Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 85.3% → 99.8%

Time: 10.9s

Alternatives: 23

Speedup: 2.5×

• Could not converge on a ground truth

  1. x = 4.2030247991168977e+204
  2. y = 4.758176530662374e+260
  3. z = 1.3376963209053383e-184

• 64.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.3% 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

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := y\_m \cdot \cosh x\_m\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{y\_m}{x\_m} \cdot \cosh x\_m}{z\_m} \leq 5 \cdot 10^{-54}:\\ \;\;\;\;\frac{t\_0}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* y_m (cosh x_m))))
   (*
    x_s
    (*
     y_s
     (*
      z_s
      (if (<= (/ (* (/ y_m x_m) (cosh x_m)) z_m) 5e-54)
        (/ t_0 (* z_m x_m))
        (/ (/ t_0 z_m) x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * cosh(x_m);
	double tmp;
	if ((((y_m / x_m) * cosh(x_m)) / z_m) <= 5e-54) {
		tmp = t_0 / (z_m * x_m);
	} else {
		tmp = (t_0 / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * cosh(x_m)
    if ((((y_m / x_m) * cosh(x_m)) / z_m) <= 5d-54) then
        tmp = t_0 / (z_m * x_m)
    else
        tmp = (t_0 / z_m) / x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * Math.cosh(x_m);
	double tmp;
	if ((((y_m / x_m) * Math.cosh(x_m)) / z_m) <= 5e-54) {
		tmp = t_0 / (z_m * x_m);
	} else {
		tmp = (t_0 / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	t_0 = y_m * math.cosh(x_m)
	tmp = 0
	if (((y_m / x_m) * math.cosh(x_m)) / z_m) <= 5e-54:
		tmp = t_0 / (z_m * x_m)
	else:
		tmp = (t_0 / z_m) / x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	t_0 = Float64(y_m * cosh(x_m))
	tmp = 0.0
	if (Float64(Float64(Float64(y_m / x_m) * cosh(x_m)) / z_m) <= 5e-54)
		tmp = Float64(t_0 / Float64(z_m * x_m));
	else
		tmp = Float64(Float64(t_0 / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m)
	t_0 = y_m * cosh(x_m);
	tmp = 0.0;
	if ((((y_m / x_m) * cosh(x_m)) / z_m) <= 5e-54)
		tmp = t_0 / (z_m * x_m);
	else
		tmp = (t_0 / z_m) / x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(y$95$m * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5e-54], N[(t$95$0 / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.00000000000000015e-54

   1. Initial program 94.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 86.6

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

   4. Applied rewrites 86.6%

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

   if 5.00000000000000015e-54 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 73.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 5 \cdot 10^{-54}:\\ \;\;\;\;\frac{y \cdot \cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \cosh x}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot y\_m, x\_m, \frac{y\_m}{x\_m}\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z\_m}{\mathsf{fma}\left(t\_0, x\_m \cdot x\_m, 1\right)} \cdot x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0
         (fma
          (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
          (* x_m x_m)
          0.5)))
   (*
    x_s
    (*
     y_s
     (*
      z_s
      (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+155)
        (/ (fma (* t_0 y_m) x_m (/ y_m x_m)) z_m)
        (/ y_m (* (/ z_m (fma t_0 (* x_m x_m) 1.0)) x_m))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double t_0 = fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5);
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+155) {
		tmp = fma((t_0 * y_m), x_m, (y_m / x_m)) / z_m;
	} else {
		tmp = y_m / ((z_m / fma(t_0, (x_m * x_m), 1.0)) * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	t_0 = fma(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664), Float64(x_m * x_m), 0.5)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+155)
		tmp = Float64(fma(Float64(t_0 * y_m), x_m, Float64(y_m / x_m)) / z_m);
	else
		tmp = Float64(y_m / Float64(Float64(z_m / fma(t_0, Float64(x_m * x_m), 1.0)) * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+155], N[(N[(N[(t$95$0 * y$95$m), $MachinePrecision] * x$95$m + N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m / N[(N[(z$95$m / N[(t$95$0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000001e155

   1. Initial program 95.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 68.4

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

   5. Applied rewrites 68.4%

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

   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 89.2%

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

   if 2.00000000000000001e155 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 70.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 N/A

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

   7. Applied rewrites 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. *-commutative N/A

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

      10. *-lft-identity N/A

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

      11. times-frac N/A

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

      12. /-rgt-identity N/A

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

      13. div-inv N/A

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

      14. lift-/.f64 N/A

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

      15. clear-num N/A

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

      16. neg-mul-1 N/A

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

   9. Applied rewrites 78.3%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 92.2

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

   12. Applied rewrites 92.2%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot y, x, \frac{y}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot \frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z\_m}{\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), x\_m \cdot x\_m, 1\right)} \cdot x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+155)
      (/
       (*
        (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)
        (/ y_m x_m))
       z_m)
      (/
       y_m
       (*
        (/
         z_m
         (fma (fma (* x_m x_m) 0.041666666666666664 0.5) (* x_m x_m) 1.0))
        x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+155) {
		tmp = (fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) * (y_m / x_m)) / z_m;
	} else {
		tmp = y_m / ((z_m / fma(fma((x_m * x_m), 0.041666666666666664, 0.5), (x_m * x_m), 1.0)) * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+155)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * Float64(y_m / x_m)) / z_m);
	else
		tmp = Float64(y_m / Float64(Float64(z_m / fma(fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), Float64(x_m * x_m), 1.0)) * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+155], N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m / N[(N[(z$95$m / N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000001e155

   1. Initial program 95.2%

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

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if 2.00000000000000001e155 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 70.9%

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

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 60.5

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

   5. Applied rewrites 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 89.6

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

   7. Applied rewrites 89.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. *-lft-identity N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-rgt-neg-out N/A

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

      12. frac-2neg N/A

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

      13. lower-/.f64 N/A

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

   9. Applied rewrites 90.4%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right)} \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z\_m}{\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), x\_m \cdot x\_m, 1\right)} \cdot x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+155)
      (/
       (*
        (/
         (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)
         x_m)
        y_m)
       z_m)
      (/
       y_m
       (*
        (/
         z_m
         (fma (fma (* x_m x_m) 0.041666666666666664 0.5) (* x_m x_m) 1.0))
        x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+155) {
		tmp = ((fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y_m) / z_m;
	} else {
		tmp = y_m / ((z_m / fma(fma((x_m * x_m), 0.041666666666666664, 0.5), (x_m * x_m), 1.0)) * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+155)
		tmp = Float64(Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) * y_m) / z_m);
	else
		tmp = Float64(y_m / Float64(Float64(z_m / fma(fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), Float64(x_m * x_m), 1.0)) * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+155], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m / N[(N[(z$95$m / N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000001e155

   1. Initial program 95.2%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 88.4%

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

   if 2.00000000000000001e155 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 70.9%

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

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 60.5

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

   5. Applied rewrites 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 89.6

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

   7. Applied rewrites 89.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. *-lft-identity N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-rgt-neg-out N/A

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

      12. frac-2neg N/A

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

      13. lower-/.f64 N/A

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

   9. Applied rewrites 90.4%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right)} \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m, \frac{1}{x\_m}\right) \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z\_m}{\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), x\_m \cdot x\_m, 1\right)} \cdot x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+155)
      (/
       (* (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m (/ 1.0 x_m)) y_m)
       z_m)
      (/
       y_m
       (*
        (/
         z_m
         (fma (fma (* x_m x_m) 0.041666666666666664 0.5) (* x_m x_m) 1.0))
        x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+155) {
		tmp = (fma(fma(0.041666666666666664, (x_m * x_m), 0.5), x_m, (1.0 / x_m)) * y_m) / z_m;
	} else {
		tmp = y_m / ((z_m / fma(fma((x_m * x_m), 0.041666666666666664, 0.5), (x_m * x_m), 1.0)) * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+155)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), x_m, Float64(1.0 / x_m)) * y_m) / z_m);
	else
		tmp = Float64(y_m / Float64(Float64(z_m / fma(fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), Float64(x_m * x_m), 1.0)) * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+155], N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m / N[(N[(z$95$m / N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000001e155

   1. Initial program 95.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 88.4%

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

   if 2.00000000000000001e155 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 70.9%

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

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 60.5

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

   5. Applied rewrites 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 89.6

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

   7. Applied rewrites 89.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. *-lft-identity N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-rgt-neg-out N/A

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

      12. frac-2neg N/A

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

      13. lower-/.f64 N/A

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

   9. Applied rewrites 90.4%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right)} \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, x\_m, \frac{1}{x\_m}\right) \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0, x\_m \cdot x\_m, 1\right)}{z\_m}}{x\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (fma 0.041666666666666664 (* x_m x_m) 0.5)))
   (*
    x_s
    (*
     y_s
     (*
      z_s
      (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+155)
        (/ (* (fma t_0 x_m (/ 1.0 x_m)) y_m) z_m)
        (* (/ (/ (fma t_0 (* x_m x_m) 1.0) z_m) x_m) y_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double t_0 = fma(0.041666666666666664, (x_m * x_m), 0.5);
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+155) {
		tmp = (fma(t_0, x_m, (1.0 / x_m)) * y_m) / z_m;
	} else {
		tmp = ((fma(t_0, (x_m * x_m), 1.0) / z_m) / x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	t_0 = fma(0.041666666666666664, Float64(x_m * x_m), 0.5)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+155)
		tmp = Float64(Float64(fma(t_0, x_m, Float64(1.0 / x_m)) * y_m) / z_m);
	else
		tmp = Float64(Float64(Float64(fma(t_0, Float64(x_m * x_m), 1.0) / z_m) / x_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+155], N[(N[(N[(t$95$0 * x$95$m + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[(t$95$0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000001e155

   1. Initial program 95.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 88.4%

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

   if 2.00000000000000001e155 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 70.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.4%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{y\_m}{x\_m} \cdot \cosh x\_m}{z\_m} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right) \cdot \frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5 \cdot x\_m, x\_m, 1\right)}{z\_m} \cdot y\_m}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (/ y_m x_m) (cosh x_m)) z_m) 2e+148)
      (/ (* (fma (* x_m x_m) 0.5 1.0) (/ y_m x_m)) z_m)
      (/ (* (/ (fma (* 0.5 x_m) x_m 1.0) z_m) y_m) x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((((y_m / x_m) * cosh(x_m)) / z_m) <= 2e+148) {
		tmp = (fma((x_m * x_m), 0.5, 1.0) * (y_m / x_m)) / z_m;
	} else {
		tmp = ((fma((0.5 * x_m), x_m, 1.0) / z_m) * y_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(y_m / x_m) * cosh(x_m)) / z_m) <= 2e+148)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) * Float64(y_m / x_m)) / z_m);
	else
		tmp = Float64(Float64(Float64(fma(Float64(0.5 * x_m), x_m, 1.0) / z_m) * y_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 2e+148], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 2.0000000000000001e148

   1. Initial program 95.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

   if 2.0000000000000001e148 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 70.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 53.9

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

   5. Applied rewrites 53.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 N/A

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

   7. Applied rewrites 84.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. *-commutative N/A

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

      10. *-lft-identity N/A

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

      11. times-frac N/A

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

      12. /-rgt-identity N/A

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

      13. div-inv N/A

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

      14. lift-/.f64 N/A

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

      15. clear-num N/A

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

      16. neg-mul-1 N/A

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

   9. Applied rewrites 75.2%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-/r* N/A

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

       5. lower-/.f64 N/A

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

   11. Applied rewrites 84.2%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5 \cdot x, x, 1\right)}{z} \cdot y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{y\_m}{x\_m} \cdot \cosh x\_m}{z\_m} \leq 10^{+103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5 \cdot x\_m, x\_m, 1\right)}{z\_m} \cdot y\_m}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (/ y_m x_m) (cosh x_m)) z_m) 1e+103)
      (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) (* z_m x_m))
      (/ (* (/ (fma (* 0.5 x_m) x_m 1.0) z_m) y_m) x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((((y_m / x_m) * cosh(x_m)) / z_m) <= 1e+103) {
		tmp = (fma(0.5, (x_m * x_m), 1.0) * y_m) / (z_m * x_m);
	} else {
		tmp = ((fma((0.5 * x_m), x_m, 1.0) / z_m) * y_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(y_m / x_m) * cosh(x_m)) / z_m) <= 1e+103)
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / Float64(z_m * x_m));
	else
		tmp = Float64(Float64(Float64(fma(Float64(0.5 * x_m), x_m, 1.0) / z_m) * y_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 1e+103], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1e103

   1. Initial program 95.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 77.0

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

   7. Applied rewrites 77.0%

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

   if 1e103 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 71.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 54.7

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

   5. Applied rewrites 54.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 N/A

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

   7. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. *-commutative N/A

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

      10. *-lft-identity N/A

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

      11. times-frac N/A

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

      12. /-rgt-identity N/A

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

      13. div-inv N/A

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

      14. lift-/.f64 N/A

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

      15. clear-num N/A

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

      16. neg-mul-1 N/A

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

   9. Applied rewrites 74.8%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-/r* N/A

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

       5. lower-/.f64 N/A

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

   11. Applied rewrites 84.5%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 10^{+103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5 \cdot x, x, 1\right)}{z} \cdot y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m, \frac{1}{x\_m}\right) \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z\_m}}{x\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+155)
      (/ (* (fma 0.5 x_m (/ 1.0 x_m)) y_m) z_m)
      (* (/ (/ (fma (* x_m x_m) 0.5 1.0) z_m) x_m) y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+155) {
		tmp = (fma(0.5, x_m, (1.0 / x_m)) * y_m) / z_m;
	} else {
		tmp = ((fma((x_m * x_m), 0.5, 1.0) / z_m) / x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+155)
		tmp = Float64(Float64(fma(0.5, x_m, Float64(1.0 / x_m)) * y_m) / z_m);
	else
		tmp = Float64(Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / z_m) / x_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+155], N[(N[(N[(0.5 * x$95$m + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000001e155

   1. Initial program 95.2%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-out N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. unpow2 N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-/.f64 77.7

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

   5. Applied rewrites 77.7%

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

   if 2.00000000000000001e155 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 70.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 N/A

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

   7. Applied rewrites 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. metadata-eval N/A

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

      14. *-rgt-identity N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 79.1%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{z}}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 78.4

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

   7. Applied rewrites 78.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 0.5

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

   5. Applied rewrites 0.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 N/A

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

   7. Applied rewrites 84.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. times-frac N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lift-neg.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. metadata-eval N/A

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

      18. *-rgt-identity N/A

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

      19. lower-*.f64 N/A

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

   9. Applied rewrites 46.9%

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

4. Final simplification 75.2%

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

Alternative 11: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m, \frac{1}{x\_m}\right) \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z\_m \cdot x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+155)
      (/ (* (fma 0.5 x_m (/ 1.0 x_m)) y_m) z_m)
      (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) (* z_m x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+155) {
		tmp = (fma(0.5, x_m, (1.0 / x_m)) * y_m) / z_m;
	} else {
		tmp = (fma(0.5, (x_m * x_m), 1.0) * y_m) / (z_m * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+155)
		tmp = Float64(Float64(fma(0.5, x_m, Float64(1.0 / x_m)) * y_m) / z_m);
	else
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / Float64(z_m * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+155], N[(N[(N[(0.5 * x$95$m + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000001e155

   1. Initial program 95.2%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-out N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. unpow2 N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-/.f64 77.7

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

   5. Applied rewrites 77.7%

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

   if 2.00000000000000001e155 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 70.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 62.3

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

   7. Applied rewrites 62.3%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 10^{+301}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z\_m \cdot x\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (* (/ y_m x_m) (cosh x_m)) 1e+301)
      (/ (/ y_m x_m) z_m)
      (* (/ (fma (* x_m x_m) 0.5 1.0) (* z_m x_m)) y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 1e+301) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (fma((x_m * x_m), 0.5, 1.0) / (z_m * x_m)) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 1e+301)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / Float64(z_m * x_m)) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 1e+301], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.00000000000000005e301

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 72.1

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

   5. Applied rewrites 72.1%

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

   if 1.00000000000000005e301 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 65.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 43.3

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

   5. Applied rewrites 43.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 N/A

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

   7. Applied rewrites 76.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. times-frac N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lift-neg.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. metadata-eval N/A

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

      18. *-rgt-identity N/A

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

      19. lower-*.f64 N/A

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

   9. Applied rewrites 54.7%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 10^{+301}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{z \cdot x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \mathbf{elif}\;x\_m \leq 1.7 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m} \cdot \cosh x\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z\_m}}{x\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 2e-29)
      (/ (/ y_m z_m) x_m)
      (if (<= x_m 1.7e+62)
        (/ (* (/ y_m x_m) (cosh x_m)) z_m)
        (*
         (/
          (/
           (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)
           z_m)
          x_m)
         y_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 2e-29) {
		tmp = (y_m / z_m) / x_m;
	} else if (x_m <= 1.7e+62) {
		tmp = ((y_m / x_m) * cosh(x_m)) / z_m;
	} else {
		tmp = ((fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) / z_m) / x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 2e-29)
		tmp = Float64(Float64(y_m / z_m) / x_m);
	elseif (x_m <= 1.7e+62)
		tmp = Float64(Float64(Float64(y_m / x_m) * cosh(x_m)) / z_m);
	else
		tmp = Float64(Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / z_m) / x_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 2e-29], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], If[LessEqual[x$95$m, 1.7e+62], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.99999999999999989e-29

   1. Initial program 85.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 97.1

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 71.6

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

   7. Applied rewrites 71.6%

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

   if 1.99999999999999989e-29 < x < 1.70000000000000007e62

   1. Initial program 99.8%

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

   if 1.70000000000000007e62 < x

   1. Initial program 72.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.12 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \mathbf{elif}\;x\_m \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\frac{y\_m \cdot \cosh x\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z\_m}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)} \cdot x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 1.12e-39)
      (/ (/ y_m z_m) x_m)
      (if (<= x_m 7e+51)
        (/ (* y_m (cosh x_m)) (* z_m x_m))
        (/
         y_m
         (*
          (/
           z_m
           (fma
            (fma
             (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
             (* x_m x_m)
             0.5)
            (* x_m x_m)
            1.0))
          x_m))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.12e-39) {
		tmp = (y_m / z_m) / x_m;
	} else if (x_m <= 7e+51) {
		tmp = (y_m * cosh(x_m)) / (z_m * x_m);
	} else {
		tmp = y_m / ((z_m / fma(fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0)) * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 1.12e-39)
		tmp = Float64(Float64(y_m / z_m) / x_m);
	elseif (x_m <= 7e+51)
		tmp = Float64(Float64(y_m * cosh(x_m)) / Float64(z_m * x_m));
	else
		tmp = Float64(y_m / Float64(Float64(z_m / fma(fma(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0)) * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 1.12e-39], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], If[LessEqual[x$95$m, 7e+51], N[(N[(y$95$m * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(N[(z$95$m / N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.12e-39

   1. Initial program 86.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 97.1

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 71.3

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

   7. Applied rewrites 71.3%

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

   if 1.12e-39 < x < 7e51

   1. Initial program 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 89.5

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

   4. Applied rewrites 89.5%

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

   if 7e51 < x

   1. Initial program 74.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 N/A

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

   7. Applied rewrites 85.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. *-commutative N/A

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

      10. *-lft-identity N/A

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

      11. times-frac N/A

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

      12. /-rgt-identity N/A

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

      13. div-inv N/A

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

      14. lift-/.f64 N/A

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

      15. clear-num N/A

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

      16. neg-mul-1 N/A

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

   9. Applied rewrites 77.5%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 100.0

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

   12. Applied rewrites 100.0%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\frac{y \cdot \cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 8.4 \cdot 10^{+147}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \cosh x\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x\_m}{\frac{z\_m}{y\_m} \cdot x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= y_m 8.4e+147)
      (/ (/ (* y_m (cosh x_m)) x_m) z_m)
      (/ (cosh x_m) (* (/ z_m y_m) x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 8.4e+147) {
		tmp = ((y_m * cosh(x_m)) / x_m) / z_m;
	} else {
		tmp = cosh(x_m) / ((z_m / y_m) * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 8.4d+147) then
        tmp = ((y_m * cosh(x_m)) / x_m) / z_m
    else
        tmp = cosh(x_m) / ((z_m / y_m) * x_m)
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 8.4e+147) {
		tmp = ((y_m * Math.cosh(x_m)) / x_m) / z_m;
	} else {
		tmp = Math.cosh(x_m) / ((z_m / y_m) * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 8.4e+147:
		tmp = ((y_m * math.cosh(x_m)) / x_m) / z_m
	else:
		tmp = math.cosh(x_m) / ((z_m / y_m) * x_m)
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 8.4e+147)
		tmp = Float64(Float64(Float64(y_m * cosh(x_m)) / x_m) / z_m);
	else
		tmp = Float64(cosh(x_m) / Float64(Float64(z_m / y_m) * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 8.4e+147)
		tmp = ((y_m * cosh(x_m)) / x_m) / z_m;
	else
		tmp = cosh(x_m) / ((z_m / y_m) * x_m);
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[y$95$m, 8.4e+147], N[(N[(N[(y$95$m * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[Cosh[x$95$m], $MachinePrecision] / N[(N[(z$95$m / y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 8.40000000000000024e147

   1. Initial program 83.6%

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

      1. lift-*.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. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.9

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

   4. Applied rewrites 94.9%

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

   if 8.40000000000000024e147 < y

   1. Initial program 92.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cosh.f64 N/A

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

      6. times-frac N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

      11. lift-cosh.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.9

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

   6. Applied rewrites 99.9%

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

Alternative 16: 91.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.3 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot y\_m, x\_m, \frac{y\_m}{x\_m}\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m} \cdot \mathsf{fma}\left(t\_0, x\_m \cdot x\_m, 1\right)\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0
         (fma
          (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
          (* x_m x_m)
          0.5)))
   (*
    x_s
    (*
     y_s
     (*
      z_s
      (if (<= y_m 1.3e+150)
        (/ (fma (* t_0 y_m) x_m (/ y_m x_m)) z_m)
        (* (/ (/ y_m z_m) x_m) (fma t_0 (* x_m x_m) 1.0))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double t_0 = fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5);
	double tmp;
	if (y_m <= 1.3e+150) {
		tmp = fma((t_0 * y_m), x_m, (y_m / x_m)) / z_m;
	} else {
		tmp = ((y_m / z_m) / x_m) * fma(t_0, (x_m * x_m), 1.0);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	t_0 = fma(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664), Float64(x_m * x_m), 0.5)
	tmp = 0.0
	if (y_m <= 1.3e+150)
		tmp = Float64(fma(Float64(t_0 * y_m), x_m, Float64(y_m / x_m)) / z_m);
	else
		tmp = Float64(Float64(Float64(y_m / z_m) / x_m) * fma(t_0, Float64(x_m * x_m), 1.0));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[y$95$m, 1.3e+150], N[(N[(N[(t$95$0 * y$95$m), $MachinePrecision] * x$95$m + N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] * N[(t$95$0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.30000000000000003e150

   1. Initial program 83.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 52.0

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

   5. Applied rewrites 52.0%

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

   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 87.9%

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

   if 1.30000000000000003e150 < y

   1. Initial program 92.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.0

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

   5. Applied rewrites 85.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 92.5

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

   7. Applied rewrites 92.5%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 96.2

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

   10. Applied rewrites 96.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot y, x, \frac{y}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 91.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.4 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right) \cdot y\_m, x\_m, \frac{y\_m}{x\_m}\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z\_m}{\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), x\_m \cdot x\_m, 1\right)} \cdot x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 3.4e+146)
      (/
       (fma
        (*
         (fma
          (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
          (* x_m x_m)
          0.5)
         y_m)
        x_m
        (/ y_m x_m))
       z_m)
      (/
       y_m
       (*
        (/
         z_m
         (fma (fma (* x_m x_m) 0.041666666666666664 0.5) (* x_m x_m) 1.0))
        x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 3.4e+146) {
		tmp = fma((fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5) * y_m), x_m, (y_m / x_m)) / z_m;
	} else {
		tmp = y_m / ((z_m / fma(fma((x_m * x_m), 0.041666666666666664, 0.5), (x_m * x_m), 1.0)) * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 3.4e+146)
		tmp = Float64(fma(Float64(fma(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664), Float64(x_m * x_m), 0.5) * y_m), x_m, Float64(y_m / x_m)) / z_m);
	else
		tmp = Float64(y_m / Float64(Float64(z_m / fma(fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), Float64(x_m * x_m), 1.0)) * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 3.4e+146], N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m + N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m / N[(N[(z$95$m / N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.39999999999999991e146

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 55.7

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

   5. Applied rewrites 55.7%

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

   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 91.4%

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

   if 3.39999999999999991e146 < z

   1. Initial program 67.6%

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

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 55.7

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

   5. Applied rewrites 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 86.0

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

   7. Applied rewrites 86.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. *-lft-identity N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-rgt-neg-out N/A

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

      12. frac-2neg N/A

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

      13. lower-/.f64 N/A

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

   9. Applied rewrites 87.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot y, x, \frac{y}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right)} \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 85.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5 \cdot x\_m, x\_m, 1\right)}{z\_m} \cdot y\_m}{x\_m}\\ \mathbf{elif}\;z\_m \leq 2.5 \cdot 10^{+84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z\_m}{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)} \cdot x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 6.6e-52)
      (/ (* (/ (fma (* 0.5 x_m) x_m 1.0) z_m) y_m) x_m)
      (if (<= z_m 2.5e+84)
        (/
         (*
          (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)
          y_m)
         (* z_m x_m))
        (/ y_m (* (/ z_m (fma (* x_m x_m) 0.5 1.0)) x_m))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 6.6e-52) {
		tmp = ((fma((0.5 * x_m), x_m, 1.0) / z_m) * y_m) / x_m;
	} else if (z_m <= 2.5e+84) {
		tmp = (fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) * y_m) / (z_m * x_m);
	} else {
		tmp = y_m / ((z_m / fma((x_m * x_m), 0.5, 1.0)) * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 6.6e-52)
		tmp = Float64(Float64(Float64(fma(Float64(0.5 * x_m), x_m, 1.0) / z_m) * y_m) / x_m);
	elseif (z_m <= 2.5e+84)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * y_m) / Float64(z_m * x_m));
	else
		tmp = Float64(y_m / Float64(Float64(z_m / fma(Float64(x_m * x_m), 0.5, 1.0)) * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 6.6e-52], N[(N[(N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], If[LessEqual[z$95$m, 2.5e+84], N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(N[(z$95$m / N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 6.5999999999999999e-52

   1. Initial program 86.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(y\right)}{\mathsf{neg}\left(x\right)}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{\mathsf{neg}\left(x\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{\mathsf{neg}\left(x\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}}{\mathsf{neg}\left(x\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(-y\right)} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{\mathsf{neg}\left(x\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\left(-y\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}}{\mathsf{neg}\left(x\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \frac{\left(-y\right) \cdot \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right)}{z}}{\color{blue}{-x}} \]

   7. Applied rewrites 86.3%

      \[\leadsto \color{blue}{\frac{\left(-y\right) \cdot \frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z}}{-x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}{-x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}{-x} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(-y\right) \cdot \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}{-x}} \]

      4. clear-num N/A

         \[\leadsto \left(-y\right) \cdot \color{blue}{\frac{1}{\frac{-x}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{-y}{\frac{-x}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-y}{\frac{-x}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{-y}{\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

      8. neg-mul-1 N/A

         \[\leadsto \frac{-y}{\frac{\color{blue}{-1 \cdot x}}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{-y}{\frac{\color{blue}{x \cdot -1}}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

      10. *-lft-identity N/A

          \[\leadsto \frac{-y}{\frac{x \cdot -1}{\color{blue}{1 \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      11. times-frac N/A

          \[\leadsto \frac{-y}{\color{blue}{\frac{x}{1} \cdot \frac{-1}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      12. /-rgt-identity N/A

          \[\leadsto \frac{-y}{\color{blue}{x} \cdot \frac{-1}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

      13. div-inv N/A

          \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(-1 \cdot \frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}\right)}} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{-y}{x \cdot \left(-1 \cdot \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}\right)} \]

      15. clear-num N/A

          \[\leadsto \frac{-y}{x \cdot \left(-1 \cdot \color{blue}{\frac{z}{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}}\right)} \]

      16. neg-mul-1 N/A

          \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{z}{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}\right)\right)}} \]

   9. Applied rewrites 79.8%

      \[\leadsto \color{blue}{\frac{-y}{x \cdot \frac{-z}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-y}{x \cdot \frac{-z}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}}} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{-y}{\color{blue}{x \cdot \frac{-z}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}}} \]

       3. *-commutative N/A

          \[\leadsto \frac{-y}{\color{blue}{\frac{-z}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)} \cdot x}} \]

       4. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{-y}{\frac{-z}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}}}{x}} \]

       5. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{-y}{\frac{-z}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}}}{x}} \]

   11. Applied rewrites 86.3%

       \[\leadsto \color{blue}{\frac{y \cdot \frac{\mathsf{fma}\left(0.5 \cdot x, x, 1\right)}{z}}{x}} \]

   if 6.5999999999999999e-52 < z < 2.5e84

   1. Initial program 89.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 74.7

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 74.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{z \cdot x}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{z \cdot x}} \]

      8. lower-*.f64 89.1

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}}{z \cdot x} \]

   7. Applied rewrites 89.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z \cdot x}} \]

   if 2.5e84 < z

   1. Initial program 74.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 59.9

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 59.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}} \]

      5. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z} \]

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(y\right)}{\mathsf{neg}\left(x\right)}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{\mathsf{neg}\left(x\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{\mathsf{neg}\left(x\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}}{\mathsf{neg}\left(x\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(-y\right)} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{\mathsf{neg}\left(x\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\left(-y\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}}{\mathsf{neg}\left(x\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \frac{\left(-y\right) \cdot \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right)}{z}}{\color{blue}{-x}} \]

   7. Applied rewrites 79.7%

      \[\leadsto \color{blue}{\frac{\left(-y\right) \cdot \frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z}}{-x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}{-x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}{-x} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(-y\right) \cdot \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}{-x}} \]

      4. clear-num N/A

         \[\leadsto \left(-y\right) \cdot \color{blue}{\frac{1}{\frac{-x}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{-y}{\frac{-x}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-y}{\frac{-x}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{-y}{\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

      8. neg-mul-1 N/A

         \[\leadsto \frac{-y}{\frac{\color{blue}{-1 \cdot x}}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{-y}{\frac{\color{blue}{x \cdot -1}}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

      10. *-lft-identity N/A

          \[\leadsto \frac{-y}{\frac{x \cdot -1}{\color{blue}{1 \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      11. times-frac N/A

          \[\leadsto \frac{-y}{\color{blue}{\frac{x}{1} \cdot \frac{-1}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      12. /-rgt-identity N/A

          \[\leadsto \frac{-y}{\color{blue}{x} \cdot \frac{-1}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

      13. div-inv N/A

          \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(-1 \cdot \frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}\right)}} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{-y}{x \cdot \left(-1 \cdot \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}\right)} \]

      15. clear-num N/A

          \[\leadsto \frac{-y}{x \cdot \left(-1 \cdot \color{blue}{\frac{z}{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}}\right)} \]

      16. neg-mul-1 N/A

          \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{z}{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}\right)\right)}} \]

   9. Applied rewrites 81.0%

      \[\leadsto \color{blue}{\frac{-y}{x \cdot \frac{-z}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5 \cdot x, x, 1\right)}{z} \cdot y}{x}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 88.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m, \frac{1}{x\_m}\right)\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3 \cdot 10^{+81}:\\ \;\;\;\;\frac{t\_0 \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m (/ 1.0 x_m))))
   (*
    x_s
    (*
     y_s
     (* z_s (if (<= y_m 3e+81) (/ (* t_0 y_m) z_m) (* t_0 (/ y_m z_m))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double t_0 = fma(fma(0.041666666666666664, (x_m * x_m), 0.5), x_m, (1.0 / x_m));
	double tmp;
	if (y_m <= 3e+81) {
		tmp = (t_0 * y_m) / z_m;
	} else {
		tmp = t_0 * (y_m / z_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	t_0 = fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), x_m, Float64(1.0 / x_m))
	tmp = 0.0
	if (y_m <= 3e+81)
		tmp = Float64(Float64(t_0 * y_m) / z_m);
	else
		tmp = Float64(t_0 * Float64(y_m / z_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[y$95$m, 3e+81], N[(N[(t$95$0 * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(t$95$0 * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.99999999999999997e81

   1. Initial program 82.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
   4. Step-by-step derivation

      1. lower-/.f64 52.6

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   5. Applied rewrites 52.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}}}{z} \]

   8. Applied rewrites 83.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x, \frac{1}{x}\right) \cdot y}}{z} \]

   if 2.99999999999999997e81 < y

   1. Initial program 95.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cosh x \cdot \frac{y}{x}\right)} \cdot \frac{1}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\cosh x \cdot \color{blue}{\frac{y}{x}}\right) \cdot \frac{1}{z} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x}} \cdot \frac{1}{z} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{z}}{x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{z}}{x}} \]

      8. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{z}}}{x} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{z}}}{x} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot \cosh x}}{z}}{x} \]

      11. lower-*.f64 100.0

          \[\leadsto \frac{\frac{\color{blue}{y \cdot \cosh x}}{z}}{x} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\frac{y \cdot \cosh x}{z}}{x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{1}{2} \cdot \frac{y}{z}\right) + \frac{y}{z}}{x}} \]

   7. Applied rewrites 87.5%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x, \frac{1}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x, \frac{1}{x}\right) \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 66.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.235:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\_m\right) \cdot y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 0.235) (/ (/ y_m z_m) x_m) (/ (* (* 0.5 x_m) y_m) z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.235) {
		tmp = (y_m / z_m) / x_m;
	} else {
		tmp = ((0.5 * x_m) * y_m) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.235d0) then
        tmp = (y_m / z_m) / x_m
    else
        tmp = ((0.5d0 * x_m) * y_m) / z_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.235) {
		tmp = (y_m / z_m) / x_m;
	} else {
		tmp = ((0.5 * x_m) * y_m) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.235:
		tmp = (y_m / z_m) / x_m
	else:
		tmp = ((0.5 * x_m) * y_m) / z_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.235)
		tmp = Float64(Float64(y_m / z_m) / x_m);
	else
		tmp = Float64(Float64(Float64(0.5 * x_m) * y_m) / z_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.235)
		tmp = (y_m / z_m) / x_m;
	else
		tmp = ((0.5 * x_m) * y_m) / z_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 0.235], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.23499999999999999

   1. Initial program 86.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cosh x \cdot \frac{y}{x}\right)} \cdot \frac{1}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\cosh x \cdot \color{blue}{\frac{y}{x}}\right) \cdot \frac{1}{z} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x}} \cdot \frac{1}{z} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{z}}{x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{z}}{x}} \]

      8. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{z}}}{x} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{z}}}{x} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot \cosh x}}{z}}{x} \]

      11. lower-*.f64 97.1

          \[\leadsto \frac{\frac{\color{blue}{y \cdot \cosh x}}{z}}{x} \]

   4. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\frac{\frac{y \cdot \cosh x}{z}}{x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{z}}}{x} \]
   6. Step-by-step derivation

      1. lower-/.f64 72.2

         \[\leadsto \frac{\color{blue}{\frac{y}{z}}}{x} \]

   7. Applied rewrites 72.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{z}}}{x} \]

   if 0.23499999999999999 < x

   1. Initial program 78.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot y} + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{1 \cdot y + \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x}}{z} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{x}}{z} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)}}{z} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \frac{y}{x} \cdot 1}}{z} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{\frac{y}{x}}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{x}} + \frac{y}{x}}{z} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x}} + \frac{y}{x}}{z} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{\color{blue}{y \cdot 1}}{x}}{z} \]

      11. associate-/l* N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \color{blue}{y \cdot \frac{1}{x}}}{z} \]

      12. distribute-lft-out N/A

          \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right)}}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

      15. associate-/l* N/A

          \[\leadsto \frac{\left(\color{blue}{\frac{1}{2} \cdot \frac{{x}^{2}}{x}} + \frac{1}{x}\right) \cdot y}{z} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{\color{blue}{x \cdot x}}{x} + \frac{1}{x}\right) \cdot y}{z} \]

      17. associate-/l* N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot \frac{x}{x}\right)} + \frac{1}{x}\right) \cdot y}{z} \]

      18. *-inverses N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \left(x \cdot \color{blue}{1}\right) + \frac{1}{x}\right) \cdot y}{z} \]

      19. *-rgt-identity N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{x} + \frac{1}{x}\right) \cdot y}{z} \]

      20. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, \frac{1}{x}\right)} \cdot y}{z} \]

      21. lower-/.f64 37.5

          \[\leadsto \frac{\mathsf{fma}\left(0.5, x, \color{blue}{\frac{1}{x}}\right) \cdot y}{z} \]

   5. Applied rewrites 37.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5, x, \frac{1}{x}\right) \cdot y}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{1}{2} \cdot x\right) \cdot y}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.5%

      \[\leadsto \frac{\left(0.5 \cdot x\right) \cdot y}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 66.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.235:\\ \;\;\;\;\frac{y\_m}{\left(-x\_m\right) \cdot \left(-z\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\_m\right) \cdot y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 0.235)
      (/ y_m (* (- x_m) (- z_m)))
      (/ (* (* 0.5 x_m) y_m) z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.235) {
		tmp = y_m / (-x_m * -z_m);
	} else {
		tmp = ((0.5 * x_m) * y_m) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.235d0) then
        tmp = y_m / (-x_m * -z_m)
    else
        tmp = ((0.5d0 * x_m) * y_m) / z_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.235) {
		tmp = y_m / (-x_m * -z_m);
	} else {
		tmp = ((0.5 * x_m) * y_m) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.235:
		tmp = y_m / (-x_m * -z_m)
	else:
		tmp = ((0.5 * x_m) * y_m) / z_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.235)
		tmp = Float64(y_m / Float64(Float64(-x_m) * Float64(-z_m)));
	else
		tmp = Float64(Float64(Float64(0.5 * x_m) * y_m) / z_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.235)
		tmp = y_m / (-x_m * -z_m);
	else
		tmp = ((0.5 * x_m) * y_m) / z_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 0.235], N[(y$95$m / N[((-x$95$m) * (-z$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.23499999999999999

   1. Initial program 86.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 76.4

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}} \]

      5. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z} \]

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(y\right)}{\mathsf{neg}\left(x\right)}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{\mathsf{neg}\left(x\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{\mathsf{neg}\left(x\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}}{\mathsf{neg}\left(x\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(-y\right)} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{\mathsf{neg}\left(x\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\left(-y\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}}{\mathsf{neg}\left(x\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \frac{\left(-y\right) \cdot \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right)}{z}}{\color{blue}{-x}} \]

   7. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\frac{\left(-y\right) \cdot \frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z}}{-x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}{-x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}{-x} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(-y\right) \cdot \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}{-x}} \]

      4. clear-num N/A

         \[\leadsto \left(-y\right) \cdot \color{blue}{\frac{1}{\frac{-x}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{-y}{\frac{-x}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-y}{\frac{-x}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{-y}{\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

      8. neg-mul-1 N/A

         \[\leadsto \frac{-y}{\frac{\color{blue}{-1 \cdot x}}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{-y}{\frac{\color{blue}{x \cdot -1}}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

      10. *-lft-identity N/A

          \[\leadsto \frac{-y}{\frac{x \cdot -1}{\color{blue}{1 \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      11. times-frac N/A

          \[\leadsto \frac{-y}{\color{blue}{\frac{x}{1} \cdot \frac{-1}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      12. /-rgt-identity N/A

          \[\leadsto \frac{-y}{\color{blue}{x} \cdot \frac{-1}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

      13. div-inv N/A

          \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(-1 \cdot \frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}\right)}} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{-y}{x \cdot \left(-1 \cdot \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}\right)} \]

      15. clear-num N/A

          \[\leadsto \frac{-y}{x \cdot \left(-1 \cdot \color{blue}{\frac{z}{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}}\right)} \]

      16. neg-mul-1 N/A

          \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{z}{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}\right)\right)}} \]

   9. Applied rewrites 82.9%

      \[\leadsto \color{blue}{\frac{-y}{x \cdot \frac{-z}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(-1 \cdot z\right)}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}} \]

       2. lower-neg.f64 65.4

          \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(-z\right)}} \]

   12. Applied rewrites 65.4%

       \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(-z\right)}} \]

   if 0.23499999999999999 < x

   1. Initial program 78.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot y} + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{1 \cdot y + \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x}}{z} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{x}}{z} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)}}{z} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \frac{y}{x} \cdot 1}}{z} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{y}{x} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{\frac{y}{x}}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{x}} + \frac{y}{x}}{z} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x}} + \frac{y}{x}}{z} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{\color{blue}{y \cdot 1}}{x}}{z} \]

      11. associate-/l* N/A

          \[\leadsto \frac{y \cdot \frac{\frac{1}{2} \cdot {x}^{2}}{x} + \color{blue}{y \cdot \frac{1}{x}}}{z} \]

      12. distribute-lft-out N/A

          \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right)}}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2}}{x} + \frac{1}{x}\right) \cdot y}}{z} \]

      15. associate-/l* N/A

          \[\leadsto \frac{\left(\color{blue}{\frac{1}{2} \cdot \frac{{x}^{2}}{x}} + \frac{1}{x}\right) \cdot y}{z} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{\color{blue}{x \cdot x}}{x} + \frac{1}{x}\right) \cdot y}{z} \]

      17. associate-/l* N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot \frac{x}{x}\right)} + \frac{1}{x}\right) \cdot y}{z} \]

      18. *-inverses N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \left(x \cdot \color{blue}{1}\right) + \frac{1}{x}\right) \cdot y}{z} \]

      19. *-rgt-identity N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{x} + \frac{1}{x}\right) \cdot y}{z} \]

      20. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, \frac{1}{x}\right)} \cdot y}{z} \]

      21. lower-/.f64 37.5

          \[\leadsto \frac{\mathsf{fma}\left(0.5, x, \color{blue}{\frac{1}{x}}\right) \cdot y}{z} \]

   5. Applied rewrites 37.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5, x, \frac{1}{x}\right) \cdot y}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{1}{2} \cdot x\right) \cdot y}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.5%

      \[\leadsto \frac{\left(0.5 \cdot x\right) \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.235:\\ \;\;\;\;\frac{y}{\left(-x\right) \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\right) \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 62.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.235:\\ \;\;\;\;\frac{y\_m}{\left(-x\_m\right) \cdot \left(-z\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\_m\right) \cdot \frac{y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 0.235)
      (/ y_m (* (- x_m) (- z_m)))
      (* (* 0.5 x_m) (/ y_m z_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.235) {
		tmp = y_m / (-x_m * -z_m);
	} else {
		tmp = (0.5 * x_m) * (y_m / z_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 0.235d0) then
        tmp = y_m / (-x_m * -z_m)
    else
        tmp = (0.5d0 * x_m) * (y_m / z_m)
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 0.235) {
		tmp = y_m / (-x_m * -z_m);
	} else {
		tmp = (0.5 * x_m) * (y_m / z_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 0.235:
		tmp = y_m / (-x_m * -z_m)
	else:
		tmp = (0.5 * x_m) * (y_m / z_m)
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 0.235)
		tmp = Float64(y_m / Float64(Float64(-x_m) * Float64(-z_m)));
	else
		tmp = Float64(Float64(0.5 * x_m) * Float64(y_m / z_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 0.235)
		tmp = y_m / (-x_m * -z_m);
	else
		tmp = (0.5 * x_m) * (y_m / z_m);
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 0.235], N[(y$95$m / N[((-x$95$m) * (-z$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * x$95$m), $MachinePrecision] * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.23499999999999999

   1. Initial program 86.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 76.4

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}} \]

      5. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z} \]

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(y\right)}{\mathsf{neg}\left(x\right)}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{\mathsf{neg}\left(x\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{\mathsf{neg}\left(x\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}}{\mathsf{neg}\left(x\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(-y\right)} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{\mathsf{neg}\left(x\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\left(-y\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}}{\mathsf{neg}\left(x\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \frac{\left(-y\right) \cdot \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right)}{z}}{\color{blue}{-x}} \]

   7. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\frac{\left(-y\right) \cdot \frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z}}{-x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}{-x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}{-x} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(-y\right) \cdot \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}{-x}} \]

      4. clear-num N/A

         \[\leadsto \left(-y\right) \cdot \color{blue}{\frac{1}{\frac{-x}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{-y}{\frac{-x}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-y}{\frac{-x}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{-y}{\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

      8. neg-mul-1 N/A

         \[\leadsto \frac{-y}{\frac{\color{blue}{-1 \cdot x}}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{-y}{\frac{\color{blue}{x \cdot -1}}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

      10. *-lft-identity N/A

          \[\leadsto \frac{-y}{\frac{x \cdot -1}{\color{blue}{1 \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      11. times-frac N/A

          \[\leadsto \frac{-y}{\color{blue}{\frac{x}{1} \cdot \frac{-1}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

      12. /-rgt-identity N/A

          \[\leadsto \frac{-y}{\color{blue}{x} \cdot \frac{-1}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

      13. div-inv N/A

          \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(-1 \cdot \frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}\right)}} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{-y}{x \cdot \left(-1 \cdot \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}\right)} \]

      15. clear-num N/A

          \[\leadsto \frac{-y}{x \cdot \left(-1 \cdot \color{blue}{\frac{z}{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}}\right)} \]

      16. neg-mul-1 N/A

          \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{z}{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}\right)\right)}} \]

   9. Applied rewrites 82.9%

      \[\leadsto \color{blue}{\frac{-y}{x \cdot \frac{-z}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(-1 \cdot z\right)}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}} \]

       2. lower-neg.f64 65.4

          \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(-z\right)}} \]

   12. Applied rewrites 65.4%

       \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(-z\right)}} \]

   if 0.23499999999999999 < x

   1. Initial program 78.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{y}{z}\right)} + \frac{y}{z}}{x} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{z}} + \frac{y}{z}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{y}{z} + \frac{y}{z}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{y}{z}\right)} + \frac{y}{z}}{x} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \frac{y}{z}} + \frac{y}{z}}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{z} + \frac{y}{z}}{x} \]

      7. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{y}{z}}}{x} \]

      8. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{z}}{x} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{\frac{y}{z}}{x}} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{\frac{y}{z}}{x} \]

      11. associate-/l/ N/A

          \[\leadsto \left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      12. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{x \cdot z} + \frac{y}{x \cdot z}} \]

   5. Applied rewrites 31.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, \frac{1}{x}\right) \cdot \frac{y}{z}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{\color{blue}{y}}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.0%

      \[\leadsto \left(0.5 \cdot x\right) \cdot \frac{\color{blue}{y}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.235:\\ \;\;\;\;\frac{y}{\left(-x\right) \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 48.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \frac{y\_m}{\left(-x\_m\right) \cdot \left(-z\_m\right)}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (* x_s (* y_s (* z_s (/ y_m (* (- x_m) (- z_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	return x_s * (y_s * (z_s * (y_m / (-x_m * -z_m))));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    code = x_s * (y_s * (z_s * (y_m / (-x_m * -z_m))))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	return x_s * (y_s * (z_s * (y_m / (-x_m * -z_m))));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	return x_s * (y_s * (z_s * (y_m / (-x_m * -z_m))))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	return Float64(x_s * Float64(y_s * Float64(z_s * Float64(y_m / Float64(Float64(-x_m) * Float64(-z_m))))))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = x_s * (y_s * (z_s * (y_m / (-x_m * -z_m))));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * N[(y$95$m / N[((-x$95$m) * (-z$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.5%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

   4. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. lower-*.f64 69.4

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

5. Applied rewrites 69.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}}{z} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}}{z} \]

   4. associate-/l* N/A

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}} \]

   5. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z} \]

   6. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(y\right)}{\mathsf{neg}\left(x\right)}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z} \]

   7. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{\mathsf{neg}\left(x\right)}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{\mathsf{neg}\left(x\right)}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}}{\mathsf{neg}\left(x\right)} \]

   10. lower-neg.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(-y\right)} \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}{\mathsf{neg}\left(x\right)} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\left(-y\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)}{z}}}{\mathsf{neg}\left(x\right)} \]

   12. lower-neg.f64 N/A

       \[\leadsto \frac{\left(-y\right) \cdot \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right)}{z}}{\color{blue}{-x}} \]

7. Applied rewrites 83.9%

   \[\leadsto \color{blue}{\frac{\left(-y\right) \cdot \frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z}}{-x}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}{-x}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}{-x} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{\left(-y\right) \cdot \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}{-x}} \]

   4. clear-num N/A

      \[\leadsto \left(-y\right) \cdot \color{blue}{\frac{1}{\frac{-x}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

   5. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{-y}{\frac{-x}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-y}{\frac{-x}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

   7. lift-neg.f64 N/A

      \[\leadsto \frac{-y}{\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

   8. neg-mul-1 N/A

      \[\leadsto \frac{-y}{\frac{\color{blue}{-1 \cdot x}}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

   9. *-commutative N/A

      \[\leadsto \frac{-y}{\frac{\color{blue}{x \cdot -1}}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

   10. *-lft-identity N/A

       \[\leadsto \frac{-y}{\frac{x \cdot -1}{\color{blue}{1 \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

   11. times-frac N/A

       \[\leadsto \frac{-y}{\color{blue}{\frac{x}{1} \cdot \frac{-1}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}} \]

   12. /-rgt-identity N/A

       \[\leadsto \frac{-y}{\color{blue}{x} \cdot \frac{-1}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}} \]

   13. div-inv N/A

       \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(-1 \cdot \frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}\right)}} \]

   14. lift-/.f64 N/A

       \[\leadsto \frac{-y}{x \cdot \left(-1 \cdot \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}{z}}}\right)} \]

   15. clear-num N/A

       \[\leadsto \frac{-y}{x \cdot \left(-1 \cdot \color{blue}{\frac{z}{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}}\right)} \]

   16. neg-mul-1 N/A

       \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{z}{\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)}\right)\right)}} \]

9. Applied rewrites 79.2%

   \[\leadsto \color{blue}{\frac{-y}{x \cdot \frac{-z}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(-1 \cdot z\right)}} \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}} \]

    2. lower-neg.f64 53.0

       \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(-z\right)}} \]

12. Applied rewrites 53.0%

    \[\leadsto \frac{-y}{x \cdot \color{blue}{\left(-z\right)}} \]

13. Final simplification 53.0%

    \[\leadsto \frac{y}{\left(-x\right) \cdot \left(-z\right)} \]
14. Add Preprocessing

Developer Target 1: 97.1% 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 2024276 
(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))