Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.6% → 98.0%

Time: 11.5s

Alternatives: 25

Speedup: 2.5×

• Could not converge on a ground truth

  1. x = 1.479899658423964e+59
  2. y = 2.867706105032871e+187
  3. z = 1.3744450948710188e-266

• 64.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.6% 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: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\cosh x}{x} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x \cdot x}{z}, \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \frac{1}{z}\right)}{x} \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.3e+80)
    (/ (* (/ (cosh x) x) y_m) z)
    (*
     (/
      (fma
       (/ (* x x) z)
       (fma
        (fma 0.001388888888888889 (* x x) 0.041666666666666664)
        (* x x)
        0.5)
       (/ 1.0 z))
      x)
     y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.3e+80) {
		tmp = ((cosh(x) / x) * y_m) / z;
	} else {
		tmp = (fma(((x * x) / z), fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (1.0 / z)) / x) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.3e+80)
		tmp = Float64(Float64(Float64(cosh(x) / x) * y_m) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x * x) / z), fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(1.0 / z)) / x) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.3e+80], N[(N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] / z), $MachinePrecision] * N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] + N[(1.0 / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.29999999999999991e80

   1. Initial program 84.0%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 98.0

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

   4. Applied rewrites 98.0%

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

   if 1.29999999999999991e80 < y

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 89.8

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

   4. Applied rewrites 89.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 62.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\cosh x}{x} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x \cdot x}{z}, \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \frac{1}{z}\right)}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x} \cdot \cosh x \leq 2 \cdot 10^{+257}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot x, x, 1\right) \cdot \frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0, x \cdot x, 1\right)}{z}}{x} \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (fma 0.041666666666666664 (* x x) 0.5)))
   (*
    y_s
    (if (<= (* (/ y_m x) (cosh x)) 2e+257)
      (/ (* (fma (* t_0 x) x 1.0) (/ y_m x)) z)
      (* (/ (/ (fma t_0 (* x x) 1.0) z) x) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = fma(0.041666666666666664, (x * x), 0.5);
	double tmp;
	if (((y_m / x) * cosh(x)) <= 2e+257) {
		tmp = (fma((t_0 * x), x, 1.0) * (y_m / x)) / z;
	} else {
		tmp = ((fma(t_0, (x * x), 1.0) / z) / x) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = fma(0.041666666666666664, Float64(x * x), 0.5)
	tmp = 0.0
	if (Float64(Float64(y_m / x) * cosh(x)) <= 2e+257)
		tmp = Float64(Float64(fma(Float64(t_0 * x), x, 1.0) * Float64(y_m / x)) / z);
	else
		tmp = Float64(Float64(Float64(fma(t_0, Float64(x * x), 1.0) / z) / x) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], 2e+257], N[(N[(N[(N[(t$95$0 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(t$95$0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000006e257

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 89.5

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

   5. Applied rewrites 89.5%

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

   7. Applied rewrites 89.5%

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

   if 2.00000000000000006e257 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 69.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 89.3%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+257}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x} \cdot \cosh x \leq 2 \cdot 10^{+257}:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{z}}{x} \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)))
   (*
    y_s
    (if (<= (* (/ y_m x) (cosh x)) 2e+257)
      (* (/ (/ y_m x) z) t_0)
      (* (/ (/ t_0 z) x) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0);
	double tmp;
	if (((y_m / x) * cosh(x)) <= 2e+257) {
		tmp = ((y_m / x) / z) * t_0;
	} else {
		tmp = ((t_0 / z) / x) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0)
	tmp = 0.0
	if (Float64(Float64(y_m / x) * cosh(x)) <= 2e+257)
		tmp = Float64(Float64(Float64(y_m / x) / z) * t_0);
	else
		tmp = Float64(Float64(Float64(t_0 / z) / x) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], 2e+257], N[(N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(t$95$0 / z), $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000006e257

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 89.5

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

   5. Applied rewrites 89.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   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. 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. lower-/.f64 84.1

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

   7. Applied rewrites 84.1%

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

   if 2.00000000000000006e257 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 69.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 89.3%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+257}:\\ \;\;\;\;\frac{\frac{y}{x}}{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x} \cdot \cosh x \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, x, \frac{1}{x}\right) \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0, x \cdot x, 1\right)}{z}}{x} \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (fma 0.041666666666666664 (* x x) 0.5)))
   (*
    y_s
    (if (<= (* (/ y_m x) (cosh x)) 2e+298)
      (/ (* (fma t_0 x (/ 1.0 x)) y_m) z)
      (* (/ (/ (fma t_0 (* x x) 1.0) z) x) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = fma(0.041666666666666664, (x * x), 0.5);
	double tmp;
	if (((y_m / x) * cosh(x)) <= 2e+298) {
		tmp = (fma(t_0, x, (1.0 / x)) * y_m) / z;
	} else {
		tmp = ((fma(t_0, (x * x), 1.0) / z) / x) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = fma(0.041666666666666664, Float64(x * x), 0.5)
	tmp = 0.0
	if (Float64(Float64(y_m / x) * cosh(x)) <= 2e+298)
		tmp = Float64(Float64(fma(t_0, x, Float64(1.0 / x)) * y_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(t_0, Float64(x * x), 1.0) / z) / x) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], 2e+298], N[(N[(N[(t$95$0 * x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(t$95$0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.9999999999999999e298

   1. Initial program 96.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in x around 0

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

      1. rgt-mult-inverse N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. times-frac N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. /-rgt-identity N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. distribute-rgt-in N/A

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

   7. Applied rewrites 89.8%

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

   if 1.9999999999999999e298 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 88.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{y\_m}{x} \cdot \cosh x}{z} \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y\_m}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y\_m}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (/ y_m x) (cosh x)) z) 5e+84)
    (/ (* (fma 0.5 (* x x) 1.0) y_m) (* z x))
    (/ (/ (* (fma (* x x) 0.5 1.0) y_m) z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((((y_m / x) * cosh(x)) / z) <= 5e+84) {
		tmp = (fma(0.5, (x * x), 1.0) * y_m) / (z * x);
	} else {
		tmp = ((fma((x * x), 0.5, 1.0) * y_m) / z) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(y_m / x) * cosh(x)) / z) <= 5e+84)
		tmp = Float64(Float64(fma(0.5, Float64(x * x), 1.0) * y_m) / Float64(z * x));
	else
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 0.5, 1.0) * y_m) / z) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[(y$95$m / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e+84], N[(N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.0000000000000001e84

   1. Initial program 96.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   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 81.2

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

   5. Applied rewrites 81.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 76.8

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

   7. Applied rewrites 76.8%

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

   if 5.0000000000000001e84 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 71.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 61.6

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

   5. Applied rewrites 61.6%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.5

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

   7. Applied rewrites 76.5%

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

   8. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-/.f64 N/A

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

   10. Applied rewrites 78.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x} \cdot \cosh x \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 0.5, \frac{1}{x}\right) \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y\_m}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* (/ y_m x) (cosh x)) 2e+298)
    (/ (* (fma x 0.5 (/ 1.0 x)) y_m) z)
    (/ (* (fma 0.5 (* x x) 1.0) y_m) (* z x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((y_m / x) * cosh(x)) <= 2e+298) {
		tmp = (fma(x, 0.5, (1.0 / x)) * y_m) / z;
	} else {
		tmp = (fma(0.5, (x * x), 1.0) * y_m) / (z * x);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x) * cosh(x)) <= 2e+298)
		tmp = Float64(Float64(fma(x, 0.5, Float64(1.0 / x)) * y_m) / z);
	else
		tmp = Float64(Float64(fma(0.5, Float64(x * x), 1.0) * y_m) / Float64(z * x));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], 2e+298], N[(N[(N[(x * 0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.9999999999999999e298

   1. Initial program 96.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-out N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 79.4%

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

   if 1.9999999999999999e298 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 47.7

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

   5. Applied rewrites 47.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   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. 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 47.4

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

   7. Applied rewrites 47.4%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 0.5, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x} \cdot \cosh x \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y\_m}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* (/ y_m x) (cosh x)) 5e+286)
    (/ (/ y_m x) z)
    (/ (* (fma 0.5 (* x x) 1.0) y_m) (* z x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((y_m / x) * cosh(x)) <= 5e+286) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (fma(0.5, (x * x), 1.0) * y_m) / (z * x);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x) * cosh(x)) <= 5e+286)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(Float64(fma(0.5, Float64(x * x), 1.0) * y_m) / Float64(z * x));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], 5e+286], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.0000000000000004e286

   1. Initial program 96.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 67.2

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

   5. Applied rewrites 67.2%

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

   if 5.0000000000000004e286 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 48.7

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

   5. Applied rewrites 48.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   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. 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 48.4

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

   7. Applied rewrites 48.4%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x} \cdot \cosh x \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{z \cdot x} \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* (/ y_m x) (cosh x)) 2e+298)
    (/ (/ y_m x) z)
    (* (/ (fma (* x x) 0.5 1.0) (* z x)) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((y_m / x) * cosh(x)) <= 2e+298) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (fma((x * x), 0.5, 1.0) / (z * x)) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x) * cosh(x)) <= 2e+298)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(Float64(fma(Float64(x * x), 0.5, 1.0) / Float64(z * x)) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], 2e+298], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.9999999999999999e298

   1. Initial program 96.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 67.6

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

   5. Applied rewrites 67.6%

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

   if 1.9999999999999999e298 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 70.9

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

   4. Applied rewrites 70.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.5

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

   7. Applied rewrites 47.5%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{z \cdot x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\cosh x \cdot y\_m}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y\_m}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 2e+51)
    (/ (* (cosh x) y_m) (* z x))
    (/
     (*
      (/
       (fma
        (fma
         (fma 0.001388888888888889 (* x x) 0.041666666666666664)
         (* x x)
         0.5)
        (* x x)
        1.0)
       x)
      y_m)
     z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2e+51) {
		tmp = (cosh(x) * y_m) / (z * x);
	} else {
		tmp = ((fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) / x) * y_m) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2e+51)
		tmp = Float64(Float64(cosh(x) * y_m) / Float64(z * x));
	else
		tmp = Float64(Float64(Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) / x) * y_m) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2e+51], N[(N[(N[Cosh[x], $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2e51

   1. Initial program 87.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 87.0

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

   4. Applied rewrites 87.0%

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

   if 2e51 < x

   1. Initial program 76.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\cosh x}{z \cdot x} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y\_m}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 2e+51)
    (* (/ (cosh x) (* z x)) y_m)
    (/
     (*
      (/
       (fma
        (fma
         (fma 0.001388888888888889 (* x x) 0.041666666666666664)
         (* x x)
         0.5)
        (* x x)
        1.0)
       x)
      y_m)
     z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2e+51) {
		tmp = (cosh(x) / (z * x)) * y_m;
	} else {
		tmp = ((fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) / x) * y_m) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2e+51)
		tmp = Float64(Float64(cosh(x) / Float64(z * x)) * y_m);
	else
		tmp = Float64(Float64(Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) / x) * y_m) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2e+51], N[(N[(N[Cosh[x], $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2e51

   1. Initial program 87.7%

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

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 86.6

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

   4. Applied rewrites 86.6%

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

   if 2e51 < x

   1. Initial program 76.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\cosh x}{z \cdot x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0, x \cdot x, 1\right) \cdot y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x \cdot x}{z}, t\_0, \frac{1}{z}\right)}{x} \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0
         (fma
          (fma 0.001388888888888889 (* x x) 0.041666666666666664)
          (* x x)
          0.5)))
   (*
    y_s
    (if (<= y_m 1.6e+69)
      (/ (/ (* (fma t_0 (* x x) 1.0) y_m) x) z)
      (* (/ (fma (/ (* x x) z) t_0 (/ 1.0 z)) x) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5);
	double tmp;
	if (y_m <= 1.6e+69) {
		tmp = ((fma(t_0, (x * x), 1.0) * y_m) / x) / z;
	} else {
		tmp = (fma(((x * x) / z), t_0, (1.0 / z)) / x) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5)
	tmp = 0.0
	if (y_m <= 1.6e+69)
		tmp = Float64(Float64(Float64(fma(t_0, Float64(x * x), 1.0) * y_m) / x) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x * x) / z), t_0, Float64(1.0 / z)) / x) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 1.6e+69], N[(N[(N[(N[(t$95$0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] / z), $MachinePrecision] * t$95$0 + N[(1.0 / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.59999999999999992e69

   1. Initial program 83.9%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in x around 0

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

      1. rgt-mult-inverse N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. times-frac N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. /-rgt-identity N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

   7. Applied rewrites 89.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 92.2%

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

   if 1.59999999999999992e69 < y

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 88.0

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 61.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.0%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x \cdot x}{z}, \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \frac{1}{z}\right)}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.8 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y\_m}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.8e+123)
    (/
     (/
      (*
       (fma
        (fma
         (fma 0.001388888888888889 (* x x) 0.041666666666666664)
         (* x x)
         0.5)
        (* x x)
        1.0)
       y_m)
      x)
     z)
    (/ (/ (* (fma (* x x) 0.5 1.0) y_m) z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.8e+123) {
		tmp = ((fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * y_m) / x) / z;
	} else {
		tmp = ((fma((x * x), 0.5, 1.0) * y_m) / z) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.8e+123)
		tmp = Float64(Float64(Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * y_m) / x) / z);
	else
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 0.5, 1.0) * y_m) / z) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.8e+123], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.79999999999999999e123

   1. Initial program 84.4%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in x around 0

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

      1. rgt-mult-inverse N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. times-frac N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. /-rgt-identity N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

   7. Applied rewrites 90.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 92.4%

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

   if 1.79999999999999999e123 < y

   1. Initial program 89.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 87.2

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

   5. Applied rewrites 87.2%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 95.7

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

   7. Applied rewrites 95.7%

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

   8. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-/.f64 N/A

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

   10. Applied rewrites 95.6%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 93.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.9 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y\_m}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.9e+123)
    (/
     (*
      (/
       (fma
        (fma
         (fma 0.001388888888888889 (* x x) 0.041666666666666664)
         (* x x)
         0.5)
        (* x x)
        1.0)
       x)
      y_m)
     z)
    (/ (/ (* (fma (* x x) 0.5 1.0) y_m) z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.9e+123) {
		tmp = ((fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) / x) * y_m) / z;
	} else {
		tmp = ((fma((x * x), 0.5, 1.0) * y_m) / z) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.9e+123)
		tmp = Float64(Float64(Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) / x) * y_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 0.5, 1.0) * y_m) / z) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.9e+123], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.89999999999999997e123

   1. Initial program 84.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 91.4%

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

   if 1.89999999999999997e123 < y

   1. Initial program 89.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 87.2

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

   5. Applied rewrites 87.2%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 95.7

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

   7. Applied rewrites 95.7%

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

   8. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-/.f64 N/A

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

   10. Applied rewrites 95.6%

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

Alternative 14: 92.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.9 \cdot 10^{+123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x, \frac{1}{x}\right) \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y\_m}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.9e+123)
    (/
     (*
      (fma
       (fma
        (fma 0.001388888888888889 (* x x) 0.041666666666666664)
        (* x x)
        0.5)
       x
       (/ 1.0 x))
      y_m)
     z)
    (/ (/ (* (fma (* x x) 0.5 1.0) y_m) z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.9e+123) {
		tmp = (fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), x, (1.0 / x)) * y_m) / z;
	} else {
		tmp = ((fma((x * x), 0.5, 1.0) * y_m) / z) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.9e+123)
		tmp = Float64(Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), x, Float64(1.0 / x)) * y_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 0.5, 1.0) * y_m) / z) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.9e+123], N[(N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.89999999999999997e123

   1. Initial program 84.4%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in x around 0

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

      1. rgt-mult-inverse N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. times-frac N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. /-rgt-identity N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

   7. Applied rewrites 90.1%

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

   if 1.89999999999999997e123 < y

   1. Initial program 89.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 87.2

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

   5. Applied rewrites 87.2%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 95.7

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

   7. Applied rewrites 95.7%

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

   8. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-/.f64 N/A

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

   10. Applied rewrites 95.6%

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

Alternative 15: 92.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.9 \cdot 10^{+123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), x \cdot x, 0.5\right), x, \frac{1}{x}\right) \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y\_m}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.9e+123)
    (/
     (*
      (fma (fma (* 0.001388888888888889 (* x x)) (* x x) 0.5) x (/ 1.0 x))
      y_m)
     z)
    (/ (/ (* (fma (* x x) 0.5 1.0) y_m) z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.9e+123) {
		tmp = (fma(fma((0.001388888888888889 * (x * x)), (x * x), 0.5), x, (1.0 / x)) * y_m) / z;
	} else {
		tmp = ((fma((x * x), 0.5, 1.0) * y_m) / z) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.9e+123)
		tmp = Float64(Float64(fma(fma(Float64(0.001388888888888889 * Float64(x * x)), Float64(x * x), 0.5), x, Float64(1.0 / x)) * y_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 0.5, 1.0) * y_m) / z) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.9e+123], N[(N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.89999999999999997e123

   1. Initial program 84.4%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in x around 0

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

      1. rgt-mult-inverse N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. times-frac N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. /-rgt-identity N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

   7. Applied rewrites 90.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 89.5%

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

   if 1.89999999999999997e123 < y

   1. Initial program 89.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 87.2

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

   5. Applied rewrites 87.2%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 95.7

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

   7. Applied rewrites 95.7%

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

   8. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-/.f64 N/A

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

   10. Applied rewrites 95.6%

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

Alternative 16: 84.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.62 \cdot 10^{+103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y\_m}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x, \frac{1}{x}\right) \cdot y\_m}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 1.62e+103)
    (/
     (*
      (fma
       (fma
        (fma 0.001388888888888889 (* x x) 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0)
      y_m)
     (* z x))
    (/ (* (fma (fma 0.041666666666666664 (* x x) 0.5) x (/ 1.0 x)) y_m) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.62e+103) {
		tmp = (fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * y_m) / (z * x);
	} else {
		tmp = (fma(fma(0.041666666666666664, (x * x), 0.5), x, (1.0 / x)) * y_m) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.62e+103)
		tmp = Float64(Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * y_m) / Float64(z * x));
	else
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), x, Float64(1.0 / x)) * y_m) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.62e+103], N[(N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.62000000000000007e103

   1. Initial program 87.8%

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

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 87.3

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

   4. Applied rewrites 87.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 56.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.1

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

   9. Applied rewrites 57.1%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 81.9

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

   12. Applied rewrites 81.9%

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

   if 1.62000000000000007e103 < x

   1. Initial program 73.3%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. rgt-mult-inverse N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. times-frac N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. /-rgt-identity N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. distribute-rgt-in N/A

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

   7. Applied rewrites 100.0%

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

Alternative 17: 87.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 6.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{y\_m}{z} \cdot t\_0}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{z}}{x} \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)))
   (*
    y_s
    (if (<= z 6.6e-37) (/ (* (/ y_m z) t_0) x) (* (/ (/ t_0 z) x) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0);
	double tmp;
	if (z <= 6.6e-37) {
		tmp = ((y_m / z) * t_0) / x;
	} else {
		tmp = ((t_0 / z) / x) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0)
	tmp = 0.0
	if (z <= 6.6e-37)
		tmp = Float64(Float64(Float64(y_m / z) * t_0) / x);
	else
		tmp = Float64(Float64(Float64(t_0 / z) / x) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(y$95$s * If[LessEqual[z, 6.6e-37], N[(N[(N[(y$95$m / z), $MachinePrecision] * t$95$0), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(t$95$0 / z), $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.59999999999999964e-37

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 79.2

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

   5. Applied rewrites 79.2%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 87.1

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

   7. Applied rewrites 87.1%

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

   if 6.59999999999999964e-37 < z

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 87.9%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 88.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.5 \cdot 10^{+122}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x, \frac{1}{x}\right) \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y\_m}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 3.5e+122)
    (/ (* (fma (fma 0.041666666666666664 (* x x) 0.5) x (/ 1.0 x)) y_m) z)
    (/ (/ (* (fma (* x x) 0.5 1.0) y_m) z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.5e+122) {
		tmp = (fma(fma(0.041666666666666664, (x * x), 0.5), x, (1.0 / x)) * y_m) / z;
	} else {
		tmp = ((fma((x * x), 0.5, 1.0) * y_m) / z) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 3.5e+122)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), x, Float64(1.0 / x)) * y_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 0.5, 1.0) * y_m) / z) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 3.5e+122], N[(N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.50000000000000014e122

   1. Initial program 84.4%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in x around 0

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

      1. rgt-mult-inverse N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. times-frac N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. /-rgt-identity N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. distribute-rgt-in N/A

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

   7. Applied rewrites 86.7%

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

   if 3.50000000000000014e122 < y

   1. Initial program 89.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 87.2

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

   5. Applied rewrites 87.2%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 95.7

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

   7. Applied rewrites 95.7%

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

   8. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-/.f64 N/A

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

   10. Applied rewrites 95.6%

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

Alternative 19: 81.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y\_m}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y\_m}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 1.3e+154)
    (/
     (* (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) y_m)
     (* z x))
    (/ (/ (* (fma (* x x) 0.5 1.0) y_m) z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.3e+154) {
		tmp = (fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * y_m) / (z * x);
	} else {
		tmp = ((fma((x * x), 0.5, 1.0) * y_m) / z) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.3e+154)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * y_m) / Float64(z * x));
	else
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 0.5, 1.0) * y_m) / z) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.3e+154], N[(N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.29999999999999994e154

   1. Initial program 87.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 77.9

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

   5. Applied rewrites 77.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   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 77.3

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

   7. Applied rewrites 77.3%

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

   if 1.29999999999999994e154 < x

   1. Initial program 71.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 71.4

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

   5. Applied rewrites 71.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 82.9

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

   7. Applied rewrites 82.9%

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

   8. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-/.f64 N/A

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

   10. Applied rewrites 100.0%

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

Alternative 20: 80.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x, x, 1\right)}{z \cdot x} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y\_m}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 1.3e+154)
    (*
     (/ (fma (* (fma 0.041666666666666664 (* x x) 0.5) x) x 1.0) (* z x))
     y_m)
    (/ (/ (* (fma (* x x) 0.5 1.0) y_m) z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.3e+154) {
		tmp = (fma((fma(0.041666666666666664, (x * x), 0.5) * x), x, 1.0) / (z * x)) * y_m;
	} else {
		tmp = ((fma((x * x), 0.5, 1.0) * y_m) / z) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.3e+154)
		tmp = Float64(Float64(fma(Float64(fma(0.041666666666666664, Float64(x * x), 0.5) * x), x, 1.0) / Float64(z * x)) * y_m);
	else
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 0.5, 1.0) * y_m) / z) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.3e+154], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.29999999999999994e154

   1. Initial program 87.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 86.9

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

   4. Applied rewrites 86.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 76.6

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

   7. Applied rewrites 76.6%

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

   9. Applied rewrites 76.6%

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

   if 1.29999999999999994e154 < x

   1. Initial program 71.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 71.4

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

   5. Applied rewrites 71.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 82.9

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

   7. Applied rewrites 82.9%

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

   8. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-/.f64 N/A

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

   10. Applied rewrites 100.0%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x, x, 1\right)}{z \cdot x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 80.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), x \cdot x, 1\right)}{z \cdot x} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y\_m}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 1.3e+154)
    (* (/ (fma (* 0.041666666666666664 (* x x)) (* x x) 1.0) (* z x)) y_m)
    (/ (/ (* (fma (* x x) 0.5 1.0) y_m) z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.3e+154) {
		tmp = (fma((0.041666666666666664 * (x * x)), (x * x), 1.0) / (z * x)) * y_m;
	} else {
		tmp = ((fma((x * x), 0.5, 1.0) * y_m) / z) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.3e+154)
		tmp = Float64(Float64(fma(Float64(0.041666666666666664 * Float64(x * x)), Float64(x * x), 1.0) / Float64(z * x)) * y_m);
	else
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 0.5, 1.0) * y_m) / z) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.3e+154], N[(N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.29999999999999994e154

   1. Initial program 87.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 86.9

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

   4. Applied rewrites 86.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 76.6

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

   7. Applied rewrites 76.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 75.6%

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

   if 1.29999999999999994e154 < x

   1. Initial program 71.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 71.4

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

   5. Applied rewrites 71.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 82.9

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

   7. Applied rewrites 82.9%

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

   8. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-/.f64 N/A

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

   10. Applied rewrites 100.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), x \cdot x, 1\right)}{z \cdot x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 70.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y\_m}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{y\_m}{x}}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 2.7e+108)
    (/ (* (fma 0.5 (* x x) 1.0) y_m) (* z x))
    (/ (* (* 0.5 (* x x)) (/ y_m x)) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.7e+108) {
		tmp = (fma(0.5, (x * x), 1.0) * y_m) / (z * x);
	} else {
		tmp = ((0.5 * (x * x)) * (y_m / x)) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.7e+108)
		tmp = Float64(Float64(fma(0.5, Float64(x * x), 1.0) * y_m) / Float64(z * x));
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(x * x)) * Float64(y_m / x)) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.7e+108], N[(N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.7e108

   1. Initial program 87.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 70.0

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

   5. Applied rewrites 70.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 70.2

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

   7. Applied rewrites 70.2%

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

   if 2.7e108 < x

   1. Initial program 74.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   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 65.6

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

   5. Applied rewrites 65.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.6%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{y}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 57.3% accurate, 4.6× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 1.4) (/ (* 1.0 y_m) (* z x)) (/ (* (* 0.5 x) y_m) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = (1.0 * y_m) / (z * x);
	} else {
		tmp = ((0.5 * x) * y_m) / z;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = (1.0d0 * y_m) / (z * x)
    else
        tmp = ((0.5d0 * x) * y_m) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = (1.0 * y_m) / (z * x);
	} else {
		tmp = ((0.5 * x) * y_m) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 1.4:
		tmp = (1.0 * y_m) / (z * x)
	else:
		tmp = ((0.5 * x) * y_m) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(Float64(1.0 * y_m) / Float64(z * x));
	else
		tmp = Float64(Float64(Float64(0.5 * x) * y_m) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = (1.0 * y_m) / (z * x);
	else
		tmp = ((0.5 * x) * y_m) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.4], N[(N[(1.0 * y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 86.7%

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

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 85.6

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 62.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.4

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

   9. Applied rewrites 62.4%

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

   if 1.3999999999999999 < x

   1. Initial program 81.2%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lft-mult-inverse N/A

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

      2. distribute-rgt-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. unpow2 N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. associate-*r/ N/A

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

      15. rgt-mult-inverse N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 37.5

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

   7. Applied rewrites 37.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 37.5%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{1 \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\right) \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 48.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \frac{1 \cdot y\_m}{z \cdot x} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (/ (* 1.0 y_m) (* z x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * ((1.0 * y_m) / (z * x));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * ((1.0d0 * y_m) / (z * x))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * ((1.0 * y_m) / (z * x));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * ((1.0 * y_m) / (z * x))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(Float64(1.0 * y_m) / Float64(z * x)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * ((1.0 * y_m) / (z * x));
end

Wolfram

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

Derivation

1. Initial program 85.2%

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

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 82.5

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

4. Applied rewrites 82.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 47.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 47.5

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

9. Applied rewrites 47.5%

   \[\leadsto \color{blue}{\frac{1 \cdot y}{z \cdot x}} \]
10. Add Preprocessing

Alternative 25: 48.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(\frac{1}{z \cdot x} \cdot y\_m\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (* (/ 1.0 (* z x)) y_m)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * ((1.0 / (z * x)) * y_m);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * ((1.0d0 / (z * x)) * y_m)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * ((1.0 / (z * x)) * y_m);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * ((1.0 / (z * x)) * y_m)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(Float64(1.0 / Float64(z * x)) * y_m))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * ((1.0 / (z * x)) * y_m);
end

Wolfram

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

Derivation

1. Initial program 85.2%

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

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

   5. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\color{blue}{y \cdot \cosh x}}{z \cdot x} \]

   7. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{z \cdot x}} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{z \cdot x}} \]

   9. lower-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{\cosh x}{z \cdot x}} \]

   10. lower-*.f64 82.5

       \[\leadsto y \cdot \frac{\cosh x}{\color{blue}{z \cdot x}} \]

4. Applied rewrites 82.5%

   \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{z \cdot x}} \]

5. Taylor expanded in x around 0

   \[\leadsto y \cdot \frac{\color{blue}{1}}{z \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 47.2%

   \[\leadsto y \cdot \frac{\color{blue}{1}}{z \cdot x} \]

8. Final simplification 47.2%

   \[\leadsto \frac{1}{z \cdot x} \cdot y \]
9. Add Preprocessing

Developer Target 1: 97.3% 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 2024249 
(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))