Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.8% → 99.5%

Time: 11.4s

Alternatives: 25

Speedup: 2.6×

• Could not converge on a ground truth

  1. x = 3.6743789190157625e+103
  2. y = 2.7796911534866125e-82
  3. z = 2.2256927229236908e-194

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 4 \cdot 10^{+287}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right) \cdot x\_m, x\_m, 1\right) \cdot \frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \cosh x\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (/ y_m x_m) (cosh x_m)) 4e+287)
     (/
      (*
       (fma (* (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m) x_m 1.0)
       (/ y_m x_m))
      z)
     (/ (/ (* y_m (cosh x_m)) z) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 4e+287) {
		tmp = (fma((fma(0.041666666666666664, (x_m * x_m), 0.5) * x_m), x_m, 1.0) * (y_m / x_m)) / z;
	} else {
		tmp = ((y_m * cosh(x_m)) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 4e+287)
		tmp = Float64(Float64(fma(Float64(fma(0.041666666666666664, Float64(x_m * x_m), 0.5) * x_m), x_m, 1.0) * Float64(y_m / x_m)) / z);
	else
		tmp = Float64(Float64(Float64(y_m * cosh(x_m)) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 4e+287], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y$95$m * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 4.0000000000000003e287

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   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 89.8

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

   5. Applied rewrites 89.8%

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

   7. Applied rewrites 89.8%

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

   if 4.0000000000000003e287 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 67.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 4 \cdot 10^{+287}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \cosh x}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right) \cdot x\_m, x\_m, 1\right) \cdot \frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)} \cdot x\_m} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+268)
     (/
      (*
       (fma (* (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m) x_m 1.0)
       (/ y_m x_m))
      z)
     (*
      (/
       1.0
       (*
        (/
         z
         (fma
          (fma
           (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
           (* x_m x_m)
           0.5)
          (* x_m x_m)
          1.0))
        x_m))
      y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+268) {
		tmp = (fma((fma(0.041666666666666664, (x_m * x_m), 0.5) * x_m), x_m, 1.0) * (y_m / x_m)) / z;
	} else {
		tmp = (1.0 / ((z / fma(fma(fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0)) * x_m)) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+268)
		tmp = Float64(Float64(fma(Float64(fma(0.041666666666666664, Float64(x_m * x_m), 0.5) * x_m), x_m, 1.0) * Float64(y_m / x_m)) / z);
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(z / fma(fma(fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0)) * x_m)) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+268], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(1.0 / N[(N[(z / N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.9999999999999999e268

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   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 89.8

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

   5. Applied rewrites 89.8%

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

   7. Applied rewrites 89.8%

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

   if 1.9999999999999999e268 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 57.1

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

   7. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. div-inv N/A

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

      7. *-lft-identity N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 88.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right) \cdot x\_m, x\_m, 1\right) \cdot \frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)} \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+268)
     (/
      (*
       (fma (* (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m) x_m 1.0)
       (/ y_m x_m))
      z)
     (/
      y_m
      (*
       (/
        z
        (fma
         (fma
          (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
          (* x_m x_m)
          0.5)
         (* x_m x_m)
         1.0))
       x_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+268) {
		tmp = (fma((fma(0.041666666666666664, (x_m * x_m), 0.5) * x_m), x_m, 1.0) * (y_m / x_m)) / z;
	} else {
		tmp = y_m / ((z / fma(fma(fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0)) * x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+268)
		tmp = Float64(Float64(fma(Float64(fma(0.041666666666666664, Float64(x_m * x_m), 0.5) * x_m), x_m, 1.0) * Float64(y_m / x_m)) / z);
	else
		tmp = Float64(y_m / Float64(Float64(z / fma(fma(fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0)) * x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+268], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(N[(z / N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.9999999999999999e268

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   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 89.8

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

   5. Applied rewrites 89.8%

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

   7. Applied rewrites 89.8%

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

   if 1.9999999999999999e268 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 57.1

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

   7. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 88.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot x\_m, x\_m, 1\right) \cdot \frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z}{\mathsf{fma}\left(t\_0, x\_m \cdot x\_m, 1\right)} \cdot x\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (fma 0.041666666666666664 (* x_m x_m) 0.5)))
   (*
    x_s
    (*
     y_s
     (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+268)
       (/ (* (fma (* t_0 x_m) x_m 1.0) (/ y_m x_m)) z)
       (/ y_m (* (/ z (fma t_0 (* x_m x_m) 1.0)) x_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = fma(0.041666666666666664, (x_m * x_m), 0.5);
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+268) {
		tmp = (fma((t_0 * x_m), x_m, 1.0) * (y_m / x_m)) / z;
	} else {
		tmp = y_m / ((z / fma(t_0, (x_m * x_m), 1.0)) * x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	t_0 = fma(0.041666666666666664, Float64(x_m * x_m), 0.5)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+268)
		tmp = Float64(Float64(fma(Float64(t_0 * x_m), x_m, 1.0) * Float64(y_m / x_m)) / z);
	else
		tmp = Float64(y_m / Float64(Float64(z / fma(t_0, Float64(x_m * x_m), 1.0)) * x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+268], N[(N[(N[(N[(t$95$0 * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(N[(z / N[(t$95$0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.9999999999999999e268

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   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 89.8

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

   5. Applied rewrites 89.8%

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

   7. Applied rewrites 89.8%

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

   if 1.9999999999999999e268 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 57.1

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

   7. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 88.0%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 85.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 10^{+186}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, x\_m, \frac{1}{x\_m}\right) \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z}{\mathsf{fma}\left(t\_0, x\_m \cdot x\_m, 1\right)} \cdot x\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (fma 0.041666666666666664 (* x_m x_m) 0.5)))
   (*
    x_s
    (*
     y_s
     (if (<= (* (/ y_m x_m) (cosh x_m)) 1e+186)
       (/ (* (fma t_0 x_m (/ 1.0 x_m)) y_m) z)
       (/ y_m (* (/ z (fma t_0 (* x_m x_m) 1.0)) x_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = fma(0.041666666666666664, (x_m * x_m), 0.5);
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 1e+186) {
		tmp = (fma(t_0, x_m, (1.0 / x_m)) * y_m) / z;
	} else {
		tmp = y_m / ((z / fma(t_0, (x_m * x_m), 1.0)) * x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	t_0 = fma(0.041666666666666664, Float64(x_m * x_m), 0.5)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 1e+186)
		tmp = Float64(Float64(fma(t_0, x_m, Float64(1.0 / x_m)) * y_m) / z);
	else
		tmp = Float64(y_m / Float64(Float64(z / fma(t_0, Float64(x_m * x_m), 1.0)) * x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 1e+186], N[(N[(N[(t$95$0 * x$95$m + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(N[(z / N[(t$95$0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 9.9999999999999998e185

   1. Initial program 95.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 60.5

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.1%

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

   if 9.9999999999999998e185 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 70.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 60.3

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

   7. Applied rewrites 60.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 88.9%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 87.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 10^{+186}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 4 \cdot 10^{+287}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, x\_m, \frac{1}{x\_m}\right) \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0, x\_m \cdot x\_m, 1\right)}{z}}{x\_m} \cdot y\_m\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (fma 0.041666666666666664 (* x_m x_m) 0.5)))
   (*
    x_s
    (*
     y_s
     (if (<= (* (/ y_m x_m) (cosh x_m)) 4e+287)
       (/ (* (fma t_0 x_m (/ 1.0 x_m)) y_m) z)
       (* (/ (/ (fma t_0 (* x_m x_m) 1.0) z) x_m) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = fma(0.041666666666666664, (x_m * x_m), 0.5);
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 4e+287) {
		tmp = (fma(t_0, x_m, (1.0 / x_m)) * y_m) / z;
	} else {
		tmp = ((fma(t_0, (x_m * x_m), 1.0) / z) / x_m) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	t_0 = fma(0.041666666666666664, Float64(x_m * x_m), 0.5)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 4e+287)
		tmp = Float64(Float64(fma(t_0, x_m, Float64(1.0 / x_m)) * y_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(t_0, Float64(x_m * x_m), 1.0) / z) / x_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 4e+287], N[(N[(N[(t$95$0 * x$95$m + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(t$95$0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 4.0000000000000003e287

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.7%

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

   if 4.0000000000000003e287 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 67.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 85.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 4 \cdot 10^{+287}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 4 \cdot 10^{+287}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right) \cdot \frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (/ y_m x_m) (cosh x_m)) 4e+287)
     (/ (* (fma (* x_m x_m) 0.5 1.0) (/ y_m x_m)) z)
     (/ (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) z) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 4e+287) {
		tmp = (fma((x_m * x_m), 0.5, 1.0) * (y_m / x_m)) / z;
	} else {
		tmp = ((fma(0.5, (x_m * x_m), 1.0) * y_m) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 4e+287)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) * Float64(y_m / x_m)) / z);
	else
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 4e+287], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 4.0000000000000003e287

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if 4.0000000000000003e287 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 67.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 44.5

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

   5. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 4 \cdot 10^{+287}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right) \cdot \frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)}{z} \cdot y\_m}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (/ y_m x_m) (cosh x_m)) 4e+293)
     (/ (* (fma (* x_m x_m) 0.5 1.0) (/ y_m x_m)) z)
     (/ (* (/ (fma 0.5 (* x_m x_m) 1.0) z) y_m) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 4e+293) {
		tmp = (fma((x_m * x_m), 0.5, 1.0) * (y_m / x_m)) / z;
	} else {
		tmp = ((fma(0.5, (x_m * x_m), 1.0) / z) * y_m) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 4e+293)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) * Float64(y_m / x_m)) / z);
	else
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) / z) * y_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 4e+293], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 3.9999999999999997e293

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if 3.9999999999999997e293 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 67.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 44.5

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

   5. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.5

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

   9. Applied rewrites 76.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z} \cdot y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m \cdot x\_m, 0.5, \frac{y\_m}{x\_m}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)}{z} \cdot y\_m}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (/ y_m x_m) (cosh x_m)) 4e+293)
     (/ (fma (* y_m x_m) 0.5 (/ y_m x_m)) z)
     (/ (* (/ (fma 0.5 (* x_m x_m) 1.0) z) y_m) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 4e+293) {
		tmp = fma((y_m * x_m), 0.5, (y_m / x_m)) / z;
	} else {
		tmp = ((fma(0.5, (x_m * x_m), 1.0) / z) * y_m) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 4e+293)
		tmp = Float64(fma(Float64(y_m * x_m), 0.5, Float64(y_m / x_m)) / z);
	else
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) / z) * y_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 4e+293], N[(N[(N[(y$95$m * x$95$m), $MachinePrecision] * 0.5 + N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 3.9999999999999997e293

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-rgt-identity N/A

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

      13. associate-*r/ N/A

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

      14. rgt-mult-inverse N/A

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

      15. *-rgt-identity N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-in N/A

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

   8. Applied rewrites 81.9%

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

   if 3.9999999999999997e293 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 67.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 44.5

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

   5. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.5

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

   9. Applied rewrites 76.5%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 0.5, \frac{y}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z} \cdot y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m \cdot x\_m, 0.5, \frac{y\_m}{x\_m}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z}{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)} \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+268)
     (/ (fma (* y_m x_m) 0.5 (/ y_m x_m)) z)
     (/ y_m (* (/ z (fma (* x_m x_m) 0.5 1.0)) x_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+268) {
		tmp = fma((y_m * x_m), 0.5, (y_m / x_m)) / z;
	} else {
		tmp = y_m / ((z / fma((x_m * x_m), 0.5, 1.0)) * x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+268)
		tmp = Float64(fma(Float64(y_m * x_m), 0.5, Float64(y_m / x_m)) / z);
	else
		tmp = Float64(y_m / Float64(Float64(z / fma(Float64(x_m * x_m), 0.5, 1.0)) * x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+268], N[(N[(N[(y$95$m * x$95$m), $MachinePrecision] * 0.5 + N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(N[(z / N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.9999999999999999e268

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 62.1

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-rgt-identity N/A

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

      13. associate-*r/ N/A

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

      14. rgt-mult-inverse N/A

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

      15. *-rgt-identity N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-in N/A

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

   8. Applied rewrites 81.8%

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

   if 1.9999999999999999e268 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 57.1

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

   7. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 88.0%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 72.9

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

   12. Applied rewrites 72.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 0.5, \frac{y}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m \cdot x\_m, 0.5, \frac{y\_m}{x\_m}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{x\_m}{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)} \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+268)
     (/ (fma (* y_m x_m) 0.5 (/ y_m x_m)) z)
     (/ y_m (* (/ x_m (fma 0.5 (* x_m x_m) 1.0)) z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+268) {
		tmp = fma((y_m * x_m), 0.5, (y_m / x_m)) / z;
	} else {
		tmp = y_m / ((x_m / fma(0.5, (x_m * x_m), 1.0)) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+268)
		tmp = Float64(fma(Float64(y_m * x_m), 0.5, Float64(y_m / x_m)) / z);
	else
		tmp = Float64(y_m / Float64(Float64(x_m / fma(0.5, Float64(x_m * x_m), 1.0)) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+268], N[(N[(N[(y$95$m * x$95$m), $MachinePrecision] * 0.5 + N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(N[(x$95$m / N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.9999999999999999e268

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 62.1

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-rgt-identity N/A

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

      13. associate-*r/ N/A

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

      14. rgt-mult-inverse N/A

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

      15. *-rgt-identity N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-in N/A

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

   8. Applied rewrites 81.8%

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

   if 1.9999999999999999e268 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 45.1

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

   5. Applied rewrites 45.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 64.6

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

   9. Applied rewrites 64.6%

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

       1. lift-/.f64 N/A

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

       2. div-inv N/A

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

       3. lift-/.f64 N/A

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

       4. clear-num N/A

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

       5. lift-/.f64 N/A

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

       6. clear-num N/A

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

       7. frac-times N/A

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

       8. *-lft-identity N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 70.9

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

   11. Applied rewrites 70.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 0.5, \frac{y}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{x}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.0

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

   5. Applied rewrites 77.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 76.5

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

   7. Applied rewrites 76.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 0.3

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

   5. Applied rewrites 0.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 80.4

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

   7. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 48.4

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

   9. Applied rewrites 48.4%

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

4. Final simplification 73.8%

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

Alternative 13: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 10^{+186}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (/ y_m x_m) (cosh x_m)) 1e+186)
     (/ (/ y_m x_m) z)
     (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) (* z x_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 1e+186) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = (fma(0.5, (x_m * x_m), 1.0) * y_m) / (z * x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 1e+186)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / Float64(z * x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 1e+186], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 9.9999999999999998e185

   1. Initial program 95.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 60.5

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

   5. Applied rewrites 60.5%

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

   if 9.9999999999999998e185 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 70.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 49.2

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

   5. Applied rewrites 49.2%

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

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 58.4%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 10^{+186}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)}{z \cdot x\_m} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (/ y_m x_m) (cosh x_m)) 5e+255)
     (/ (/ y_m x_m) z)
     (* (/ (fma 0.5 (* x_m x_m) 1.0) (* z x_m)) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 5e+255) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = (fma(0.5, (x_m * x_m), 1.0) / (z * x_m)) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 5e+255)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) / Float64(z * x_m)) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 5e+255], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.0000000000000002e255

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if 5.0000000000000002e255 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 69.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.1

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

   5. Applied rewrites 47.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 77.4

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

   7. Applied rewrites 77.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 54.8

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

   9. Applied rewrites 54.8%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}{z \cdot x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 4 \cdot 10^{+287}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (/ y_m x_m) (cosh x_m)) 4e+287)
     (/ (/ y_m x_m) z)
     (/ (/ y_m z) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 4e+287) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = (y_m / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((y_m / x_m) * cosh(x_m)) <= 4d+287) then
        tmp = (y_m / x_m) / z
    else
        tmp = (y_m / z) / x_m
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m / x_m) * Math.cosh(x_m)) <= 4e+287) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = (y_m / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if ((y_m / x_m) * math.cosh(x_m)) <= 4e+287:
		tmp = (y_m / x_m) / z
	else:
		tmp = (y_m / z) / x_m
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 4e+287)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(Float64(y_m / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (((y_m / x_m) * cosh(x_m)) <= 4e+287)
		tmp = (y_m / x_m) / z;
	else
		tmp = (y_m / z) / x_m;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 4e+287], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 4.0000000000000003e287

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 62.4

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

   5. Applied rewrites 62.4%

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

   if 4.0000000000000003e287 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 67.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 28.0

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

   7. Applied rewrites 28.0%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 4 \cdot 10^{+287}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 52.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+268)
     (/ (/ y_m x_m) z)
     (/ y_m (* z x_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+268) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = y_m / (z * x_m);
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((y_m / x_m) * cosh(x_m)) <= 2d+268) then
        tmp = (y_m / x_m) / z
    else
        tmp = y_m / (z * x_m)
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m / x_m) * Math.cosh(x_m)) <= 2e+268) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = y_m / (z * x_m);
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if ((y_m / x_m) * math.cosh(x_m)) <= 2e+268:
		tmp = (y_m / x_m) / z
	else:
		tmp = y_m / (z * x_m)
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+268)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(y_m / Float64(z * x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+268)
		tmp = (y_m / x_m) / z;
	else
		tmp = y_m / (z * x_m);
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+268], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.9999999999999999e268

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 62.1

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

   5. Applied rewrites 62.1%

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

   if 1.9999999999999999e268 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 57.1

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

   7. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 88.0%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 25.8

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

   12. Applied rewrites 25.8%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2e51

   1. Initial program 87.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 88.5

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

   4. Applied rewrites 88.5%

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

   if 2e51 < x

   1. Initial program 76.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

Alternative 18: 93.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 210000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= y_m 210000000.0)
     (/
      (*
       (/
        (fma
         (fma (* 0.001388888888888889 (* x_m x_m)) (* x_m x_m) 0.5)
         (* x_m x_m)
         1.0)
        x_m)
       y_m)
      z)
     (*
      (/ y_m z)
      (/
       (fma
        (fma
         (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
         (* x_m x_m)
         0.5)
        (* x_m x_m)
        1.0)
       x_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 210000000.0) {
		tmp = ((fma(fma((0.001388888888888889 * (x_m * x_m)), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y_m) / z;
	} else {
		tmp = (y_m / z) * (fma(fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 210000000.0)
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.001388888888888889 * Float64(x_m * x_m)), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) * y_m) / z);
	else
		tmp = Float64(Float64(y_m / z) * Float64(fma(fma(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[y$95$m, 210000000.0], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.1e8

   1. Initial program 83.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 88.6%

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

   if 2.1e8 < y

   1. Initial program 90.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 87.5

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

   5. Applied rewrites 87.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 92.9

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

   7. Applied rewrites 92.9%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 210000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 92.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot \frac{y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= y_m 1.5e-9)
     (/
      (*
       (/
        (fma
         (fma (* 0.001388888888888889 (* x_m x_m)) (* x_m x_m) 0.5)
         (* x_m x_m)
         1.0)
        x_m)
       y_m)
      z)
     (/
      (*
       (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)
       (/ y_m z))
      x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1.5e-9) {
		tmp = ((fma(fma((0.001388888888888889 * (x_m * x_m)), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y_m) / z;
	} else {
		tmp = (fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) * (y_m / z)) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 1.5e-9)
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.001388888888888889 * Float64(x_m * x_m)), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) * y_m) / z);
	else
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * Float64(y_m / z)) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[y$95$m, 1.5e-9], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.49999999999999999e-9

   1. Initial program 83.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 88.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 88.4%

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

   if 1.49999999999999999e-9 < y

   1. Initial program 90.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. lft-mult-inverse N/A

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

      12. distribute-rgt-in N/A

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

      13. distribute-rgt-out N/A

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

   7. Applied rewrites 93.1%

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

Alternative 20: 89.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 600000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m, \frac{1}{x\_m}\right) \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664, x\_m \cdot x\_m, 1\right) \cdot \frac{y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= y_m 600000000.0)
     (/
      (* (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m (/ 1.0 x_m)) y_m)
      z)
     (/
      (* (fma (* (* x_m x_m) 0.041666666666666664) (* x_m x_m) 1.0) (/ y_m z))
      x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 600000000.0) {
		tmp = (fma(fma(0.041666666666666664, (x_m * x_m), 0.5), x_m, (1.0 / x_m)) * y_m) / z;
	} else {
		tmp = (fma(((x_m * x_m) * 0.041666666666666664), (x_m * x_m), 1.0) * (y_m / z)) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 600000000.0)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), x_m, Float64(1.0 / x_m)) * y_m) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * 0.041666666666666664), Float64(x_m * x_m), 1.0) * Float64(y_m / z)) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[y$95$m, 600000000.0], N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6e8

   1. Initial program 83.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 49.2

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

   5. Applied rewrites 49.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.6%

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

   if 6e8 < y

   1. Initial program 90.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. lft-mult-inverse N/A

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

      12. distribute-rgt-in N/A

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

      13. distribute-rgt-out N/A

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

   7. Applied rewrites 92.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 92.7%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 600000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x \cdot x, 1\right) \cdot \frac{y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 88.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m, \frac{1}{x\_m}\right) \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= y_m 2.5e+137)
     (/
      (* (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m (/ 1.0 x_m)) y_m)
      z)
     (/ (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) z) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 2.5e+137) {
		tmp = (fma(fma(0.041666666666666664, (x_m * x_m), 0.5), x_m, (1.0 / x_m)) * y_m) / z;
	} else {
		tmp = ((fma(0.5, (x_m * x_m), 1.0) * y_m) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 2.5e+137)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), x_m, Float64(1.0 / x_m)) * y_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[y$95$m, 2.5e+137], N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.5000000000000001e137

   1. Initial program 84.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.0%

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

   if 2.5000000000000001e137 < y

   1. Initial program 89.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   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 80.1

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

   5. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 97.7

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

   7. Applied rewrites 97.7%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 85.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{x\_m}{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)} \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 1.35e+154)
     (/
      (* (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0) y_m)
      (* z x_m))
     (/ y_m (* (/ x_m (fma 0.5 (* x_m x_m) 1.0)) z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.35e+154) {
		tmp = (fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) * y_m) / (z * x_m);
	} else {
		tmp = y_m / ((x_m / fma(0.5, (x_m * x_m), 1.0)) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.35e+154)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * y_m) / Float64(z * x_m));
	else
		tmp = Float64(y_m / Float64(Float64(x_m / fma(0.5, Float64(x_m * x_m), 1.0)) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.35e+154], N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(N[(x$95$m / N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   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 77.2

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

   5. Applied rewrites 77.2%

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

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.1

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

   7. Applied rewrites 78.1%

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

   if 1.35000000000000003e154 < x

   1. Initial program 73.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 73.7

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

   5. Applied rewrites 73.7%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 92.1

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

   9. Applied rewrites 92.1%

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

       1. lift-/.f64 N/A

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

       2. div-inv N/A

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

       3. lift-/.f64 N/A

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

       4. clear-num N/A

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

       5. lift-/.f64 N/A

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

       6. clear-num N/A

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

       7. frac-times N/A

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

       8. *-lft-identity N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 23: 69.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \frac{y\_m}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 5.2e+41)
     (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) (* z x_m))
     (/ (* (* 0.5 (* x_m x_m)) (/ y_m x_m)) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 5.2e+41) {
		tmp = (fma(0.5, (x_m * x_m), 1.0) * y_m) / (z * x_m);
	} else {
		tmp = ((0.5 * (x_m * x_m)) * (y_m / x_m)) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 5.2e+41)
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / Float64(z * x_m));
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(x_m * x_m)) * Float64(y_m / x_m)) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 5.2e+41], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.2000000000000001e41

   1. Initial program 87.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 71.9

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 71.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{z \cdot x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\color{blue}{x \cdot z}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{x \cdot z}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}}{x \cdot z} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right) \cdot y}{\color{blue}{z \cdot x}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right) \cdot y}{\color{blue}{z \cdot x}} \]

   7. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}} \]

   if 5.2000000000000001e41 < x

   1. Initial program 76.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 60.3

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 60.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{y}{x}}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.3%

      \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{0.5}\right) \cdot \frac{y}{x}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{y}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 68.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 6 \cdot 10^{+125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, 0.5, \frac{1}{x\_m}\right) \cdot y\_m}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z 6e+125)
     (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) (* z x_m))
     (/ (* (fma x_m 0.5 (/ 1.0 x_m)) y_m) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 6e+125) {
		tmp = (fma(0.5, (x_m * x_m), 1.0) * y_m) / (z * x_m);
	} else {
		tmp = (fma(x_m, 0.5, (1.0 / x_m)) * y_m) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 6e+125)
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / Float64(z * x_m));
	else
		tmp = Float64(Float64(fma(x_m, 0.5, Float64(1.0 / x_m)) * y_m) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z, 6e+125], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * 0.5 + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.0000000000000003e125

   1. Initial program 85.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 68.8

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 68.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. 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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\color{blue}{x \cdot z}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{x \cdot z}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}}{x \cdot z} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right) \cdot y}{\color{blue}{z \cdot x}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right) \cdot y}{\color{blue}{z \cdot x}} \]

   7. Applied rewrites 71.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}} \]

   if 6.0000000000000003e125 < z

   1. Initial program 83.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   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 69.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 0.5, \frac{1}{x}\right) \cdot y}}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 48.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \frac{y\_m}{z \cdot x\_m}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (/ y_m (* z x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (y_m / (z * x_m)));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x_s * (y_s * (y_m / (z * x_m)))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (y_m / (z * x_m)));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	return x_s * (y_s * (y_m / (z * x_m)))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(y_m / Float64(z * x_m))))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, y_s, x_m, y_m, z)
	tmp = x_s * (y_s * (y_m / (z * x_m)));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(y$95$m / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.4%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{y}{x}}{z} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

   4. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

   9. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   14. lower-*.f64 78.6

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

5. Applied rewrites 78.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}{z}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}}{z} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

   5. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{z \cdot x}} \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{z} \cdot \frac{y}{x}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{z} \cdot \color{blue}{\frac{y}{x}} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{z} \cdot \frac{y}{x}} \]

   9. lower-/.f64 78.9

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}} \cdot \frac{y}{x} \]

7. Applied rewrites 78.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{z} \cdot \frac{y}{x}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{z} \cdot \frac{y}{x}} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{z}} \cdot \frac{y}{x} \]

   3. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}}} \cdot \frac{y}{x} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{1}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}} \cdot \color{blue}{\frac{y}{x}} \]

   5. frac-times N/A

      \[\leadsto \color{blue}{\frac{1 \cdot y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)} \cdot x}} \]

   6. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{y}}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)} \cdot x} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)} \cdot x}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{y}{\color{blue}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)} \cdot x}} \]

9. Applied rewrites 90.3%

   \[\leadsto \color{blue}{\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot x}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

    2. lower-*.f64 48.2

       \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

12. Applied rewrites 48.2%

    \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]
13. Add Preprocessing

Developer Target 1: 97.0% 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 2024277 
(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))