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

Percentage Accurate: 88.6% → 98.8%

Time: 8.1s

Alternatives: 12

Speedup: 1.1×

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: 88.6% 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: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\frac{--1}{\left(\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot x\_m\right) \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(\left(-z\_m\right) \cdot y\_m\right) \cdot \left(x\_m \cdot z\_m\right)}\\ \end{array}\right) \end{array} \]

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 5e+143)
     (/ (- -1.0) (* (* (fma z_m z_m 1.0) x_m) y_m))
     (/ -1.0 (* (* (- z_m) y_m) (* x_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 <= 5e+143) {
		tmp = -(-1.0) / ((fma(z_m, z_m, 1.0) * x_m) * y_m);
	} else {
		tmp = -1.0 / ((-z_m * y_m) * (x_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 <= 5e+143)
		tmp = Float64(Float64(-(-1.0)) / Float64(Float64(fma(z_m, z_m, 1.0) * x_m) * y_m));
	else
		tmp = Float64(-1.0 / Float64(Float64(Float64(-z_m) * y_m) * Float64(x_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, 5e+143], N[((--1.0) / N[(N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[((-z$95$m) * y$95$m), $MachinePrecision] * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.00000000000000012e143

   1. Initial program 93.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 93.5

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

   4. Applied rewrites 93.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 92.0

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. remove-double-div N/A

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

      17. lift-/.f64 N/A

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

      18. div-inv N/A

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

   6. Applied rewrites 93.1%

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

   if 5.00000000000000012e143 < z

   1. Initial program 85.1%

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

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. *-rgt-identity N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 81.9

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lower-fma.f64 81.9

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

   4. Applied rewrites 81.9%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-out N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 94.6

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

   7. Applied rewrites 94.6%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\frac{--1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(\left(-z\right) \cdot y\right) \cdot \left(x \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{1}{x\_m}}{\left(z\_m \cdot z\_m + 1\right) \cdot y\_m} \leq 0:\\ \;\;\;\;\frac{y\_m}{\left(y\_m \cdot y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \end{array}\right) \end{array} \]

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 (<= (/ (/ 1.0 x_m) (* (+ (* z_m z_m) 1.0) y_m)) 0.0)
     (/ y_m (* (* y_m y_m) x_m))
     (/ (/ 1.0 x_m) 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) {
	double tmp;
	if (((1.0 / x_m) / (((z_m * z_m) + 1.0) * y_m)) <= 0.0) {
		tmp = y_m / ((y_m * y_m) * x_m);
	} else {
		tmp = (1.0 / x_m) / y_m;
	}
	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 (((1.0d0 / x_m) / (((z_m * z_m) + 1.0d0) * y_m)) <= 0.0d0) then
        tmp = y_m / ((y_m * y_m) * x_m)
    else
        tmp = (1.0d0 / x_m) / y_m
    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 (((1.0 / x_m) / (((z_m * z_m) + 1.0) * y_m)) <= 0.0) {
		tmp = y_m / ((y_m * y_m) * x_m);
	} else {
		tmp = (1.0 / x_m) / y_m;
	}
	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 ((1.0 / x_m) / (((z_m * z_m) + 1.0) * y_m)) <= 0.0:
		tmp = y_m / ((y_m * y_m) * x_m)
	else:
		tmp = (1.0 / x_m) / y_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 (Float64(Float64(1.0 / x_m) / Float64(Float64(Float64(z_m * z_m) + 1.0) * y_m)) <= 0.0)
		tmp = Float64(y_m / Float64(Float64(y_m * y_m) * x_m));
	else
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	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 (((1.0 / x_m) / (((z_m * z_m) + 1.0) * y_m)) <= 0.0)
		tmp = y_m / ((y_m * y_m) * x_m);
	else
		tmp = (1.0 / x_m) / y_m;
	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[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision], 0.0], N[(y$95$m / N[(N[(y$95$m * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 1 binary64) x) (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))) < 0.0

   1. Initial program 89.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 51.9

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

   5. Applied rewrites 51.9%

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

   7. Applied rewrites 51.9%

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

   9. Applied rewrites 52.8%

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

   if 0.0 < (/.f64 (/.f64 #s(literal 1 binary64) x) (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))))

   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 81.5

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

   5. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{1}{x}}{\left(z \cdot z + 1\right) \cdot y} \leq 0:\\ \;\;\;\;\frac{y}{\left(y \cdot y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{1}{x\_m}}{\left(z\_m \cdot z\_m + 1\right) \cdot y\_m} \leq 0:\\ \;\;\;\;\frac{y\_m}{\left(y\_m \cdot y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m \cdot x\_m}\\ \end{array}\right) \end{array} \]

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 (<= (/ (/ 1.0 x_m) (* (+ (* z_m z_m) 1.0) y_m)) 0.0)
     (/ y_m (* (* y_m y_m) x_m))
     (/ 1.0 (* y_m x_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 (((1.0 / x_m) / (((z_m * z_m) + 1.0) * y_m)) <= 0.0) {
		tmp = y_m / ((y_m * y_m) * x_m);
	} else {
		tmp = 1.0 / (y_m * x_m);
	}
	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 (((1.0d0 / x_m) / (((z_m * z_m) + 1.0d0) * y_m)) <= 0.0d0) then
        tmp = y_m / ((y_m * y_m) * x_m)
    else
        tmp = 1.0d0 / (y_m * x_m)
    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 (((1.0 / x_m) / (((z_m * z_m) + 1.0) * y_m)) <= 0.0) {
		tmp = y_m / ((y_m * y_m) * x_m);
	} else {
		tmp = 1.0 / (y_m * x_m);
	}
	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 ((1.0 / x_m) / (((z_m * z_m) + 1.0) * y_m)) <= 0.0:
		tmp = y_m / ((y_m * y_m) * x_m)
	else:
		tmp = 1.0 / (y_m * x_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 (Float64(Float64(1.0 / x_m) / Float64(Float64(Float64(z_m * z_m) + 1.0) * y_m)) <= 0.0)
		tmp = Float64(y_m / Float64(Float64(y_m * y_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(y_m * x_m));
	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 (((1.0 / x_m) / (((z_m * z_m) + 1.0) * y_m)) <= 0.0)
		tmp = y_m / ((y_m * y_m) * x_m);
	else
		tmp = 1.0 / (y_m * x_m);
	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[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision], 0.0], N[(y$95$m / N[(N[(y$95$m * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 1 binary64) x) (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))) < 0.0

   1. Initial program 89.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 51.9

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

   5. Applied rewrites 51.9%

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

   7. Applied rewrites 51.9%

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

   9. Applied rewrites 52.8%

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

   if 0.0 < (/.f64 (/.f64 #s(literal 1 binary64) x) (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))))

   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 81.5

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 81.0%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{1}{x}}{\left(z \cdot z + 1\right) \cdot y} \leq 0:\\ \;\;\;\;\frac{y}{\left(y \cdot y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z\_m \cdot z\_m + 1\right) \cdot y\_m \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m \cdot z\_m, z\_m, y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(\left(-z\_m\right) \cdot y\_m\right) \cdot \left(x\_m \cdot z\_m\right)}\\ \end{array}\right) \end{array} \]

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) 1.0) y_m) 2e+306)
     (/ 1.0 (* (fma (* y_m z_m) z_m y_m) x_m))
     (/ -1.0 (* (* (- z_m) y_m) (* x_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 * z_m) + 1.0) * y_m) <= 2e+306) {
		tmp = 1.0 / (fma((y_m * z_m), z_m, y_m) * x_m);
	} else {
		tmp = -1.0 / ((-z_m * y_m) * (x_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 (Float64(Float64(Float64(z_m * z_m) + 1.0) * y_m) <= 2e+306)
		tmp = Float64(1.0 / Float64(fma(Float64(y_m * z_m), z_m, y_m) * x_m));
	else
		tmp = Float64(-1.0 / Float64(Float64(Float64(-z_m) * y_m) * Float64(x_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[N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision], 2e+306], N[(1.0 / N[(N[(N[(y$95$m * z$95$m), $MachinePrecision] * z$95$m + y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[((-z$95$m) * y$95$m), $MachinePrecision] * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 2.00000000000000003e306

   1. Initial program 94.9%

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

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 94.5

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 94.5

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

   4. Applied rewrites 94.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 96.7

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

   6. Applied rewrites 96.7%

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

   if 2.00000000000000003e306 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 81.6%

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

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. *-rgt-identity N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 86.0

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lower-fma.f64 86.0

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

   4. Applied rewrites 86.0%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-out N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 93.6

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

   7. Applied rewrites 93.6%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z + 1\right) \cdot y \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot z, z, y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(\left(-z\right) \cdot y\right) \cdot \left(x \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.9% accurate, 0.9× speedup

Math

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

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 z_m) 0.5)
     (/ 1.0 (* (* (fma z_m z_m 1.0) y_m) x_m))
     (/ 1.0 (* (* (* z_m z_m) x_m) 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) {
	double tmp;
	if ((z_m * z_m) <= 0.5) {
		tmp = 1.0 / ((fma(z_m, z_m, 1.0) * y_m) * x_m);
	} else {
		tmp = 1.0 / (((z_m * z_m) * x_m) * y_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 (Float64(z_m * z_m) <= 0.5)
		tmp = Float64(1.0 / Float64(Float64(fma(z_m, z_m, 1.0) * y_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z_m * z_m) * x_m) * y_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[N[(z$95$m * z$95$m), $MachinePrecision], 0.5], N[(1.0 / N[(N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.5

   1. Initial program 99.7%

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

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.3

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

   if 0.5 < (*.f64 z z)

   1. Initial program 86.0%

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

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 85.7

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 85.7

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   6. 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 85.0

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

   7. Applied rewrites 85.0%

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

4. Final simplification 92.2%

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

Alternative 6: 91.7% accurate, 0.9× speedup

Math

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

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 z_m) 0.5)
     (/ (/ 1.0 x_m) y_m)
     (/ 1.0 (* (* (* z_m z_m) x_m) 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) {
	double tmp;
	if ((z_m * z_m) <= 0.5) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (((z_m * z_m) * x_m) * y_m);
	}
	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) <= 0.5d0) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = 1.0d0 / (((z_m * z_m) * x_m) * y_m)
    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) <= 0.5) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (((z_m * z_m) * x_m) * y_m);
	}
	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) <= 0.5:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = 1.0 / (((z_m * z_m) * x_m) * y_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 (Float64(z_m * z_m) <= 0.5)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z_m * z_m) * x_m) * y_m));
	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) <= 0.5)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = 1.0 / (((z_m * z_m) * x_m) * y_m);
	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], 0.5], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.5

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.5 < (*.f64 z z)

   1. Initial program 86.0%

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

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 85.7

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 85.7

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   6. 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 85.0

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

   7. Applied rewrites 85.0%

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

Alternative 7: 87.9% accurate, 0.9× speedup

Math

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

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 z_m) 0.5)
     (/ (/ 1.0 x_m) y_m)
     (/ 1.0 (* (* (* z_m z_m) y_m) x_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 * z_m) <= 0.5) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (((z_m * z_m) * y_m) * x_m);
	}
	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) <= 0.5d0) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = 1.0d0 / (((z_m * z_m) * y_m) * x_m)
    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) <= 0.5) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (((z_m * z_m) * y_m) * x_m);
	}
	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) <= 0.5:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = 1.0 / (((z_m * z_m) * y_m) * x_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 (Float64(z_m * z_m) <= 0.5)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z_m * z_m) * y_m) * x_m));
	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) <= 0.5)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = 1.0 / (((z_m * z_m) * y_m) * x_m);
	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], 0.5], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.5

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.5 < (*.f64 z z)

   1. Initial program 86.0%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 85.7

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

   5. Applied rewrites 85.7%

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

Alternative 8: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(\left(\left(-x\_m\right) \cdot z\_m\right) \cdot z\_m\right) \cdot y\_m}\\ \end{array}\right) \end{array} \]

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 3.6e+27)
     (/ 1.0 (* (* (fma z_m z_m 1.0) y_m) x_m))
     (/ -1.0 (* (* (* (- x_m) z_m) z_m) 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) {
	double tmp;
	if (z_m <= 3.6e+27) {
		tmp = 1.0 / ((fma(z_m, z_m, 1.0) * y_m) * x_m);
	} else {
		tmp = -1.0 / (((-x_m * z_m) * z_m) * y_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 <= 3.6e+27)
		tmp = Float64(1.0 / Float64(Float64(fma(z_m, z_m, 1.0) * y_m) * x_m));
	else
		tmp = Float64(-1.0 / Float64(Float64(Float64(Float64(-x_m) * z_m) * z_m) * y_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, 3.6e+27], N[(1.0 / N[(N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(N[((-x$95$m) * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.59999999999999983e27

   1. Initial program 93.7%

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

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 93.2

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 93.2

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

   4. Applied rewrites 93.2%

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

   if 3.59999999999999983e27 < z

   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

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. *-rgt-identity N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 86.8

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lower-fma.f64 86.8

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 91.8

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

   6. Applied rewrites 91.8%

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

   7. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-lft-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 93.7

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

   9. Applied rewrites 93.7%

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

   11. Applied rewrites 90.3%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(\left(\left(-x\right) \cdot z\right) \cdot z\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 0.01:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m \cdot z\_m, z\_m, y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot \left(y\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

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 0.01)
     (/ 1.0 (* (fma (* y_m z_m) z_m y_m) x_m))
     (/ 1.0 (* (fma z_m z_m 1.0) (* y_m x_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 (y_m <= 0.01) {
		tmp = 1.0 / (fma((y_m * z_m), z_m, y_m) * x_m);
	} else {
		tmp = 1.0 / (fma(z_m, z_m, 1.0) * (y_m * x_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 (y_m <= 0.01)
		tmp = Float64(1.0 / Float64(fma(Float64(y_m * z_m), z_m, y_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(fma(z_m, z_m, 1.0) * Float64(y_m * x_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[y$95$m, 0.01], N[(1.0 / N[(N[(N[(y$95$m * z$95$m), $MachinePrecision] * z$95$m + y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 0.0100000000000000002

   1. Initial program 91.4%

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

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 91.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 91.1

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

   4. Applied rewrites 91.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 94.9

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

   6. Applied rewrites 94.9%

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

   if 0.0100000000000000002 < y

   1. Initial program 96.5%

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

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 95.9

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 95.9

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

   4. Applied rewrites 95.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 98.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.3

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

   6. Applied rewrites 98.3%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.01:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot z, z, y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(y \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z\_m, z\_m, 1\right) \cdot \left(y\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

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 2e+99)
     (/ 1.0 (* (* (fma z_m z_m 1.0) y_m) x_m))
     (/ 1.0 (* (fma z_m z_m 1.0) (* y_m x_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 (y_m <= 2e+99) {
		tmp = 1.0 / ((fma(z_m, z_m, 1.0) * y_m) * x_m);
	} else {
		tmp = 1.0 / (fma(z_m, z_m, 1.0) * (y_m * x_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 (y_m <= 2e+99)
		tmp = Float64(1.0 / Float64(Float64(fma(z_m, z_m, 1.0) * y_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(fma(z_m, z_m, 1.0) * Float64(y_m * x_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[y$95$m, 2e+99], N[(1.0 / N[(N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(z$95$m * z$95$m + 1.0), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.9999999999999999e99

   1. Initial program 91.9%

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

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 91.7

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 91.7

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

   4. Applied rewrites 91.7%

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

   if 1.9999999999999999e99 < y

   1. Initial program 96.4%

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

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 95.6

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 95.6

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

   4. Applied rewrites 95.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 99.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.0

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

   6. Applied rewrites 99.0%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(y \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.2% accurate, 1.1× speedup

Math

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

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 (/ 1.0 (fma (* (* x_m z_m) z_m) y_m (* y_m x_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 * (1.0 / fma(((x_m * z_m) * z_m), y_m, (y_m * x_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(1.0 / fma(Float64(Float64(x_m * z_m) * z_m), y_m, Float64(y_m * x_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[(1.0 / N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * y$95$m + N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.9%

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

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 92.5

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 92.5

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

4. Applied rewrites 92.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-lft-in N/A

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

   8. *-rgt-identity N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r* N/A

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 95.5

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 95.5

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

6. Applied rewrites 95.5%

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

7. Final simplification 95.5%

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

Alternative 12: 59.0% accurate, 2.1× speedup

Math

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

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 (/ 1.0 (* y_m x_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 * (1.0 / (y_m * x_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 * (1.0d0 / (y_m * x_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 * (1.0 / (y_m * x_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 * (1.0 / (y_m * x_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(1.0 / Float64(y_m * x_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 * (1.0 / (y_m * x_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[(1.0 / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.9%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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 60.9

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

5. Applied rewrites 60.9%

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

7. Applied rewrites 61.9%

   \[\leadsto \frac{1}{\color{blue}{y \cdot x}} \]
8. Add Preprocessing

Developer Target 1: 92.6% 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 2024327 
(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)))))