Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 85.1% → 95.4%

Time: 9.5s

Alternatives: 15

Speedup: 2.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.12 \cdot 10^{+75}:\\ \;\;\;\;\frac{\cosh x\_m \cdot y}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z}}{x\_m} \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.12e+75)
    (/ (* (cosh x_m) y) (* z x_m))
    (*
     (/
      (/ (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0) z)
      x_m)
     y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.12e+75) {
		tmp = (cosh(x_m) * y) / (z * x_m);
	} else {
		tmp = ((fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) / z) / x_m) * y;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1.12e+75)
		tmp = Float64(Float64(cosh(x_m) * y) / Float64(z * x_m));
	else
		tmp = Float64(Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / z) / x_m) * y);
	end
	return Float64(x_s * tmp)
end

Wolfram

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_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 1.12e+75], N[(N[(N[Cosh[x$95$m], $MachinePrecision] * y), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.12000000000000001e75

   1. Initial program 84.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 86.9

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

   4. Applied rewrites 86.9%

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

   if 1.12000000000000001e75 < x

   1. Initial program 62.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+75}:\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x\_m} \cdot \cosh x\_m}{z} \leq 5 \cdot 10^{-175}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z}}{x\_m} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y}{z}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* (/ y x_m) (cosh x_m)) z) 5e-175)
    (*
     (/
      (/ (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0) z)
      x_m)
     y)
    (/
     (/
      (*
       (fma
        (fma
         (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
         (* x_m x_m)
         0.5)
        (* x_m x_m)
        1.0)
       y)
      z)
     x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((((y / x_m) * cosh(x_m)) / z) <= 5e-175) {
		tmp = ((fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) / z) / x_m) * y;
	} else {
		tmp = ((fma(fma(fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) * y) / z) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(y / x_m) * cosh(x_m)) / z) <= 5e-175)
		tmp = Float64(Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / z) / x_m) * y);
	else
		tmp = Float64(Float64(Float64(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) * y) / z) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

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_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(N[(y / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e-175], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(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] * y), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5e-175

   1. Initial program 94.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 83.6%

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

   if 5e-175 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   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 52.0

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

   5. Applied rewrites 52.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 74.3

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

   7. Applied rewrites 74.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 81.7

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

   9. Applied rewrites 81.7%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 94.3

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

   12. Applied rewrites 94.3%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 5 \cdot 10^{-175}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+167}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right) \cdot \frac{y}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z}}{x\_m} \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* (/ y x_m) (cosh x_m)) 2e+167)
    (/ (* (fma (* x_m x_m) 0.5 1.0) (/ y x_m)) z)
    (*
     (/
      (/ (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0) z)
      x_m)
     y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((y / x_m) * cosh(x_m)) <= 2e+167) {
		tmp = (fma((x_m * x_m), 0.5, 1.0) * (y / x_m)) / z;
	} else {
		tmp = ((fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) / z) / x_m) * y;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(y / x_m) * cosh(x_m)) <= 2e+167)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) * Float64(y / x_m)) / z);
	else
		tmp = Float64(Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / z) / x_m) * y);
	end
	return Float64(x_s * tmp)
end

Wolfram

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_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(y / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+167], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.0000000000000001e167

   1. Initial program 97.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 80.9

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

   5. Applied rewrites 80.9%

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

   if 2.0000000000000001e167 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 59.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 88.2%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+167}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x\_m} \cdot \cosh x\_m}{z} \leq 10^{-20}:\\ \;\;\;\;\frac{y}{z \cdot x\_m} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot y}{z}}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (fma 0.5 (* x_m x_m) 1.0)))
   (*
    x_s
    (if (<= (/ (* (/ y x_m) (cosh x_m)) z) 1e-20)
      (* (/ y (* z x_m)) t_0)
      (/ (/ (* t_0 y) z) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = fma(0.5, (x_m * x_m), 1.0);
	double tmp;
	if ((((y / x_m) * cosh(x_m)) / z) <= 1e-20) {
		tmp = (y / (z * x_m)) * t_0;
	} else {
		tmp = ((t_0 * y) / z) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = fma(0.5, Float64(x_m * x_m), 1.0)
	tmp = 0.0
	if (Float64(Float64(Float64(y / x_m) * cosh(x_m)) / z) <= 1e-20)
		tmp = Float64(Float64(y / Float64(z * x_m)) * t_0);
	else
		tmp = Float64(Float64(Float64(t_0 * y) / z) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

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_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(N[(y / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e-20], N[(N[(y / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(t$95$0 * y), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 9.99999999999999945e-21

   1. Initial program 94.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

   if 9.99999999999999945e-21 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 66.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.5

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

   5. Applied rewrites 47.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   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 80.0

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

   7. Applied rewrites 80.0%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 10^{-20}:\\ \;\;\;\;\frac{y}{z \cdot x} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(y / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 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. 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 74.9

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

   7. Applied rewrites 74.9%

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

   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 0.4

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

   5. Applied rewrites 0.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 76.4

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

   7. Applied rewrites 76.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

   9. Applied rewrites 59.7%

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

4. Final simplification 72.7%

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

Alternative 6: 71.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m, \frac{1}{x\_m}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot x\_m, x\_m, 1\right)}{z \cdot x\_m} \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* (/ y x_m) (cosh x_m)) 2e+287)
    (/ (* (fma 0.5 x_m (/ 1.0 x_m)) y) z)
    (* (/ (fma (* 0.5 x_m) x_m 1.0) (* z x_m)) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((y / x_m) * cosh(x_m)) <= 2e+287) {
		tmp = (fma(0.5, x_m, (1.0 / x_m)) * y) / z;
	} else {
		tmp = (fma((0.5 * x_m), x_m, 1.0) / (z * x_m)) * y;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(y / x_m) * cosh(x_m)) <= 2e+287)
		tmp = Float64(Float64(fma(0.5, x_m, Float64(1.0 / x_m)) * y) / z);
	else
		tmp = Float64(Float64(fma(Float64(0.5 * x_m), x_m, 1.0) / Float64(z * x_m)) * y);
	end
	return Float64(x_s * tmp)
end

Wolfram

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_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(y / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+287], N[(N[(N[(0.5 * x$95$m + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.0000000000000002e287

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-out N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-/.f64 78.2

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

   5. Applied rewrites 78.2%

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

   if 2.0000000000000002e287 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 57.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 34.0

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

   5. Applied rewrites 34.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 61.0

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

   7. Applied rewrites 61.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

   9. Applied rewrites 59.3%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot x, x, 1\right)}{z \cdot x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\frac{\frac{y}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot x\_m, x\_m, 1\right)}{z \cdot x\_m} \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* (/ y x_m) (cosh x_m)) 2e+287)
    (/ (/ y x_m) z)
    (* (/ (fma (* 0.5 x_m) x_m 1.0) (* z x_m)) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((y / x_m) * cosh(x_m)) <= 2e+287) {
		tmp = (y / x_m) / z;
	} else {
		tmp = (fma((0.5 * x_m), x_m, 1.0) / (z * x_m)) * y;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(y / x_m) * cosh(x_m)) <= 2e+287)
		tmp = Float64(Float64(y / x_m) / z);
	else
		tmp = Float64(Float64(fma(Float64(0.5 * x_m), x_m, 1.0) / Float64(z * x_m)) * y);
	end
	return Float64(x_s * tmp)
end

Wolfram

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_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(y / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+287], N[(N[(y / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.0000000000000002e287

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 64.9

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

   5. Applied rewrites 64.9%

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

   if 2.0000000000000002e287 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 57.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 34.0

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

   5. Applied rewrites 34.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 61.0

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

   7. Applied rewrites 61.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

   9. Applied rewrites 59.3%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot x, x, 1\right)}{z \cdot x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 7e+68)
    (/
     (/
      (*
       (fma
        (fma
         (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
         (* x_m x_m)
         0.5)
        (* x_m x_m)
        1.0)
       y)
      x_m)
     z)
    (/
     (*
      (/ y z)
      (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0))
     x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 7e+68) {
		tmp = ((fma(fma(fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) * y) / x_m) / z;
	} else {
		tmp = ((y / z) * fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0)) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 7e+68)
		tmp = Float64(Float64(Float64(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) * y) / x_m) / z);
	else
		tmp = Float64(Float64(Float64(y / z) * fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0)) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

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_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 7e+68], N[(N[(N[(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] * y), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y / z), $MachinePrecision] * 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]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.99999999999999955e68

   1. Initial program 78.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 87.3%

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

   7. Applied rewrites 87.4%

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

   if 6.99999999999999955e68 < y

   1. Initial program 93.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 100.0%

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

Alternative 9: 92.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 7e+68)
    (/
     (*
      (/
       (fma
        (fma
         (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
         (* x_m x_m)
         0.5)
        (* x_m x_m)
        1.0)
       x_m)
      y)
     z)
    (/
     (*
      (/ y z)
      (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0))
     x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 7e+68) {
		tmp = ((fma(fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y) / z;
	} else {
		tmp = ((y / z) * fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0)) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 7e+68)
		tmp = Float64(Float64(Float64(fma(fma(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) * y) / z);
	else
		tmp = Float64(Float64(Float64(y / z) * fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0)) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

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_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 7e+68], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y / z), $MachinePrecision] * 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]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.99999999999999955e68

   1. Initial program 78.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 87.3%

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

   if 6.99999999999999955e68 < y

   1. Initial program 93.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 100.0%

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

Alternative 10: 85.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.4 \cdot 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664, x\_m \cdot x\_m, 1\right) \cdot \frac{y}{z}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y}{x\_m}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5.4e+152)
    (/
     (* (fma (* (* x_m x_m) 0.041666666666666664) (* x_m x_m) 1.0) (/ y z))
     x_m)
    (/ (/ (* (fma 0.5 (* x_m x_m) 1.0) y) x_m) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 5.4e+152) {
		tmp = (fma(((x_m * x_m) * 0.041666666666666664), (x_m * x_m), 1.0) * (y / z)) / x_m;
	} else {
		tmp = ((fma(0.5, (x_m * x_m), 1.0) * y) / x_m) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 5.4e+152)
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * 0.041666666666666664), Float64(x_m * x_m), 1.0) * Float64(y / z)) / x_m);
	else
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y) / x_m) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

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_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 5.4e+152], 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 / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.4000000000000003e152

   1. Initial program 84.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 83.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 83.5%

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

   if 5.4000000000000003e152 < x

   1. Initial program 48.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 48.1

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

   5. Applied rewrites 48.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.4 \cdot 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x \cdot x, 1\right) \cdot \frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.4 \cdot 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y}{x\_m}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5.4e+152)
    (/
     (* (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0) y)
     (* z x_m))
    (/ (/ (* (fma 0.5 (* x_m x_m) 1.0) y) x_m) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 5.4e+152) {
		tmp = (fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) * y) / (z * x_m);
	} else {
		tmp = ((fma(0.5, (x_m * x_m), 1.0) * y) / x_m) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 5.4e+152)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * y) / Float64(z * x_m));
	else
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y) / x_m) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

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_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 5.4e+152], 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), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.4000000000000003e152

   1. Initial program 84.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 83.6%

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

   10. Applied rewrites 76.5%

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

   if 5.4000000000000003e152 < x

   1. Initial program 48.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 48.1

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

   5. Applied rewrites 48.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 12: 80.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 7 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{y}{z}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y}{x\_m}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 7e-157)
    (/ (/ y z) x_m)
    (/ (/ (* (fma 0.5 (* x_m x_m) 1.0) y) x_m) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 7e-157) {
		tmp = (y / z) / x_m;
	} else {
		tmp = ((fma(0.5, (x_m * x_m), 1.0) * y) / x_m) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 7e-157)
		tmp = Float64(Float64(y / z) / x_m);
	else
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y) / x_m) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

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_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 7e-157], N[(N[(y / z), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7.0000000000000004e-157

   1. Initial program 81.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 57.2

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

   8. Applied rewrites 57.2%

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

   if 7.0000000000000004e-157 < x

   1. Initial program 80.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 51.7

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

   5. Applied rewrites 51.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 69.2

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

   7. Applied rewrites 69.2%

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

Alternative 13: 66.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 110:\\ \;\;\;\;\frac{\frac{y}{z}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{z} \cdot x\_m\right) \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= x_m 110.0) (/ (/ y z) x_m) (* (* (/ 0.5 z) x_m) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 110.0) {
		tmp = (y / z) / x_m;
	} else {
		tmp = ((0.5 / z) * x_m) * y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 110.0d0) then
        tmp = (y / z) / x_m
    else
        tmp = ((0.5d0 / z) * x_m) * y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 110.0) {
		tmp = (y / z) / x_m;
	} else {
		tmp = ((0.5 / z) * x_m) * y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 110.0:
		tmp = (y / z) / x_m
	else:
		tmp = ((0.5 / z) * x_m) * y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 110.0)
		tmp = Float64(Float64(y / z) / x_m);
	else
		tmp = Float64(Float64(Float64(0.5 / z) * x_m) * y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 110.0)
		tmp = (y / z) / x_m;
	else
		tmp = ((0.5 / z) * x_m) * y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

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_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 110.0], N[(N[(y / z), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(0.5 / z), $MachinePrecision] * x$95$m), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 110

   1. Initial program 83.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 87.3%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 61.9

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

   8. Applied rewrites 61.9%

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

   if 110 < x

   1. Initial program 72.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. associate-/l/ N/A

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

      8. distribute-lft1-in N/A

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

      9. associate-*r/ N/A

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

      10. times-frac N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

   5. Applied rewrites 26.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 33.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 33.2%

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

Alternative 14: 66.4% accurate, 4.6× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= x_m 1.4) (/ y (* z x_m)) (* (* (/ 0.5 z) x_m) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = y / (z * x_m);
	} else {
		tmp = ((0.5 / z) * x_m) * y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 1.4d0) then
        tmp = y / (z * x_m)
    else
        tmp = ((0.5d0 / z) * x_m) * y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = y / (z * x_m);
	} else {
		tmp = ((0.5 / z) * x_m) * y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 1.4:
		tmp = y / (z * x_m)
	else:
		tmp = ((0.5 / z) * x_m) * y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1.4)
		tmp = Float64(y / Float64(z * x_m));
	else
		tmp = Float64(Float64(Float64(0.5 / z) * x_m) * y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 1.4)
		tmp = y / (z * x_m);
	else
		tmp = ((0.5 / z) * x_m) * y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

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_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 1.4], N[(y / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / z), $MachinePrecision] * x$95$m), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 83.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.6

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

   5. Applied rewrites 56.6%

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

   if 1.3999999999999999 < x

   1. Initial program 72.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. associate-/l/ N/A

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

      8. distribute-lft1-in N/A

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

      9. associate-*r/ N/A

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

      10. times-frac N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

   5. Applied rewrites 26.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 33.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 33.2%

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

Alternative 15: 49.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{y}{z \cdot x\_m} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (/ y (* z x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (y / (z * x_m));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (y / (z * x_m))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * (y / (z * x_m));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * (y / (z * x_m))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(y / Float64(z * x_m)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (y / (z * x_m));
end

Wolfram

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_, x$95$m_, y_, z_] := N[(x$95$s * N[(y / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 45.1

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

5. Applied rewrites 45.1%

   \[\leadsto \color{blue}{\frac{y}{z \cdot x}} \]
6. Add Preprocessing

Developer Target 1: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y z) x) (cosh x))))
   (if (< y -4.618902267687042e-52)
     t_0
     (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y / z) / x) * cosh(x)
    if (y < (-4.618902267687042d-52)) then
        tmp = t_0
    else if (y < 1.038530535935153d-39) then
        tmp = ((cosh(x) * y) / x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * Math.cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((Math.cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / z) / x) * math.cosh(x)
	tmp = 0
	if y < -4.618902267687042e-52:
		tmp = t_0
	elif y < 1.038530535935153e-39:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) / x) * cosh(x))
	tmp = 0.0
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / z) / x) * cosh(x);
	tmp = 0.0;
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -4.618902267687042e-52], t$95$0, If[Less[y, 1.038530535935153e-39], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024332 
(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))