Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.8% → 99.4%

Time: 10.9s

Alternatives: 21

Speedup: 2.8×

• 64.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \cosh x\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (/ y_m x_m) (cosh x_m)) 2e+152)
     (/
      (/
       (*
        (fma
         (fma
          (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
          (* x_m x_m)
          0.5)
         (* x_m x_m)
         1.0)
        y_m)
       x_m)
      z)
     (/ (/ (* y_m (cosh x_m)) z) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 2e+152) {
		tmp = ((fma(fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) * y_m) / x_m) / z;
	} else {
		tmp = ((y_m * cosh(x_m)) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 2e+152)
		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) * y_m) / x_m) / z);
	else
		tmp = Float64(Float64(Float64(y_m * cosh(x_m)) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+152], 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] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y$95$m * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.0000000000000001e152

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 60.6

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

   5. Applied rewrites 60.6%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 91.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 92.9%

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

   if 2.0000000000000001e152 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \cosh x}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m}{x\_m} \cdot \cosh x\_m \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m \cdot x\_m, 0.5, \frac{y\_m}{x\_m}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z}}{x\_m} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (/ y_m x_m) (cosh x_m)) 5e+107)
     (/ (fma (* y_m x_m) 0.5 (/ y_m x_m)) z)
     (*
      (/
       (/ (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0) z)
       x_m)
      y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m / x_m) * cosh(x_m)) <= 5e+107) {
		tmp = fma((y_m * x_m), 0.5, (y_m / 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_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m / x_m) * cosh(x_m)) <= 5e+107)
		tmp = Float64(fma(Float64(y_m * x_m), 0.5, Float64(y_m / 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_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 5e+107], N[(N[(N[(y$95$m * x$95$m), $MachinePrecision] * 0.5 + N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(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$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.0000000000000002e107

   1. Initial program 95.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   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. associate-/l* N/A

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

      16. unpow2 N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 78.4

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

   5. Applied rewrites 78.4%

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

   7. Applied rewrites 78.4%

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

   if 5.0000000000000002e107 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 70.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 0.5, \frac{y}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.7 \cdot 10^{-194}:\\ \;\;\;\;\frac{\frac{1}{z}}{x\_m} \cdot y\_m\\ \mathbf{elif}\;x\_m \leq 2.25:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, 0.5, {x\_m}^{-1}\right) \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right) \cdot x\_m}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 1.7e-194)
     (* (/ (/ 1.0 z) x_m) y_m)
     (if (<= x_m 2.25)
       (/ (* (fma x_m 0.5 (pow x_m -1.0)) y_m) z)
       (* (/ (* (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m) z) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.7e-194) {
		tmp = ((1.0 / z) / x_m) * y_m;
	} else if (x_m <= 2.25) {
		tmp = (fma(x_m, 0.5, pow(x_m, -1.0)) * y_m) / z;
	} else {
		tmp = ((fma(0.041666666666666664, (x_m * x_m), 0.5) * x_m) / z) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.7e-194)
		tmp = Float64(Float64(Float64(1.0 / z) / x_m) * y_m);
	elseif (x_m <= 2.25)
		tmp = Float64(Float64(fma(x_m, 0.5, (x_m ^ -1.0)) * y_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(0.041666666666666664, Float64(x_m * x_m), 0.5) * x_m) / z) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.7e-194], N[(N[(N[(1.0 / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[x$95$m, 2.25], N[(N[(N[(x$95$m * 0.5 + N[Power[x$95$m, -1.0], $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.70000000000000005e-194

   1. Initial program 81.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.7%

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

   if 1.70000000000000005e-194 < x < 2.25

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. 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. associate-/l* N/A

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

      16. unpow2 N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 2.25 < x

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left({x}^{3} \cdot \left(\frac{1}{24} \cdot \frac{1}{z} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot z}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 84.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 86.0%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-194}:\\ \;\;\;\;\frac{\frac{1}{z}}{x} \cdot y\\ \mathbf{elif}\;x \leq 2.25:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 0.5, {x}^{-1}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4.6 \cdot 10^{+49}:\\ \;\;\;\;\frac{y\_m \cdot \cosh x\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 4.6e+49)
     (/ (* y_m (cosh x_m)) (* z x_m))
     (/
      (*
       (/
        (fma
         (fma
          (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
          (* x_m x_m)
          0.5)
         (* x_m x_m)
         1.0)
        x_m)
       y_m)
      z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 4.6e+49) {
		tmp = (y_m * cosh(x_m)) / (z * x_m);
	} else {
		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_m) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 4.6e+49)
		tmp = Float64(Float64(y_m * cosh(x_m)) / Float64(z * x_m));
	else
		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_m) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 4.6e+49], N[(N[(y$95$m * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 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$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.60000000000000004e49

   1. Initial program 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 88.3

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

   4. Applied rewrites 88.3%

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

   if 4.60000000000000004e49 < x

   1. Initial program 76.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

Alternative 5: 85.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000004

   1. Initial program 84.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.1%

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

   if 1.30000000000000004 < x

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left({x}^{3} \cdot \left(\frac{1}{24} \cdot \frac{1}{z} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot z}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 84.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 86.0%

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

4. Final simplification 67.1%

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

Alternative 6: 83.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000004

   1. Initial program 84.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.1%

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

   if 1.30000000000000004 < x

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{y}{z} + \frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.8%

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

   10. Applied rewrites 84.6%

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

4. Final simplification 66.7%

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

Alternative 7: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.25:\\ \;\;\;\;\frac{{x\_m}^{-1}}{z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{x\_m \cdot x\_m}{z} \cdot y\_m\right) \cdot 0.041666666666666664\right) \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 2.25)
     (* (/ (pow x_m -1.0) z) y_m)
     (* (* (* (/ (* x_m x_m) z) y_m) 0.041666666666666664) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 2.25) {
		tmp = (pow(x_m, -1.0) / z) * y_m;
	} else {
		tmp = ((((x_m * x_m) / z) * y_m) * 0.041666666666666664) * x_m;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 2.25d0) then
        tmp = ((x_m ** (-1.0d0)) / z) * y_m
    else
        tmp = ((((x_m * x_m) / z) * y_m) * 0.041666666666666664d0) * x_m
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 2.25) {
		tmp = (Math.pow(x_m, -1.0) / z) * y_m;
	} else {
		tmp = ((((x_m * x_m) / z) * y_m) * 0.041666666666666664) * x_m;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if x_m <= 2.25:
		tmp = (math.pow(x_m, -1.0) / z) * y_m
	else:
		tmp = ((((x_m * x_m) / z) * y_m) * 0.041666666666666664) * x_m
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 2.25)
		tmp = Float64(Float64((x_m ^ -1.0) / z) * y_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x_m * x_m) / z) * y_m) * 0.041666666666666664) * x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 2.25)
		tmp = ((x_m ^ -1.0) / z) * y_m;
	else
		tmp = ((((x_m * x_m) / z) * y_m) * 0.041666666666666664) * x_m;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 2.25], N[(N[(N[Power[x$95$m, -1.0], $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   1. Initial program 84.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.1%

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

   if 2.25 < x

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{y}{z} + \frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 83.2%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{{x}^{-1}}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{x \cdot x}{z} \cdot y\right) \cdot 0.041666666666666664\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.0% accurate, 1.0× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 1.4) (* (/ (pow x_m -1.0) z) y_m) (* (* (/ x_m z) y_m) 0.5)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.4], N[(N[(N[Power[x$95$m, -1.0], $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 84.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.1%

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

   if 1.3999999999999999 < x

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{y}{z} + \frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 39.1%

       \[\leadsto \left(\frac{y}{z} \cdot x\right) \cdot 0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 48.6%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{{x}^{-1}}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.3 \cdot 10^{-185}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z \cdot x\_m}\\ \mathbf{elif}\;x\_m \leq 4.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z} \cdot \frac{y\_m}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right) \cdot x\_m\right) \cdot x\_m}{z}}{x\_m} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 1.3e-185)
     (/
      (* (fma (* 0.041666666666666664 (* x_m x_m)) (* x_m x_m) 1.0) y_m)
      (* z x_m))
     (if (<= x_m 4.8e+76)
       (*
        (/
         (fma
          (fma
           (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
           (* x_m x_m)
           0.5)
          (* x_m x_m)
          1.0)
         z)
        (/ y_m x_m))
       (*
        (/
         (/ (* (* (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m) x_m) z)
         x_m)
        y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.3e-185) {
		tmp = (fma((0.041666666666666664 * (x_m * x_m)), (x_m * x_m), 1.0) * y_m) / (z * x_m);
	} else if (x_m <= 4.8e+76) {
		tmp = (fma(fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / z) * (y_m / x_m);
	} else {
		tmp = ((((fma(0.041666666666666664, (x_m * x_m), 0.5) * x_m) * x_m) / z) / x_m) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.3e-185)
		tmp = Float64(Float64(fma(Float64(0.041666666666666664 * Float64(x_m * x_m)), Float64(x_m * x_m), 1.0) * y_m) / Float64(z * x_m));
	elseif (x_m <= 4.8e+76)
		tmp = 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) / z) * Float64(y_m / x_m));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(0.041666666666666664, Float64(x_m * x_m), 0.5) * x_m) * x_m) / z) / x_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.3e-185], N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 4.8e+76], 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] / z), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.29999999999999992e-185

   1. Initial program 81.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 73.3

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.9

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

   10. Applied rewrites 78.9%

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

   if 1.29999999999999992e-185 < x < 4.8e76

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 88.5%

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

   if 4.8e76 < x

   1. Initial program 74.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

Alternative 10: 92.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5 \cdot x\_m, x\_m, 1\right) \cdot y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= y_m 1.4e+49)
     (/
      (/
       (*
        (fma
         (fma
          (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
          (* x_m x_m)
          0.5)
         (* x_m x_m)
         1.0)
        y_m)
       x_m)
      z)
     (/ (/ (* (fma (* 0.5 x_m) x_m 1.0) y_m) z) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1.4e+49) {
		tmp = ((fma(fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) * y_m) / x_m) / z;
	} else {
		tmp = ((fma((0.5 * x_m), x_m, 1.0) * y_m) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 1.4e+49)
		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) * y_m) / x_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(Float64(0.5 * x_m), x_m, 1.0) * y_m) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[y$95$m, 1.4e+49], 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] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.3999999999999999e49

   1. Initial program 78.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 43.1

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

   5. Applied rewrites 43.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 88.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 90.9%

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

   if 1.3999999999999999e49 < y

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5 \cdot x, x, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5 \cdot x\_m, x\_m, 1\right) \cdot y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= y_m 1.4e+49)
     (/
      (*
       (/
        (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_m)
      z)
     (/ (/ (* (fma (* 0.5 x_m) x_m 1.0) y_m) z) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1.4e+49) {
		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_m) / z;
	} else {
		tmp = ((fma((0.5 * x_m), x_m, 1.0) * y_m) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 1.4e+49)
		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_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(Float64(0.5 * x_m), x_m, 1.0) * y_m) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[y$95$m, 1.4e+49], 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$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.3999999999999999e49

   1. Initial program 78.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.8%

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

   if 1.3999999999999999e49 < y

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5 \cdot x, x, 1\right) \cdot y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 88.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.7 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right) \cdot x\_m\right) \cdot x\_m}{z}}{x\_m} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 3.7e+56)
     (/
      (* (fma (fma (* x_m x_m) 0.041666666666666664 0.5) (* x_m x_m) 1.0) y_m)
      (* z x_m))
     (*
      (/ (/ (* (* (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m) x_m) z) x_m)
      y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 3.7e+56) {
		tmp = (fma(fma((x_m * x_m), 0.041666666666666664, 0.5), (x_m * x_m), 1.0) * y_m) / (z * x_m);
	} else {
		tmp = ((((fma(0.041666666666666664, (x_m * x_m), 0.5) * x_m) * x_m) / z) / x_m) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 3.7e+56)
		tmp = Float64(Float64(fma(fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), Float64(x_m * x_m), 1.0) * y_m) / Float64(z * x_m));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(0.041666666666666664, Float64(x_m * x_m), 0.5) * x_m) * x_m) / z) / x_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 3.7e+56], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.69999999999999997e56

   1. Initial program 85.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right)\right)\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right)\right)\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

   7. Applied rewrites 79.3%

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

   if 3.69999999999999997e56 < x

   1. Initial program 75.4%

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

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

Alternative 13: 87.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5 \cdot 10^{+97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right) \cdot x\_m}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 1.5e+97)
     (/
      (* (fma (fma (* x_m x_m) 0.041666666666666664 0.5) (* x_m x_m) 1.0) y_m)
      (* z x_m))
     (* (/ (* (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m) z) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.5e+97) {
		tmp = (fma(fma((x_m * x_m), 0.041666666666666664, 0.5), (x_m * x_m), 1.0) * y_m) / (z * x_m);
	} else {
		tmp = ((fma(0.041666666666666664, (x_m * x_m), 0.5) * x_m) / z) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.5e+97)
		tmp = Float64(Float64(fma(fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), Float64(x_m * x_m), 1.0) * y_m) / Float64(z * x_m));
	else
		tmp = Float64(Float64(Float64(fma(0.041666666666666664, Float64(x_m * x_m), 0.5) * x_m) / z) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.5e+97], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.4999999999999999e97

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 75.7

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

   5. Applied rewrites 75.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right)\right)\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right)\right)\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

   7. Applied rewrites 79.6%

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

   if 1.4999999999999999e97 < x

   1. Initial program 72.0%

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

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

   6. Taylor expanded in x around inf

      \[\leadsto \left({x}^{3} \cdot \left(\frac{1}{24} \cdot \frac{1}{z} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot z}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 96.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 98.0%

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

Alternative 14: 86.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5 \cdot 10^{+97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right) \cdot x\_m}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 1.5e+97)
     (/
      (* (fma (* 0.041666666666666664 (* x_m x_m)) (* x_m x_m) 1.0) y_m)
      (* z x_m))
     (* (/ (* (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m) z) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.5e+97) {
		tmp = (fma((0.041666666666666664 * (x_m * x_m)), (x_m * x_m), 1.0) * y_m) / (z * x_m);
	} else {
		tmp = ((fma(0.041666666666666664, (x_m * x_m), 0.5) * x_m) / z) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.5e+97)
		tmp = Float64(Float64(fma(Float64(0.041666666666666664 * Float64(x_m * x_m)), Float64(x_m * x_m), 1.0) * y_m) / Float64(z * x_m));
	else
		tmp = Float64(Float64(Float64(fma(0.041666666666666664, Float64(x_m * x_m), 0.5) * x_m) / z) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.5e+97], N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.4999999999999999e97

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.9

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

   10. Applied rewrites 78.9%

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

   if 1.4999999999999999e97 < x

   1. Initial program 72.0%

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

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

   6. Taylor expanded in x around inf

      \[\leadsto \left({x}^{3} \cdot \left(\frac{1}{24} \cdot \frac{1}{z} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot z}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 96.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 98.0%

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

Alternative 15: 85.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.7 \cdot 10^{-194}:\\ \;\;\;\;\frac{\frac{1}{z}}{x\_m} \cdot y\_m\\ \mathbf{elif}\;x\_m \leq 2.25:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m \cdot x\_m, 0.5, \frac{y\_m}{x\_m}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right) \cdot x\_m}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 1.7e-194)
     (* (/ (/ 1.0 z) x_m) y_m)
     (if (<= x_m 2.25)
       (/ (fma (* y_m x_m) 0.5 (/ y_m x_m)) z)
       (* (/ (* (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m) z) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.7e-194) {
		tmp = ((1.0 / z) / x_m) * y_m;
	} else if (x_m <= 2.25) {
		tmp = fma((y_m * x_m), 0.5, (y_m / x_m)) / z;
	} else {
		tmp = ((fma(0.041666666666666664, (x_m * x_m), 0.5) * x_m) / z) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.7e-194)
		tmp = Float64(Float64(Float64(1.0 / z) / x_m) * y_m);
	elseif (x_m <= 2.25)
		tmp = Float64(fma(Float64(y_m * x_m), 0.5, Float64(y_m / x_m)) / z);
	else
		tmp = Float64(Float64(Float64(fma(0.041666666666666664, Float64(x_m * x_m), 0.5) * x_m) / z) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.7e-194], N[(N[(N[(1.0 / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[x$95$m, 2.25], N[(N[(N[(y$95$m * x$95$m), $MachinePrecision] * 0.5 + N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.70000000000000005e-194

   1. Initial program 81.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.7%

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

   if 1.70000000000000005e-194 < x < 2.25

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. 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. associate-/l* N/A

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

      16. unpow2 N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

   if 2.25 < x

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left({x}^{3} \cdot \left(\frac{1}{24} \cdot \frac{1}{z} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot z}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 84.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 86.0%

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

Alternative 16: 85.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.25:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)}{z}}{x\_m} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right) \cdot x\_m}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 2.25)
     (* (/ (/ (fma 0.5 (* x_m x_m) 1.0) z) x_m) y_m)
     (* (/ (* (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m) z) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 2.25) {
		tmp = ((fma(0.5, (x_m * x_m), 1.0) / z) / x_m) * y_m;
	} else {
		tmp = ((fma(0.041666666666666664, (x_m * x_m), 0.5) * x_m) / z) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 2.25)
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) / z) / x_m) * y_m);
	else
		tmp = Float64(Float64(Float64(fma(0.041666666666666664, Float64(x_m * x_m), 0.5) * x_m) / z) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 2.25], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   1. Initial program 84.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.7%

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

   if 2.25 < x

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left({x}^{3} \cdot \left(\frac{1}{24} \cdot \frac{1}{z} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot z}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 84.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 86.0%

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

Alternative 17: 65.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4.9 \cdot 10^{-231}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x\_m}\\ \mathbf{elif}\;x\_m \leq 1.4:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x\_m}{z} \cdot y\_m\right) \cdot 0.5\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 4.9e-231)
     (/ (/ y_m z) x_m)
     (if (<= x_m 1.4) (/ (/ y_m x_m) z) (* (* (/ x_m z) y_m) 0.5))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 4.9e-231) {
		tmp = (y_m / z) / x_m;
	} else if (x_m <= 1.4) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = ((x_m / z) * y_m) * 0.5;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 4.9d-231) then
        tmp = (y_m / z) / x_m
    else if (x_m <= 1.4d0) then
        tmp = (y_m / x_m) / z
    else
        tmp = ((x_m / z) * y_m) * 0.5d0
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 4.9e-231) {
		tmp = (y_m / z) / x_m;
	} else if (x_m <= 1.4) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = ((x_m / z) * y_m) * 0.5;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if x_m <= 4.9e-231:
		tmp = (y_m / z) / x_m
	elif x_m <= 1.4:
		tmp = (y_m / x_m) / z
	else:
		tmp = ((x_m / z) * y_m) * 0.5
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 4.9e-231)
		tmp = Float64(Float64(y_m / z) / x_m);
	elseif (x_m <= 1.4)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(Float64(Float64(x_m / z) * y_m) * 0.5);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 4.9e-231)
		tmp = (y_m / z) / x_m;
	elseif (x_m <= 1.4)
		tmp = (y_m / x_m) / z;
	else
		tmp = ((x_m / z) * y_m) * 0.5;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 4.9e-231], N[(N[(y$95$m / z), $MachinePrecision] / x$95$m), $MachinePrecision], If[LessEqual[x$95$m, 1.4], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 4.90000000000000003e-231

   1. Initial program 81.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 94.5

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

   4. Applied rewrites 94.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 53.6

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

   7. Applied rewrites 53.6%

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

   if 4.90000000000000003e-231 < x < 1.3999999999999999

   1. Initial program 97.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 96.0

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

   5. Applied rewrites 96.0%

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

   if 1.3999999999999999 < x

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{y}{z} + \frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 39.1%

       \[\leadsto \left(\frac{y}{z} \cdot x\right) \cdot 0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 48.6%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.9 \cdot 10^{-231}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 65.2% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.4], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 84.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 58.7

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

   5. Applied rewrites 58.7%

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

   if 1.3999999999999999 < x

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{y}{z} + \frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 39.1%

       \[\leadsto \left(\frac{y}{z} \cdot x\right) \cdot 0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 48.6%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 30.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{-26}:\\ \;\;\;\;\left(\frac{x\_m}{z} \cdot y\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\_m\right) \cdot y\_m}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= y_m 4e-26) (* (* (/ x_m z) y_m) 0.5) (/ (* (* 0.5 x_m) y_m) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 4e-26) {
		tmp = ((x_m / z) * y_m) * 0.5;
	} else {
		tmp = ((0.5 * x_m) * y_m) / z;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 4d-26) then
        tmp = ((x_m / z) * y_m) * 0.5d0
    else
        tmp = ((0.5d0 * x_m) * y_m) / z
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 4e-26) {
		tmp = ((x_m / z) * y_m) * 0.5;
	} else {
		tmp = ((0.5 * x_m) * y_m) / z;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if y_m <= 4e-26:
		tmp = ((x_m / z) * y_m) * 0.5
	else:
		tmp = ((0.5 * x_m) * y_m) / z
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 4e-26)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) * 0.5);
	else
		tmp = Float64(Float64(Float64(0.5 * x_m) * y_m) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (y_m <= 4e-26)
		tmp = ((x_m / z) * y_m) * 0.5;
	else
		tmp = ((0.5 * x_m) * y_m) / z;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[y$95$m, 4e-26], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.0000000000000002e-26

   1. Initial program 77.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{y}{z} + \frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 40.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 21.4%

       \[\leadsto \left(\frac{y}{z} \cdot x\right) \cdot 0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 26.7%

       \[\leadsto \left(y \cdot \frac{x}{z}\right) \cdot 0.5 \]

   if 4.0000000000000002e-26 < y

   1. Initial program 98.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   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. associate-/l* N/A

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

      16. unpow2 N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 86.7

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 54.8%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-26}:\\ \;\;\;\;\left(\frac{x}{z} \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\right) \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 29.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{-26}:\\ \;\;\;\;\left(\frac{x\_m}{z} \cdot y\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y\_m}{z} \cdot x\_m\right) \cdot 0.5\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= y_m 4e-26) (* (* (/ x_m z) y_m) 0.5) (* (* (/ y_m z) x_m) 0.5)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 4e-26) {
		tmp = ((x_m / z) * y_m) * 0.5;
	} else {
		tmp = ((y_m / z) * x_m) * 0.5;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 4d-26) then
        tmp = ((x_m / z) * y_m) * 0.5d0
    else
        tmp = ((y_m / z) * x_m) * 0.5d0
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 4e-26) {
		tmp = ((x_m / z) * y_m) * 0.5;
	} else {
		tmp = ((y_m / z) * x_m) * 0.5;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if y_m <= 4e-26:
		tmp = ((x_m / z) * y_m) * 0.5
	else:
		tmp = ((y_m / z) * x_m) * 0.5
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 4e-26)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) * 0.5);
	else
		tmp = Float64(Float64(Float64(y_m / z) * x_m) * 0.5);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (y_m <= 4e-26)
		tmp = ((x_m / z) * y_m) * 0.5;
	else
		tmp = ((y_m / z) * x_m) * 0.5;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[y$95$m, 4e-26], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(y$95$m / z), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.0000000000000002e-26

   1. Initial program 77.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{y}{z} + \frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 40.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 21.4%

       \[\leadsto \left(\frac{y}{z} \cdot x\right) \cdot 0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 26.7%

       \[\leadsto \left(y \cdot \frac{x}{z}\right) \cdot 0.5 \]

   if 4.0000000000000002e-26 < y

   1. Initial program 98.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 95.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{y}{z} + \frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 49.0%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-26}:\\ \;\;\;\;\left(\frac{x}{z} \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{z} \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 26.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(\left(\frac{x\_m}{z} \cdot y\_m\right) \cdot 0.5\right)\right) \end{array} \]

FPCore

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

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (((x_m / z) * y_m) * 0.5));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x_s * (y_s * (((x_m / z) * y_m) * 0.5d0))
end function

Java

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

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	return x_s * (y_s * (((x_m / z) * y_m) * 0.5))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(Float64(Float64(x_m / z) * y_m) * 0.5)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, y_s, x_m, y_m, z)
	tmp = x_s * (y_s * (((x_m / z) * y_m) * 0.5));
end

Wolfram

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

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 89.7%

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

6. Taylor expanded in x around inf

   \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{y}{z} + \frac{1}{2} \cdot \frac{y}{{x}^{2} \cdot z}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 47.7%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 28.4%

    \[\leadsto \left(\frac{y}{z} \cdot x\right) \cdot 0.5 \]
12. Step-by-step derivation

13. Applied rewrites 30.9%

    \[\leadsto \left(y \cdot \frac{x}{z}\right) \cdot 0.5 \]

14. Final simplification 30.9%

    \[\leadsto \left(\frac{x}{z} \cdot y\right) \cdot 0.5 \]
15. Add Preprocessing

Developer Target 1: 97.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024271 
(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))