float64(Float64(y_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0)) / Float64(x_m * z));
	else
		tmp = Float64(Float64(Float64(y_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.041666666666666664), 0.5), 1.0)) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e-69], N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $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) < 1.9999999999999999e-69

   1. Initial program 94.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 91.5

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

   5. Applied rewrites 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 91.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. remove-double-div N/A

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

      9. associate-/l* N/A

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

   9. Applied rewrites 86.4%

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

   if 1.9999999999999999e-69 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 74.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 65.3

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

   5. Applied rewrites 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 90.0%

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

Alternative 5: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;y\_m \cdot \frac{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 2e-69)
     (*
      y_m
      (/
       (fma
        x_m
        (*
         x_m
         (fma
          (* x_m x_m)
          (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
          0.5))
        1.0)
       (* x_m z)))
     (/
      (/
       (* y_m (fma (* x_m x_m) (fma x_m (* x_m 0.041666666666666664) 0.5) 1.0))
       z)
      x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 2e-69) {
		tmp = y_m * (fma(x_m, (x_m * fma((x_m * x_m), fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0) / (x_m * z));
	} else {
		tmp = ((y_m * fma((x_m * x_m), fma(x_m, (x_m * 0.041666666666666664), 0.5), 1.0)) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 2e-69)
		tmp = Float64(y_m * Float64(fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0) / Float64(x_m * z)));
	else
		tmp = Float64(Float64(Float64(y_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.041666666666666664), 0.5), 1.0)) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e-69], N[(y$95$m * N[(N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $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) < 1.9999999999999999e-69

   1. Initial program 94.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 91.5

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

   5. Applied rewrites 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 91.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

   9. Applied rewrites 91.4%

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

   10. Taylor expanded in z around 0

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 85.6%

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

   if 1.9999999999999999e-69 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 74.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 65.3

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

   5. Applied rewrites 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 90.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, y\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), y\_m\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (cosh x_m) (/ y_m x_m)) 2e+230)
     (/
      (/
       (fma
        (* x_m x_m)
        (*
         y_m
         (fma
          (* x_m x_m)
          (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
          0.5))
        y_m)
       x_m)
      z)
     (*
      y_m
      (/
       (/
        (fma
         (* x_m x_m)
         (fma
          x_m
          (* x_m (fma x_m (* x_m 0.001388888888888889) 0.041666666666666664))
          0.5)
         1.0)
        x_m)
       z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 2e+230) {
		tmp = (fma((x_m * x_m), (y_m * fma((x_m * x_m), fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5)), y_m) / x_m) / z;
	} else {
		tmp = y_m * ((fma((x_m * x_m), fma(x_m, (x_m * fma(x_m, (x_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) / x_m) / z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 2e+230)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), Float64(y_m * fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5)), y_m) / x_m) / z);
	else
		tmp = Float64(y_m * Float64(Float64(fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) / x_m) / z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 2e+230], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.0000000000000002e230

   1. Initial program 94.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 88.9%

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

   if 2.0000000000000002e230 < (*.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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

   9. Applied rewrites 94.0%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), y\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m} \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (cosh x_m) (/ y_m x_m)) 2e+230)
     (/
      (*
       (/ y_m x_m)
       (fma
        x_m
        (*
         x_m
         (fma
          (* x_m x_m)
          (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
          0.5))
        1.0))
      z)
     (*
      y_m
      (/
       (/
        (fma
         (* x_m x_m)
         (fma
          x_m
          (* x_m (fma x_m (* x_m 0.001388888888888889) 0.041666666666666664))
          0.5)
         1.0)
        x_m)
       z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 2e+230) {
		tmp = ((y_m / x_m) * fma(x_m, (x_m * fma((x_m * x_m), fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0)) / z;
	} else {
		tmp = y_m * ((fma((x_m * x_m), fma(x_m, (x_m * fma(x_m, (x_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) / x_m) / z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 2e+230)
		tmp = Float64(Float64(Float64(y_m / x_m) * fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0)) / z);
	else
		tmp = Float64(y_m * Float64(Float64(fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) / x_m) / z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 2e+230], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.0000000000000002e230

   1. Initial program 94.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if 2.0000000000000002e230 < (*.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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

   9. Applied rewrites 94.0%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, 0.041666666666666664\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\frac{y\_m}{x\_m} \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, t\_0, 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot t\_0, 0.5\right), 1\right)}{x\_m}}{z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (fma x_m (* x_m 0.001388888888888889) 0.041666666666666664)))
   (*
    x_s
    (*
     y_s
     (if (<= (* (cosh x_m) (/ y_m x_m)) 2e+230)
       (* (/ y_m x_m) (/ (fma (* x_m x_m) (fma (* x_m x_m) t_0 0.5) 1.0) z))
       (*
        y_m
        (/ (/ (fma (* x_m x_m) (fma x_m (* x_m t_0) 0.5) 1.0) x_m) z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = fma(x_m, (x_m * 0.001388888888888889), 0.041666666666666664);
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 2e+230) {
		tmp = (y_m / x_m) * (fma((x_m * x_m), fma((x_m * x_m), t_0, 0.5), 1.0) / z);
	} else {
		tmp = y_m * ((fma((x_m * x_m), fma(x_m, (x_m * t_0), 0.5), 1.0) / x_m) / z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	t_0 = fma(x_m, Float64(x_m * 0.001388888888888889), 0.041666666666666664)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 2e+230)
		tmp = Float64(Float64(y_m / x_m) * Float64(fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), t_0, 0.5), 1.0) / z));
	else
		tmp = Float64(y_m * Float64(Float64(fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * t_0), 0.5), 1.0) / 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_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 2e+230], N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * t$95$0), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.0000000000000002e230

   1. Initial program 94.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

   7. Applied rewrites 91.2%

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

   if 2.0000000000000002e230 < (*.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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

   9. Applied rewrites 94.0%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m, x\_m \cdot \left(y\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right)\right), y\_m\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (cosh x_m) (/ y_m x_m)) 2e+230)
     (/
      (/
       (fma x_m (* x_m (* y_m (fma x_m (* x_m 0.041666666666666664) 0.5))) y_m)
       x_m)
      z)
     (*
      y_m
      (/
       (/
        (fma
         (* x_m x_m)
         (fma
          x_m
          (* x_m (fma x_m (* x_m 0.001388888888888889) 0.041666666666666664))
          0.5)
         1.0)
        x_m)
       z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 2e+230) {
		tmp = (fma(x_m, (x_m * (y_m * fma(x_m, (x_m * 0.041666666666666664), 0.5))), y_m) / x_m) / z;
	} else {
		tmp = y_m * ((fma((x_m * x_m), fma(x_m, (x_m * fma(x_m, (x_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) / x_m) / z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 2e+230)
		tmp = Float64(Float64(fma(x_m, Float64(x_m * Float64(y_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5))), y_m) / x_m) / z);
	else
		tmp = Float64(y_m * Float64(Float64(fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) / x_m) / z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 2e+230], N[(N[(N[(x$95$m * N[(x$95$m * N[(y$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.0000000000000002e230

   1. Initial program 94.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 88.1%

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

   if 2.0000000000000002e230 < (*.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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

   9. Applied rewrites 94.0%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right), y\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m} \cdot t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot t\_0}{z}}{x\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (fma (* x_m x_m) (fma x_m (* x_m 0.041666666666666664) 0.5) 1.0)))
   (*
    x_s
    (*
     y_s
     (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 2e-69)
       (/ (* (/ y_m x_m) t_0) z)
       (/ (/ (* y_m t_0) z) x_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = fma((x_m * x_m), fma(x_m, (x_m * 0.041666666666666664), 0.5), 1.0);
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 2e-69) {
		tmp = ((y_m / x_m) * t_0) / z;
	} else {
		tmp = ((y_m * t_0) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	t_0 = fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.041666666666666664), 0.5), 1.0)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 2e-69)
		tmp = Float64(Float64(Float64(y_m / x_m) * t_0) / z);
	else
		tmp = Float64(Float64(Float64(y_m * t_0) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e-69], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y$95$m * t$95$0), $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) < 1.9999999999999999e-69

   1. Initial program 94.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if 1.9999999999999999e-69 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 74.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 65.3

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

   5. Applied rewrites 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 90.0%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 10^{+287}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m} \cdot t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{x\_m}{t\_0}}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (fma (* x_m x_m) (fma x_m (* x_m 0.041666666666666664) 0.5) 1.0)))
   (*
    x_s
    (*
     y_s
     (if (<= (* (cosh x_m) (/ y_m x_m)) 1e+287)
       (/ (* (/ y_m x_m) t_0) z)
       (/ y_m (* z (/ x_m t_0))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = fma((x_m * x_m), fma(x_m, (x_m * 0.041666666666666664), 0.5), 1.0);
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 1e+287) {
		tmp = ((y_m / x_m) * t_0) / z;
	} else {
		tmp = y_m / (z * (x_m / t_0));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	t_0 = fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.041666666666666664), 0.5), 1.0)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 1e+287)
		tmp = Float64(Float64(Float64(y_m / x_m) * t_0) / z);
	else
		tmp = Float64(y_m / Float64(z * Float64(x_m / t_0)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 1e+287], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(z * N[(x$95$m / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.0000000000000001e287

   1. Initial program 94.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 89.2

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

   5. Applied rewrites 89.2%

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

   if 1.0000000000000001e287 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 63.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 54.9

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

   5. Applied rewrites 54.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 81.5

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

   7. Applied rewrites 81.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 89.3

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

   9. Applied rewrites 89.3%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 10^{+287}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\cosh x\_m}{x\_m}\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+20}:\\ \;\;\;\;t\_0 \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{t\_0}{z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (cosh x_m) x_m)))
   (* x_s (* y_s (if (<= z 2e+20) (* t_0 (/ y_m z)) (* y_m (/ t_0 z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = cosh(x_m) / x_m;
	double tmp;
	if (z <= 2e+20) {
		tmp = t_0 * (y_m / z);
	} else {
		tmp = y_m * (t_0 / z);
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x_m) / x_m
    if (z <= 2d+20) then
        tmp = t_0 * (y_m / z)
    else
        tmp = y_m * (t_0 / z)
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = Math.cosh(x_m) / x_m;
	double tmp;
	if (z <= 2e+20) {
		tmp = t_0 * (y_m / z);
	} else {
		tmp = y_m * (t_0 / z);
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	t_0 = math.cosh(x_m) / x_m
	tmp = 0
	if z <= 2e+20:
		tmp = t_0 * (y_m / z)
	else:
		tmp = y_m * (t_0 / z)
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(cosh(x_m) / x_m)
	tmp = 0.0
	if (z <= 2e+20)
		tmp = Float64(t_0 * Float64(y_m / z));
	else
		tmp = Float64(y_m * Float64(t_0 / z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = cosh(x_m) / x_m;
	tmp = 0.0;
	if (z <= 2e+20)
		tmp = t_0 * (y_m / z);
	else
		tmp = y_m * (t_0 / z);
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Cosh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[z, 2e+20], N[(t$95$0 * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2e20

   1. Initial program 83.5%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. div-inv N/A

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

      16. lower-/.f64 92.5

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

   4. Applied rewrites 92.5%

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

   if 2e20 < z

   1. Initial program 84.6%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\cosh x\_m \cdot y\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

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 7.2e+51)
     (/ (* (cosh x_m) y_m) (* x_m z))
     (*
      y_m
      (/
       (/
        (fma
         (* x_m x_m)
         (fma
          x_m
          (* x_m (fma x_m (* x_m 0.001388888888888889) 0.041666666666666664))
          0.5)
         1.0)
        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 <= 7.2e+51) {
		tmp = (cosh(x_m) * y_m) / (x_m * z);
	} else {
		tmp = y_m * ((fma((x_m * x_m), fma(x_m, (x_m * fma(x_m, (x_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) / 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 <= 7.2e+51)
		tmp = Float64(Float64(cosh(x_m) * y_m) / Float64(x_m * z));
	else
		tmp = Float64(y_m * Float64(Float64(fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) / 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, 7.2e+51], N[(N[(N[Cosh[x$95$m], $MachinePrecision] * y$95$m), $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7.20000000000000022e51

   1. Initial program 84.7%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 94.9

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

   4. Applied rewrites 94.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 89.4

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

   6. Applied rewrites 89.4%

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

   if 7.20000000000000022e51 < x

   1. Initial program 80.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 80.6

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

   5. Applied rewrites 80.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

   9. Applied rewrites 100.0%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;y\_m \cdot \frac{\cosh x\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

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 7.2e+51)
     (* y_m (/ (cosh x_m) (* x_m z)))
     (*
      y_m
      (/
       (/
        (fma
         (* x_m x_m)
         (fma
          x_m
          (* x_m (fma x_m (* x_m 0.001388888888888889) 0.041666666666666664))
          0.5)
         1.0)
        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 <= 7.2e+51) {
		tmp = y_m * (cosh(x_m) / (x_m * z));
	} else {
		tmp = y_m * ((fma((x_m * x_m), fma(x_m, (x_m * fma(x_m, (x_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) / 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 <= 7.2e+51)
		tmp = Float64(y_m * Float64(cosh(x_m) / Float64(x_m * z)));
	else
		tmp = Float64(y_m * Float64(Float64(fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.001388888888888889), 0.041666666666666664)), 0.5), 1.0) / 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, 7.2e+51], N[(y$95$m * N[(N[Cosh[x$95$m], $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7.20000000000000022e51

   1. Initial program 84.7%

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

      1. lift-cosh.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 88.7

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

   4. Applied rewrites 88.7%

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

   if 7.20000000000000022e51 < x

   1. Initial program 80.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 80.6

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

   5. Applied rewrites 80.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

   9. Applied rewrites 100.0%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 86.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 10^{-25}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{t\_0}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{x\_m}{t\_0}}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (fma (* x_m x_m) (fma x_m (* x_m 0.041666666666666664) 0.5) 1.0)))
   (*
    x_s
    (*
     y_s
     (if (<= z 1e-25) (* (/ y_m z) (/ t_0 x_m)) (/ y_m (* z (/ x_m t_0))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = fma((x_m * x_m), fma(x_m, (x_m * 0.041666666666666664), 0.5), 1.0);
	double tmp;
	if (z <= 1e-25) {
		tmp = (y_m / z) * (t_0 / x_m);
	} else {
		tmp = y_m / (z * (x_m / t_0));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	t_0 = fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.041666666666666664), 0.5), 1.0)
	tmp = 0.0
	if (z <= 1e-25)
		tmp = Float64(Float64(y_m / z) * Float64(t_0 / x_m));
	else
		tmp = Float64(y_m / Float64(z * Float64(x_m / t_0)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[z, 1e-25], N[(N[(y$95$m / z), $MachinePrecision] * N[(t$95$0 / x$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(z * N[(x$95$m / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.00000000000000004e-25

   1. Initial program 83.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 86.6

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

   7. Applied rewrites 86.6%

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

   if 1.00000000000000004e-25 < z

   1. Initial program 83.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 78.2

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 78.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)} \cdot \frac{y}{x}}{z} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{x}}}{z} \]

      6. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{z \cdot x}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{\color{blue}{x \cdot z}} \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot \frac{y}{z}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot \frac{y}{z}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x}} \cdot \frac{y}{z} \]

      11. lower-/.f64 80.2

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot \color{blue}{\frac{y}{z}} \]

   7. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot \frac{y}{z}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right) + \frac{1}{2}\right) + 1}{x} \cdot \frac{y}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}\right) + 1}{x} \cdot \frac{y}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)} + 1}{x} \cdot \frac{y}{z} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}}{x} \cdot \frac{y}{z} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}}} \cdot \frac{y}{z} \]

      6. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot y}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)} \cdot z}} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{y}}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)} \cdot z} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)} \cdot z}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)} \cdot z}} \]

      10. lower-/.f64 94.7

          \[\leadsto \frac{y}{\color{blue}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}} \cdot z} \]

   9. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\frac{y}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{-25}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 87.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{y\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot \left(y\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664\right)\right)\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 1.5e+103)
     (/
      (* y_m (fma (* x_m x_m) (fma x_m (* x_m 0.041666666666666664) 0.5) 1.0))
      (* x_m z))
     (/
      (/ (* x_m (* y_m (* x_m (* (* x_m x_m) 0.041666666666666664)))) 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 (x_m <= 1.5e+103) {
		tmp = (y_m * fma((x_m * x_m), fma(x_m, (x_m * 0.041666666666666664), 0.5), 1.0)) / (x_m * z);
	} else {
		tmp = ((x_m * (y_m * (x_m * ((x_m * x_m) * 0.041666666666666664)))) / 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 (x_m <= 1.5e+103)
		tmp = Float64(Float64(y_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.041666666666666664), 0.5), 1.0)) / Float64(x_m * z));
	else
		tmp = Float64(Float64(Float64(x_m * Float64(y_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.041666666666666664)))) / 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[x$95$m, 1.5e+103], N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * N[(y$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e103

   1. Initial program 85.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 77.0

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 77.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)} \cdot \frac{y}{x}}{z} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \frac{\frac{y}{x}}{z}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \frac{\color{blue}{\frac{y}{x}}}{z} \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{\color{blue}{x \cdot z}} \]

      10. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{x \cdot z}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{x \cdot z}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\color{blue}{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}}{x \cdot z} \]

      13. lower-*.f64 80.5

          \[\leadsto \frac{\color{blue}{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}}{x \cdot z} \]

   7. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z}} \]

   if 1.5e103 < x

   1. Initial program 77.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 77.8

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 77.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right)} \cdot \frac{y}{x}}{z} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot \frac{y}{x}}{z} \]

      2. pow-sqr N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}\right) \cdot \frac{y}{x}}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right) \cdot \frac{y}{x}}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}\right) \cdot \frac{y}{x}}{z} \]

      10. unpow2 N/A

          \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24}\right)\right) \cdot \frac{y}{x}}{z} \]

      11. lower-*.f64 77.8

          \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.041666666666666664\right)\right) \cdot \frac{y}{x}}{z} \]

   8. Applied rewrites 77.8%

      \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \cdot \frac{y}{x}}{z} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right) \cdot \frac{y}{x}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)}\right) \cdot \frac{y}{x}}{z} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)}\right)\right) \cdot \frac{y}{x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)}\right)\right) \cdot \frac{y}{x}}{z} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)}\right) \cdot \frac{y}{x}}{z} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24}\right)\right) \cdot \frac{y}{x}}{z} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)}\right) \cdot \frac{y}{x}}{z} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)} \cdot \frac{y}{x}}{z} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right) \cdot \frac{y}{x}}}{z} \]

      11. div-inv N/A

          \[\leadsto \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right) \cdot \frac{y}{x}\right) \cdot \frac{1}{z}} \]

      12. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right) \cdot \frac{y}{x}\right)} \cdot \frac{1}{z} \]

      13. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right) \cdot \color{blue}{\frac{y}{x}}\right) \cdot \frac{1}{z} \]

      14. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right) \cdot y}{x}} \cdot \frac{1}{z} \]

   10. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\frac{\frac{x \cdot \left(\left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right) \cdot y\right)}{z}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(y \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 85.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 8 \cdot 10^{+120}:\\ \;\;\;\;\frac{y\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(0.041666666666666664 \cdot \left(y\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 8e+120)
     (/
      (* y_m (fma (* x_m x_m) (fma x_m (* x_m 0.041666666666666664) 0.5) 1.0))
      (* x_m z))
     (/ (* x_m (* 0.041666666666666664 (* y_m (* x_m x_m)))) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 8e+120) {
		tmp = (y_m * fma((x_m * x_m), fma(x_m, (x_m * 0.041666666666666664), 0.5), 1.0)) / (x_m * z);
	} else {
		tmp = (x_m * (0.041666666666666664 * (y_m * (x_m * x_m)))) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 8e+120)
		tmp = Float64(Float64(y_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.041666666666666664), 0.5), 1.0)) / Float64(x_m * z));
	else
		tmp = Float64(Float64(x_m * Float64(0.041666666666666664 * Float64(y_m * Float64(x_m * x_m)))) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 8e+120], N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(0.041666666666666664 * N[(y$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7.9999999999999998e120

   1. Initial program 84.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 76.7

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 76.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)} \cdot \frac{y}{x}}{z} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \frac{\frac{y}{x}}{z}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \frac{\color{blue}{\frac{y}{x}}}{z} \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{\color{blue}{x \cdot z}} \]

      10. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{x \cdot z}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{x \cdot z}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\color{blue}{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}}{x \cdot z} \]

      13. lower-*.f64 80.1

          \[\leadsto \frac{\color{blue}{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}}{x \cdot z} \]

   7. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z}} \]

   if 7.9999999999999998e120 < x

   1. Initial program 79.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 79.2

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 79.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot y\right) \cdot \frac{1}{24}}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(y \cdot \frac{1}{24}\right)}}{z} \]

      3. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(y \cdot \frac{1}{24}\right)}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(y \cdot \frac{1}{24}\right)}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot y\right)}}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} \cdot y\right)\right)}}{z} \]

      7. associate-*r* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left({x}^{2} \cdot \frac{1}{24}\right) \cdot y\right)}}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)} \cdot y\right)}{z} \]

      9. associate-*r* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \left(\frac{1}{24} \cdot \color{blue}{\left({x}^{2} \cdot y\right)}\right)}{z} \]

      13. unpow2 N/A

          \[\leadsto \frac{x \cdot \left(\frac{1}{24} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

      14. lower-*.f64 100.0

          \[\leadsto \frac{x \cdot \left(0.041666666666666664 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{x \cdot \left(0.041666666666666664 \cdot \left(\left(x \cdot x\right) \cdot y\right)\right)}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+120}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(0.041666666666666664 \cdot \left(y \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 83.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.6 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \frac{y\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(0.041666666666666664 \cdot \left(y\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 2.6e+120)
     (*
      (fma (* x_m x_m) (fma x_m (* x_m 0.041666666666666664) 0.5) 1.0)
      (/ y_m (* x_m z)))
     (/ (* x_m (* 0.041666666666666664 (* y_m (* x_m x_m)))) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 2.6e+120) {
		tmp = fma((x_m * x_m), fma(x_m, (x_m * 0.041666666666666664), 0.5), 1.0) * (y_m / (x_m * z));
	} else {
		tmp = (x_m * (0.041666666666666664 * (y_m * (x_m * x_m)))) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 2.6e+120)
		tmp = Float64(fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.041666666666666664), 0.5), 1.0) * Float64(y_m / Float64(x_m * z)));
	else
		tmp = Float64(Float64(x_m * Float64(0.041666666666666664 * Float64(y_m * Float64(x_m * x_m)))) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 2.6e+120], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(0.041666666666666664 * N[(y$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.5999999999999999e120

   1. Initial program 84.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 76.7

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 76.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)} \cdot \frac{y}{x}}{z} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \frac{\frac{y}{x}}{z}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \frac{\color{blue}{\frac{y}{x}}}{z} \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{\color{blue}{x \cdot z}} \]

      10. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      11. lower-*.f64 78.2

          \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \frac{y}{x \cdot z}} \]

   7. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \frac{y}{x \cdot z}} \]

   if 2.5999999999999999e120 < x

   1. Initial program 79.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 79.2

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 79.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot y\right) \cdot \frac{1}{24}}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(y \cdot \frac{1}{24}\right)}}{z} \]

      3. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(y \cdot \frac{1}{24}\right)}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(y \cdot \frac{1}{24}\right)}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot y\right)}}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} \cdot y\right)\right)}}{z} \]

      7. associate-*r* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left({x}^{2} \cdot \frac{1}{24}\right) \cdot y\right)}}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)} \cdot y\right)}{z} \]

      9. associate-*r* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \left(\frac{1}{24} \cdot \color{blue}{\left({x}^{2} \cdot y\right)}\right)}{z} \]

      13. unpow2 N/A

          \[\leadsto \frac{x \cdot \left(\frac{1}{24} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

      14. lower-*.f64 100.0

          \[\leadsto \frac{x \cdot \left(0.041666666666666664 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{x \cdot \left(0.041666666666666664 \cdot \left(\left(x \cdot x\right) \cdot y\right)\right)}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(0.041666666666666664 \cdot \left(y \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 83.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.7:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, \left(x\_m \cdot x\_m\right) \cdot 0.5, y\_m\right)}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(0.041666666666666664 \cdot \left(y\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 3.7)
     (/ (fma y_m (* (* x_m x_m) 0.5) y_m) (* x_m z))
     (/ (* x_m (* 0.041666666666666664 (* y_m (* x_m x_m)))) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 3.7) {
		tmp = fma(y_m, ((x_m * x_m) * 0.5), y_m) / (x_m * z);
	} else {
		tmp = (x_m * (0.041666666666666664 * (y_m * (x_m * x_m)))) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 3.7)
		tmp = Float64(fma(y_m, Float64(Float64(x_m * x_m) * 0.5), y_m) / Float64(x_m * z));
	else
		tmp = Float64(Float64(x_m * Float64(0.041666666666666664 * Float64(y_m * Float64(x_m * x_m)))) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 3.7], N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(0.041666666666666664 * N[(y$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.7000000000000002

   1. Initial program 84.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{{x}^{2} \cdot y}{z} \cdot \frac{1}{2}} + \frac{y}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z}} + \frac{y}{z}}{x} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{y}{z}\right)} + \frac{y}{z}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{z}} + \frac{y}{z}}{x} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{y}{z}}}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{z}}{x} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot {x}^{2}}{x} \cdot \frac{y}{z}} \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot y}{x \cdot z}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot y}{x \cdot z} \]

      10. distribute-rgt1-in N/A

          \[\leadsto \frac{\color{blue}{y + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x \cdot z} \]

      11. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}}{x \cdot z} \]

      12. *-commutative N/A

          \[\leadsto \frac{y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot {x}^{2}\right)}}{x \cdot z} \]

      13. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}}{x \cdot z} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y + \left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}{x \cdot z}} \]

   5. Applied rewrites 79.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 0.5 \cdot \left(x \cdot x\right), y\right)}{x \cdot z}} \]

   if 3.7000000000000002 < x

   1. Initial program 80.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 73.0

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 73.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot y\right) \cdot \frac{1}{24}}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(y \cdot \frac{1}{24}\right)}}{z} \]

      3. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(y \cdot \frac{1}{24}\right)}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(y \cdot \frac{1}{24}\right)}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot y\right)}}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} \cdot y\right)\right)}}{z} \]

      7. associate-*r* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left({x}^{2} \cdot \frac{1}{24}\right) \cdot y\right)}}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)} \cdot y\right)}{z} \]

      9. associate-*r* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \left(\frac{1}{24} \cdot \color{blue}{\left({x}^{2} \cdot y\right)}\right)}{z} \]

      13. unpow2 N/A

          \[\leadsto \frac{x \cdot \left(\frac{1}{24} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

      14. lower-*.f64 85.4

          \[\leadsto \frac{x \cdot \left(0.041666666666666664 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

   8. Applied rewrites 85.4%

      \[\leadsto \frac{\color{blue}{x \cdot \left(0.041666666666666664 \cdot \left(\left(x \cdot x\right) \cdot y\right)\right)}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \left(x \cdot x\right) \cdot 0.5, y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(0.041666666666666664 \cdot \left(y \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 83.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(0.041666666666666664 \cdot \left(y\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 2.2)
     (/ y_m (* x_m z))
     (/ (* x_m (* 0.041666666666666664 (* y_m (* x_m x_m)))) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = (x_m * (0.041666666666666664 * (y_m * (x_m * x_m)))) / z;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 2.2d0) then
        tmp = y_m / (x_m * z)
    else
        tmp = (x_m * (0.041666666666666664d0 * (y_m * (x_m * x_m)))) / z
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = (x_m * (0.041666666666666664 * (y_m * (x_m * x_m)))) / z;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if x_m <= 2.2:
		tmp = y_m / (x_m * z)
	else:
		tmp = (x_m * (0.041666666666666664 * (y_m * (x_m * x_m)))) / z
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 2.2)
		tmp = Float64(y_m / Float64(x_m * z));
	else
		tmp = Float64(Float64(x_m * Float64(0.041666666666666664 * Float64(y_m * Float64(x_m * x_m)))) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 2.2)
		tmp = y_m / (x_m * z);
	else
		tmp = (x_m * (0.041666666666666664 * (y_m * (x_m * x_m)))) / z;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 2.2], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(0.041666666666666664 * N[(y$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 84.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. lower-*.f64 65.2

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

   5. Applied rewrites 65.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 2.2000000000000002 < x

   1. Initial program 80.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 72.3

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 72.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot y\right) \cdot \frac{1}{24}}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(y \cdot \frac{1}{24}\right)}}{z} \]

      3. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(y \cdot \frac{1}{24}\right)}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(y \cdot \frac{1}{24}\right)}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot y\right)}}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} \cdot y\right)\right)}}{z} \]

      7. associate-*r* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left({x}^{2} \cdot \frac{1}{24}\right) \cdot y\right)}}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)} \cdot y\right)}{z} \]

      9. associate-*r* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \left(\frac{1}{24} \cdot \color{blue}{\left({x}^{2} \cdot y\right)}\right)}{z} \]

      13. unpow2 N/A

          \[\leadsto \frac{x \cdot \left(\frac{1}{24} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

      14. lower-*.f64 84.4

          \[\leadsto \frac{x \cdot \left(0.041666666666666664 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

   8. Applied rewrites 84.4%

      \[\leadsto \frac{\color{blue}{x \cdot \left(0.041666666666666664 \cdot \left(\left(x \cdot x\right) \cdot y\right)\right)}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(0.041666666666666664 \cdot \left(y \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 82.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(x\_m \cdot \frac{0.041666666666666664}{z}\right)\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 2.2)
     (/ y_m (* x_m z))
     (* (* y_m (* x_m x_m)) (* x_m (/ 0.041666666666666664 z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = (y_m * (x_m * x_m)) * (x_m * (0.041666666666666664 / z));
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 2.2d0) then
        tmp = y_m / (x_m * z)
    else
        tmp = (y_m * (x_m * x_m)) * (x_m * (0.041666666666666664d0 / z))
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = (y_m * (x_m * x_m)) * (x_m * (0.041666666666666664 / z));
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if x_m <= 2.2:
		tmp = y_m / (x_m * z)
	else:
		tmp = (y_m * (x_m * x_m)) * (x_m * (0.041666666666666664 / z))
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 2.2)
		tmp = Float64(y_m / Float64(x_m * z));
	else
		tmp = Float64(Float64(y_m * Float64(x_m * x_m)) * Float64(x_m * Float64(0.041666666666666664 / z)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 2.2)
		tmp = y_m / (x_m * z);
	else
		tmp = (y_m * (x_m * x_m)) * (x_m * (0.041666666666666664 / z));
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 2.2], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * N[(0.041666666666666664 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 84.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. lower-*.f64 65.2

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

   5. Applied rewrites 65.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 2.2000000000000002 < x

   1. Initial program 80.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 72.3

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 72.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right)} \cdot \frac{y}{x}}{z} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot \frac{y}{x}}{z} \]

      2. pow-sqr N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}\right) \cdot \frac{y}{x}}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right) \cdot \frac{y}{x}}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}\right) \cdot \frac{y}{x}}{z} \]

      10. unpow2 N/A

          \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24}\right)\right) \cdot \frac{y}{x}}{z} \]

      11. lower-*.f64 72.2

          \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.041666666666666664\right)\right) \cdot \frac{y}{x}}{z} \]

   8. Applied rewrites 72.2%

      \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \cdot \frac{y}{x}}{z} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}{z}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot y\right) \cdot \frac{1}{24}}}{z} \]

       3. associate-/l* N/A

          \[\leadsto \color{blue}{\left({x}^{3} \cdot y\right) \cdot \frac{\frac{1}{24}}{z}} \]

       4. cube-mult N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot y\right) \cdot \frac{\frac{1}{24}}{z} \]

       5. unpow2 N/A

          \[\leadsto \left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot y\right) \cdot \frac{\frac{1}{24}}{z} \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{\left(x \cdot \left({x}^{2} \cdot y\right)\right)} \cdot \frac{\frac{1}{24}}{z} \]

       7. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left({x}^{2} \cdot y\right) \cdot x\right)} \cdot \frac{\frac{1}{24}}{z} \]

       8. metadata-eval N/A

          \[\leadsto \left(\left({x}^{2} \cdot y\right) \cdot x\right) \cdot \frac{\color{blue}{\frac{1}{24} \cdot 1}}{z} \]

       9. associate-*r/ N/A

          \[\leadsto \left(\left({x}^{2} \cdot y\right) \cdot x\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{1}{z}\right)} \]

       10. associate-*l* N/A

           \[\leadsto \color{blue}{\left({x}^{2} \cdot y\right) \cdot \left(x \cdot \left(\frac{1}{24} \cdot \frac{1}{z}\right)\right)} \]

       11. *-commutative N/A

           \[\leadsto \left({x}^{2} \cdot y\right) \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot \frac{1}{z}\right) \cdot x\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \color{blue}{\left({x}^{2} \cdot y\right) \cdot \left(\left(\frac{1}{24} \cdot \frac{1}{z}\right) \cdot x\right)} \]

       13. lower-*.f64 N/A

           \[\leadsto \color{blue}{\left({x}^{2} \cdot y\right)} \cdot \left(\left(\frac{1}{24} \cdot \frac{1}{z}\right) \cdot x\right) \]

       14. unpow2 N/A

           \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right) \cdot \left(\left(\frac{1}{24} \cdot \frac{1}{z}\right) \cdot x\right) \]

       15. lower-*.f64 N/A

           \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right) \cdot \left(\left(\frac{1}{24} \cdot \frac{1}{z}\right) \cdot x\right) \]

       16. *-commutative N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot y\right) \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot \frac{1}{z}\right)\right)} \]

       17. lower-*.f64 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot y\right) \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot \frac{1}{z}\right)\right)} \]

       18. associate-*r/ N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot y\right) \cdot \left(x \cdot \color{blue}{\frac{\frac{1}{24} \cdot 1}{z}}\right) \]

       19. metadata-eval N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot y\right) \cdot \left(x \cdot \frac{\color{blue}{\frac{1}{24}}}{z}\right) \]

       20. lower-/.f64 84.4

           \[\leadsto \left(\left(x \cdot x\right) \cdot y\right) \cdot \left(x \cdot \color{blue}{\frac{0.041666666666666664}{z}}\right) \]

   11. Applied rewrites 84.4%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot y\right) \cdot \left(x \cdot \frac{0.041666666666666664}{z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \frac{0.041666666666666664}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 80.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \left(0.041666666666666664 \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 2.2)
     (/ y_m (* x_m z))
     (* (/ y_m z) (* 0.041666666666666664 (* x_m (* x_m x_m))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = (y_m / z) * (0.041666666666666664 * (x_m * (x_m * x_m)));
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 2.2d0) then
        tmp = y_m / (x_m * z)
    else
        tmp = (y_m / z) * (0.041666666666666664d0 * (x_m * (x_m * x_m)))
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = (y_m / z) * (0.041666666666666664 * (x_m * (x_m * x_m)));
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if x_m <= 2.2:
		tmp = y_m / (x_m * z)
	else:
		tmp = (y_m / z) * (0.041666666666666664 * (x_m * (x_m * x_m)))
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 2.2)
		tmp = Float64(y_m / Float64(x_m * z));
	else
		tmp = Float64(Float64(y_m / z) * Float64(0.041666666666666664 * Float64(x_m * Float64(x_m * x_m))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 2.2)
		tmp = y_m / (x_m * z);
	else
		tmp = (y_m / z) * (0.041666666666666664 * (x_m * (x_m * x_m)));
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 2.2], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] * N[(0.041666666666666664 * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 84.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. lower-*.f64 65.2

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

   5. Applied rewrites 65.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 2.2000000000000002 < x

   1. Initial program 80.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 72.3

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 72.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)} + 1\right) \cdot \frac{y}{x}}{z} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)} \cdot \frac{y}{x}}{z} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{x}}}{z} \]

      6. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{z \cdot x}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{\color{blue}{x \cdot z}} \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot \frac{y}{z}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x} \cdot \frac{y}{z}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x}} \cdot \frac{y}{z} \]

      11. lower-/.f64 82.7

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot \color{blue}{\frac{y}{z}} \]

   7. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x} \cdot \frac{y}{z}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)} \cdot \frac{y}{z} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)} \cdot \frac{y}{z} \]

      2. cube-mult N/A

         \[\leadsto \left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right) \cdot \frac{y}{z} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right) \cdot \frac{y}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right) \cdot \frac{y}{z} \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \frac{y}{z} \]

      6. lower-*.f64 82.6

         \[\leadsto \left(0.041666666666666664 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \frac{y}{z} \]

   10. Applied rewrites 82.6%

       \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \cdot \frac{y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 66.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{z} \cdot \left(x\_m \cdot y\_m\right)\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (if (<= x_m 1.4) (/ y_m (* x_m z)) (* (/ 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 <= 1.4) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = (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 <= 1.4d0) then
        tmp = y_m / (x_m * z)
    else
        tmp = (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 <= 1.4) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = (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 <= 1.4:
		tmp = y_m / (x_m * z)
	else:
		tmp = (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 <= 1.4)
		tmp = Float64(y_m / Float64(x_m * z));
	else
		tmp = Float64(Float64(0.5 / z) * Float64(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 <= 1.4)
		tmp = y_m / (x_m * z);
	else
		tmp = (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, 1.4], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / z), $MachinePrecision] * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 84.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. lower-*.f64 65.2

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

   5. Applied rewrites 65.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 1.3999999999999999 < x

   1. Initial program 80.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\frac{y + \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x}}{z} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot y}}{x}}{z} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot y}{x}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{x}}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      6. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{x} + \frac{y}{x}}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)} + \frac{y}{x}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{x}} + \frac{y}{x}}{z} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x}} + \frac{y}{x}}{z} \]

      10. associate-/l* N/A

          \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{x}^{2}}{x}\right)} + \frac{y}{x}}{z} \]

      11. unpow2 N/A

          \[\leadsto \frac{y \cdot \left(\frac{1}{2} \cdot \frac{\color{blue}{x \cdot x}}{x}\right) + \frac{y}{x}}{z} \]

      12. associate-/l* N/A

          \[\leadsto \frac{y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot \frac{x}{x}\right)}\right) + \frac{y}{x}}{z} \]

      13. *-inverses N/A

          \[\leadsto \frac{y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \color{blue}{1}\right)\right) + \frac{y}{x}}{z} \]

      14. *-rgt-identity N/A

          \[\leadsto \frac{y \cdot \left(\frac{1}{2} \cdot \color{blue}{x}\right) + \frac{y}{x}}{z} \]

      15. *-commutative N/A

          \[\leadsto \frac{y \cdot \color{blue}{\left(x \cdot \frac{1}{2}\right)} + \frac{y}{x}}{z} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x \cdot \frac{1}{2}, \frac{y}{x}\right)}}{z} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{x \cdot \frac{1}{2}}, \frac{y}{x}\right)}{z} \]

      18. lower-/.f64 57.0

          \[\leadsto \frac{\mathsf{fma}\left(y, x \cdot 0.5, \color{blue}{\frac{y}{x}}\right)}{z} \]

   5. Applied rewrites 57.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x \cdot 0.5, \frac{y}{x}\right)}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{z} \cdot \frac{1}{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{z}\right)} \cdot \frac{1}{2} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{y}{z} \cdot \frac{1}{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{y}{z}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \frac{y}{z}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{y}{z}\right)} \]

      7. lower-/.f64 44.2

         \[\leadsto x \cdot \left(0.5 \cdot \color{blue}{\frac{y}{z}}\right) \]

   8. Applied rewrites 44.2%

      \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \frac{y}{z}\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{y}{z}}\right) \]

      2. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{y}{z} \cdot \frac{1}{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{\frac{y}{z}} \cdot \frac{1}{2}\right) \]

      4. frac-2neg N/A

         \[\leadsto x \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(y\right)}{\mathsf{neg}\left(z\right)}} \cdot \frac{1}{2}\right) \]

      5. associate-*l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{1}{2}}{\mathsf{neg}\left(z\right)}} \]

      6. neg-mul-1 N/A

         \[\leadsto x \cdot \frac{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{1}{2}}{\color{blue}{-1 \cdot z}} \]

      7. times-frac N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{\mathsf{neg}\left(y\right)}{-1} \cdot \frac{\frac{1}{2}}{z}\right)} \]

      8. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{\mathsf{neg}\left(y\right)}{\color{blue}{\mathsf{neg}\left(1\right)}} \cdot \frac{\frac{1}{2}}{z}\right) \]

      9. frac-2neg N/A

         \[\leadsto x \cdot \left(\color{blue}{\frac{y}{1}} \cdot \frac{\frac{1}{2}}{z}\right) \]

      10. /-rgt-identity N/A

          \[\leadsto x \cdot \left(\color{blue}{y} \cdot \frac{\frac{1}{2}}{z}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{\frac{1}{2}}{z}\right)} \]

      12. lower-/.f64 44.2

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\frac{0.5}{z}}\right) \]

   10. Applied rewrites 44.2%

       \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{0.5}{z}\right)} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\frac{\frac{1}{2}}{z}}\right) \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{\frac{1}{2}}{z}} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2}}{z} \cdot \left(x \cdot y\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2}}{z} \cdot \left(x \cdot y\right)} \]

       5. lower-*.f64 57.0

          \[\leadsto \frac{0.5}{z} \cdot \color{blue}{\left(x \cdot y\right)} \]

   12. Applied rewrites 57.0%

       \[\leadsto \color{blue}{\frac{0.5}{z} \cdot \left(x \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 62.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\frac{y\_m}{z} \cdot 0.5\right)\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (if (<= x_m 1.4) (/ y_m (* x_m z)) (* x_m (* (/ y_m z) 0.5))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = x_m * ((y_m / z) * 0.5);
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 1.4d0) then
        tmp = y_m / (x_m * z)
    else
        tmp = x_m * ((y_m / z) * 0.5d0)
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = x_m * ((y_m / z) * 0.5);
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if x_m <= 1.4:
		tmp = y_m / (x_m * z)
	else:
		tmp = x_m * ((y_m / z) * 0.5)
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.4)
		tmp = Float64(y_m / Float64(x_m * z));
	else
		tmp = Float64(x_m * Float64(Float64(y_m / z) * 0.5));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 1.4)
		tmp = y_m / (x_m * z);
	else
		tmp = x_m * ((y_m / z) * 0.5);
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.4], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y$95$m / z), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 84.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. lower-*.f64 65.2

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

   5. Applied rewrites 65.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if 1.3999999999999999 < x

   1. Initial program 80.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\frac{y + \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x}}{z} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot y}}{x}}{z} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot y}{x}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{x}}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      6. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{x} + \frac{y}{x}}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)} + \frac{y}{x}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{x}} + \frac{y}{x}}{z} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x}} + \frac{y}{x}}{z} \]

      10. associate-/l* N/A

          \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{x}^{2}}{x}\right)} + \frac{y}{x}}{z} \]

      11. unpow2 N/A

          \[\leadsto \frac{y \cdot \left(\frac{1}{2} \cdot \frac{\color{blue}{x \cdot x}}{x}\right) + \frac{y}{x}}{z} \]

      12. associate-/l* N/A

          \[\leadsto \frac{y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot \frac{x}{x}\right)}\right) + \frac{y}{x}}{z} \]

      13. *-inverses N/A

          \[\leadsto \frac{y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \color{blue}{1}\right)\right) + \frac{y}{x}}{z} \]

      14. *-rgt-identity N/A

          \[\leadsto \frac{y \cdot \left(\frac{1}{2} \cdot \color{blue}{x}\right) + \frac{y}{x}}{z} \]

      15. *-commutative N/A

          \[\leadsto \frac{y \cdot \color{blue}{\left(x \cdot \frac{1}{2}\right)} + \frac{y}{x}}{z} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x \cdot \frac{1}{2}, \frac{y}{x}\right)}}{z} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{x \cdot \frac{1}{2}}, \frac{y}{x}\right)}{z} \]

      18. lower-/.f64 57.0

          \[\leadsto \frac{\mathsf{fma}\left(y, x \cdot 0.5, \color{blue}{\frac{y}{x}}\right)}{z} \]

   5. Applied rewrites 57.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x \cdot 0.5, \frac{y}{x}\right)}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{z} \cdot \frac{1}{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{z}\right)} \cdot \frac{1}{2} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{y}{z} \cdot \frac{1}{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{y}{z}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \frac{y}{z}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{y}{z}\right)} \]

      7. lower-/.f64 44.2

         \[\leadsto x \cdot \left(0.5 \cdot \color{blue}{\frac{y}{z}}\right) \]

   8. Applied rewrites 44.2%

      \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \frac{y}{z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 50.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \frac{y\_m}{x\_m \cdot z}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (/ y_m (* x_m z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (y_m / (x_m * z)));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x_s * (y_s * (y_m / (x_m * z)))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (y_m / (x_m * z)));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	return x_s * (y_s * (y_m / (x_m * z)))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(y_m / Float64(x_m * z))))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, y_s, x_m, y_m, z)
	tmp = x_s * (y_s * (y_m / (x_m * z)));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.7%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
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. lower-*.f64 50.1

      \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

5. Applied rewrites 50.1%

   \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
6. Add Preprocessing

Developer Target 1: 97.7% 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 2024214 
(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))