Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Percentage Accurate: 88.0% → 98.9%

Time: 7.4s

Alternatives: 9

Speedup: 0.3×

Specification

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * 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 = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

Python

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / x) / (y * (1.0 + (z * z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * 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 = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

Python

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / x) / (y * (1.0 + (z * z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.6 \cdot 10^{+208}:\\ \;\;\;\;{\left(y\_m \cdot \mathsf{fma}\left(x\_m \cdot z\_m, z\_m, x\_m\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{\left(z\_m \cdot z\_m\right) \cdot x\_m} - \frac{-1}{x\_m}}{z\_m \cdot y\_m}}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z_m 7.6e+208)
     (pow (* y_m (fma (* x_m z_m) z_m x_m)) -1.0)
     (/ (/ (- (/ -1.0 (* (* z_m z_m) x_m)) (/ -1.0 x_m)) (* z_m y_m)) z_m)))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 7.6e+208) {
		tmp = pow((y_m * fma((x_m * z_m), z_m, x_m)), -1.0);
	} else {
		tmp = (((-1.0 / ((z_m * z_m) * x_m)) - (-1.0 / x_m)) / (z_m * y_m)) / z_m;
	}
	return x_s * (y_s * tmp);
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 7.6e+208)
		tmp = Float64(y_m * fma(Float64(x_m * z_m), z_m, x_m)) ^ -1.0;
	else
		tmp = Float64(Float64(Float64(Float64(-1.0 / Float64(Float64(z_m * z_m) * x_m)) - Float64(-1.0 / x_m)) / Float64(z_m * y_m)) / z_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z$95$m, 7.6e+208], N[Power[N[(y$95$m * N[(N[(x$95$m * z$95$m), $MachinePrecision] * z$95$m + x$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[(N[(-1.0 / N[(N[(z$95$m * z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 7.6000000000000004e208

   1. Initial program 92.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.8

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 91.8

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

   4. Applied rewrites 91.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.5

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

   6. Applied rewrites 96.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 94.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 94.8

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.8

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

   8. Applied rewrites 94.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-out N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 94.6

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

   10. Applied rewrites 94.6%

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

   if 7.6000000000000004e208 < z

   1. Initial program 68.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 99.8%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.6 \cdot 10^{+208}:\\ \;\;\;\;{\left(y \cdot \mathsf{fma}\left(x \cdot z, z, x\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{\left(z \cdot z\right) \cdot x} - \frac{-1}{x}}{z \cdot y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4 \cdot 10^{+203}:\\ \;\;\;\;{\left(y\_m \cdot \mathsf{fma}\left(x\_m \cdot z\_m, z\_m, x\_m\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(x\_m \cdot \left(\left(\frac{y\_m}{z\_m \cdot z\_m} + y\_m\right) \cdot z\_m\right)\right) \cdot z\_m\right)}^{-1}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z_m 4e+203)
     (pow (* y_m (fma (* x_m z_m) z_m x_m)) -1.0)
     (pow (* (* x_m (* (+ (/ y_m (* z_m z_m)) y_m) z_m)) z_m) -1.0)))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 4e+203) {
		tmp = pow((y_m * fma((x_m * z_m), z_m, x_m)), -1.0);
	} else {
		tmp = pow(((x_m * (((y_m / (z_m * z_m)) + y_m) * z_m)) * z_m), -1.0);
	}
	return x_s * (y_s * tmp);
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 4e+203)
		tmp = Float64(y_m * fma(Float64(x_m * z_m), z_m, x_m)) ^ -1.0;
	else
		tmp = Float64(Float64(x_m * Float64(Float64(Float64(y_m / Float64(z_m * z_m)) + y_m) * z_m)) * z_m) ^ -1.0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z$95$m, 4e+203], N[Power[N[(y$95$m * N[(N[(x$95$m * z$95$m), $MachinePrecision] * z$95$m + x$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(x$95$m * N[(N[(N[(y$95$m / N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision], -1.0], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4e203

   1. Initial program 92.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.1

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 92.1

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

   4. Applied rewrites 92.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.5

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

   6. Applied rewrites 96.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 95.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 95.2

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

   8. Applied rewrites 95.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-out N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 95.0

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

   10. Applied rewrites 95.0%

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

   if 4e203 < z

   1. Initial program 64.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 64.8

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 64.8

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

   4. Applied rewrites 64.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. distribute-lft-out N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 99.7

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

   9. Applied rewrites 99.7%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{+203}:\\ \;\;\;\;{\left(y \cdot \mathsf{fma}\left(x \cdot z, z, x\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(x \cdot \left(\left(\frac{y}{z \cdot z} + y\right) \cdot z\right)\right) \cdot z\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x\_m}^{-1}}{y\_m}\\ \mathbf{elif}\;z\_m \leq 5.1 \cdot 10^{+154}:\\ \;\;\;\;{\left(\left(\left(z\_m \cdot z\_m\right) \cdot x\_m\right) \cdot y\_m\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(y\_m \cdot z\_m\right) \cdot z\_m\right) \cdot x\_m\right)}^{-1}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z_m 5.8e-5)
     (/ (pow x_m -1.0) y_m)
     (if (<= z_m 5.1e+154)
       (pow (* (* (* z_m z_m) x_m) y_m) -1.0)
       (pow (* (* (* y_m z_m) z_m) x_m) -1.0))))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 5.8e-5) {
		tmp = pow(x_m, -1.0) / y_m;
	} else if (z_m <= 5.1e+154) {
		tmp = pow((((z_m * z_m) * x_m) * y_m), -1.0);
	} else {
		tmp = pow((((y_m * z_m) * z_m) * x_m), -1.0);
	}
	return x_s * (y_s * tmp);
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z_m)
    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_m
    real(8) :: tmp
    if (z_m <= 5.8d-5) then
        tmp = (x_m ** (-1.0d0)) / y_m
    else if (z_m <= 5.1d+154) then
        tmp = (((z_m * z_m) * x_m) * y_m) ** (-1.0d0)
    else
        tmp = (((y_m * z_m) * z_m) * x_m) ** (-1.0d0)
    end if
    code = x_s * (y_s * tmp)
end function

Java

z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m;
public static double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 5.8e-5) {
		tmp = Math.pow(x_m, -1.0) / y_m;
	} else if (z_m <= 5.1e+154) {
		tmp = Math.pow((((z_m * z_m) * x_m) * y_m), -1.0);
	} else {
		tmp = Math.pow((((y_m * z_m) * z_m) * x_m), -1.0);
	}
	return x_s * (y_s * tmp);
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(x_s, y_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 5.8e-5:
		tmp = math.pow(x_m, -1.0) / y_m
	elif z_m <= 5.1e+154:
		tmp = math.pow((((z_m * z_m) * x_m) * y_m), -1.0)
	else:
		tmp = math.pow((((y_m * z_m) * z_m) * x_m), -1.0)
	return x_s * (y_s * tmp)

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 5.8e-5)
		tmp = Float64((x_m ^ -1.0) / y_m);
	elseif (z_m <= 5.1e+154)
		tmp = Float64(Float64(Float64(z_m * z_m) * x_m) * y_m) ^ -1.0;
	else
		tmp = Float64(Float64(Float64(y_m * z_m) * z_m) * x_m) ^ -1.0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 5.8e-5)
		tmp = (x_m ^ -1.0) / y_m;
	elseif (z_m <= 5.1e+154)
		tmp = (((z_m * z_m) * x_m) * y_m) ^ -1.0;
	else
		tmp = (((y_m * z_m) * z_m) * x_m) ^ -1.0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z$95$m, 5.8e-5], N[(N[Power[x$95$m, -1.0], $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[z$95$m, 5.1e+154], N[Power[N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(y$95$m * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], -1.0], $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 5.8e-5

   1. Initial program 94.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if 5.8e-5 < z < 5.0999999999999999e154

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.1

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 89.6

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

   6. Applied rewrites 89.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 89.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 89.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 89.0

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

   8. Applied rewrites 89.0%

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

   9. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 81.1

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

   11. Applied rewrites 81.1%

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

   if 5.0999999999999999e154 < z

   1. Initial program 59.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 59.2

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

   5. Applied rewrites 59.2%

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

   7. Applied rewrites 77.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 77.9

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

   9. Applied rewrites 77.9%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{-1}}{y}\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+154}:\\ \;\;\;\;{\left(\left(\left(z \cdot z\right) \cdot x\right) \cdot y\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(y \cdot z\right) \cdot z\right) \cdot x\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3 \cdot 10^{-41}:\\ \;\;\;\;{\left(\mathsf{fma}\left(y\_m \cdot z\_m, z\_m \cdot x\_m, y\_m \cdot x\_m\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(y\_m \cdot \mathsf{fma}\left(x\_m \cdot z\_m, z\_m, x\_m\right)\right)}^{-1}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= y_m 3e-41)
     (pow (fma (* y_m z_m) (* z_m x_m) (* y_m x_m)) -1.0)
     (pow (* y_m (fma (* x_m z_m) z_m x_m)) -1.0)))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 3e-41) {
		tmp = pow(fma((y_m * z_m), (z_m * x_m), (y_m * x_m)), -1.0);
	} else {
		tmp = pow((y_m * fma((x_m * z_m), z_m, x_m)), -1.0);
	}
	return x_s * (y_s * tmp);
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 3e-41)
		tmp = fma(Float64(y_m * z_m), Float64(z_m * x_m), Float64(y_m * x_m)) ^ -1.0;
	else
		tmp = Float64(y_m * fma(Float64(x_m * z_m), z_m, x_m)) ^ -1.0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * If[LessEqual[y$95$m, 3e-41], N[Power[N[(N[(y$95$m * z$95$m), $MachinePrecision] * N[(z$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(y$95$m * N[(N[(x$95$m * z$95$m), $MachinePrecision] * z$95$m + x$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.99999999999999989e-41

   1. Initial program 90.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.1

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 90.1

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

   4. Applied rewrites 90.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.0

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

   6. Applied rewrites 97.0%

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

   if 2.99999999999999989e-41 < y

   1. Initial program 92.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.3

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 91.3

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

   4. Applied rewrites 91.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.9

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

   6. Applied rewrites 95.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 97.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 97.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 97.3

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

   8. Applied rewrites 97.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-out N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 98.8

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

   10. Applied rewrites 98.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-41}:\\ \;\;\;\;{\left(\mathsf{fma}\left(y \cdot z, z \cdot x, y \cdot x\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(y \cdot \mathsf{fma}\left(x \cdot z, z, x\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 0.3× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z_m z_m) 5e-10)
     (/ (pow x_m -1.0) y_m)
     (pow (* (* (* z_m z_m) y_m) x_m) -1.0)))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 5e-10) {
		tmp = pow(x_m, -1.0) / y_m;
	} else {
		tmp = pow((((z_m * z_m) * y_m) * x_m), -1.0);
	}
	return x_s * (y_s * tmp);
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z_m)
    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_m
    real(8) :: tmp
    if ((z_m * z_m) <= 5d-10) then
        tmp = (x_m ** (-1.0d0)) / y_m
    else
        tmp = (((z_m * z_m) * y_m) * x_m) ** (-1.0d0)
    end if
    code = x_s * (y_s * tmp)
end function

Java

z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m;
public static double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 5e-10) {
		tmp = Math.pow(x_m, -1.0) / y_m;
	} else {
		tmp = Math.pow((((z_m * z_m) * y_m) * x_m), -1.0);
	}
	return x_s * (y_s * tmp);
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(x_s, y_s, x_m, y_m, z_m):
	tmp = 0
	if (z_m * z_m) <= 5e-10:
		tmp = math.pow(x_m, -1.0) / y_m
	else:
		tmp = math.pow((((z_m * z_m) * y_m) * x_m), -1.0)
	return x_s * (y_s * tmp)

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 5e-10)
		tmp = Float64((x_m ^ -1.0) / y_m);
	else
		tmp = Float64(Float64(Float64(z_m * z_m) * y_m) * x_m) ^ -1.0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 5e-10)
		tmp = (x_m ^ -1.0) / y_m;
	else
		tmp = (((z_m * z_m) * y_m) * x_m) ^ -1.0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 5e-10], N[(N[Power[x$95$m, -1.0], $MachinePrecision] / y$95$m), $MachinePrecision], N[Power[N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], -1.0], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000031e-10

   1. Initial program 99.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 5.00000000000000031e-10 < (*.f64 z z)

   1. Initial program 80.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 80.1

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

   5. Applied rewrites 80.1%

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

4. Final simplification 90.3%

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

Alternative 6: 91.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x\_m}^{-1}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(z\_m \cdot z\_m\right) \cdot x\_m\right) \cdot y\_m\right)}^{-1}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z_m 5.8e-5)
     (/ (pow x_m -1.0) y_m)
     (pow (* (* (* z_m z_m) x_m) y_m) -1.0)))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 5.8e-5) {
		tmp = pow(x_m, -1.0) / y_m;
	} else {
		tmp = pow((((z_m * z_m) * x_m) * y_m), -1.0);
	}
	return x_s * (y_s * tmp);
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z_m)
    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_m
    real(8) :: tmp
    if (z_m <= 5.8d-5) then
        tmp = (x_m ** (-1.0d0)) / y_m
    else
        tmp = (((z_m * z_m) * x_m) * y_m) ** (-1.0d0)
    end if
    code = x_s * (y_s * tmp)
end function

Java

z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m;
public static double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 5.8e-5) {
		tmp = Math.pow(x_m, -1.0) / y_m;
	} else {
		tmp = Math.pow((((z_m * z_m) * x_m) * y_m), -1.0);
	}
	return x_s * (y_s * tmp);
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(x_s, y_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 5.8e-5:
		tmp = math.pow(x_m, -1.0) / y_m
	else:
		tmp = math.pow((((z_m * z_m) * x_m) * y_m), -1.0)
	return x_s * (y_s * tmp)

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 5.8e-5)
		tmp = Float64((x_m ^ -1.0) / y_m);
	else
		tmp = Float64(Float64(Float64(z_m * z_m) * x_m) * y_m) ^ -1.0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 5.8e-5)
		tmp = (x_m ^ -1.0) / y_m;
	else
		tmp = (((z_m * z_m) * x_m) * y_m) ^ -1.0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z$95$m, 5.8e-5], N[(N[Power[x$95$m, -1.0], $MachinePrecision] / y$95$m), $MachinePrecision], N[Power[N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], -1.0], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.8e-5

   1. Initial program 94.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if 5.8e-5 < z

   1. Initial program 78.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 78.4

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 78.4

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

   4. Applied rewrites 78.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.3

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

   6. Applied rewrites 93.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 89.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 89.5

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 89.5

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 69.7

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

   11. Applied rewrites 69.7%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{-1}}{y}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(z \cdot z\right) \cdot x\right) \cdot y\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot {\left(y\_m \cdot \mathsf{fma}\left(x\_m \cdot z\_m, z\_m, x\_m\right)\right)}^{-1}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (* x_s (* y_s (pow (* y_m (fma (* x_m z_m) z_m x_m)) -1.0))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	return x_s * (y_s * pow((y_m * fma((x_m * z_m), z_m, x_m)), -1.0));
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	return Float64(x_s * Float64(y_s * (Float64(y_m * fma(Float64(x_m * z_m), z_m, x_m)) ^ -1.0)))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[Power[N[(y$95$m * N[(N[(x$95$m * z$95$m), $MachinePrecision] * z$95$m + x$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 90.4

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. *-rgt-identity N/A

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

   12. lower-fma.f64 90.4

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

4. Applied rewrites 90.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. distribute-lft-in N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 96.7

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

6. Applied rewrites 96.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 94.7

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 94.7

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 94.7

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

8. Applied rewrites 94.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-out N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 93.5

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

10. Applied rewrites 93.5%

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

11. Final simplification 93.5%

    \[\leadsto {\left(y \cdot \mathsf{fma}\left(x \cdot z, z, x\right)\right)}^{-1} \]
12. Add Preprocessing

Alternative 8: 59.0% accurate, 0.3× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (* x_s (* y_s (/ (pow x_m -1.0) y_m))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	return x_s * (y_s * (pow(x_m, -1.0) / y_m));
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z_m)
    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_m
    code = x_s * (y_s * ((x_m ** (-1.0d0)) / y_m))
end function

Java

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

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(x_s, y_s, x_m, y_m, z_m):
	return x_s * (y_s * (math.pow(x_m, -1.0) / y_m))

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	return Float64(x_s * Float64(y_s * Float64((x_m ^ -1.0) / y_m)))
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp = code(x_s, y_s, x_m, y_m, z_m)
	tmp = x_s * (y_s * ((x_m ^ -1.0) / y_m));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(N[Power[x$95$m, -1.0], $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.7%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 60.2

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

5. Applied rewrites 60.2%

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

6. Final simplification 60.2%

   \[\leadsto \frac{{x}^{-1}}{y} \]
7. Add Preprocessing

Alternative 9: 59.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot {\left(y\_m \cdot x\_m\right)}^{-1}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z_m)
 :precision binary64
 (* x_s (* y_s (pow (* y_m x_m) -1.0))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m);
double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	return x_s * (y_s * pow((y_m * x_m), -1.0));
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z_m)
    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_m
    code = x_s * (y_s * ((y_m * x_m) ** (-1.0d0)))
end function

Java

z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m;
public static double code(double x_s, double y_s, double x_m, double y_m, double z_m) {
	return x_s * (y_s * Math.pow((y_m * x_m), -1.0));
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(x_s, y_s, x_m, y_m, z_m):
	return x_s * (y_s * math.pow((y_m * x_m), -1.0))

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_s, y_s, x_m, y_m, z_m)
	return Float64(x_s * Float64(y_s * (Float64(y_m * x_m) ^ -1.0)))
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp = code(x_s, y_s, x_m, y_m, z_m)
	tmp = x_s * (y_s * ((y_m * x_m) ^ -1.0));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[Power[N[(y$95$m * x$95$m), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 90.4

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. *-rgt-identity N/A

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

   12. lower-fma.f64 90.4

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

4. Applied rewrites 90.4%

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

5. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 60.0

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

7. Applied rewrites 60.0%

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

8. Final simplification 60.0%

   \[\leadsto {\left(y \cdot x\right)}^{-1} \]
9. Add Preprocessing

Developer Target 1: 92.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + z \cdot z\\ t_1 := y \cdot t\_0\\ t_2 := \frac{\frac{1}{y}}{t\_0 \cdot x}\\ \mathbf{if}\;t\_1 < -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{t\_0 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z z))) (t_1 (* y t_0)) (t_2 (/ (/ 1.0 y) (* t_0 x))))
   (if (< t_1 (- INFINITY))
     t_2
     (if (< t_1 8.680743250567252e+305) (/ (/ 1.0 x) (* t_0 y)) t_2))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z * z)
	t_1 = y * t_0
	t_2 = (1.0 / y) / (t_0 * x)
	tmp = 0
	if t_1 < -math.inf:
		tmp = t_2
	elif t_1 < 8.680743250567252e+305:
		tmp = (1.0 / x) / (t_0 * y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z * z))
	t_1 = Float64(y * t_0)
	t_2 = Float64(Float64(1.0 / y) / Float64(t_0 * x))
	tmp = 0.0
	if (t_1 < Float64(-Inf))
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = Float64(Float64(1.0 / x) / Float64(t_0 * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z * z);
	t_1 = y * t_0;
	t_2 = (1.0 / y) / (t_0 * x);
	tmp = 0.0;
	if (t_1 < -Inf)
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = (1.0 / x) / (t_0 * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / y), $MachinePrecision] / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$1, (-Infinity)], t$95$2, If[Less[t$95$1, 8.680743250567252e+305], N[(N[(1.0 / x), $MachinePrecision] / N[(t$95$0 * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Reproduce

herbie shell --seed 2024326 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* y (+ 1 (* z z))) -inf.0) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 868074325056725200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x)))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))