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

Percentage Accurate: 88.7% → 98.8%

Time: 12.6s

Alternatives: 16

Speedup: 0.5×

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: 88.7% 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, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(N[Power[y$95$m, -1.0], $MachinePrecision] / N[(x$95$m * N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.3%

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

   1. associate-/l/ 86.9%

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

   2. remove-double-neg 86.9%

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

   3. distribute-rgt-neg-out 86.9%

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

   4. distribute-rgt-neg-out 86.9%

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

   5. remove-double-neg 86.9%

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

   6. associate-*l* 87.0%

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

   7. *-commutative 87.0%

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

   8. sqr-neg 87.0%

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

   9. +-commutative 87.0%

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

   10. sqr-neg 87.0%

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

   11. fma-define 87.0%

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

3. Simplified 87.0%

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

   1. associate-*r* 87.5%

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

   2. *-commutative 87.5%

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

   3. associate-/r* 87.6%

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

   4. *-commutative 87.6%

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

   5. associate-/l/ 87.8%

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

   6. fma-undefine 87.8%

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

   7. +-commutative 87.8%

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

   8. associate-/r* 87.3%

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

   9. *-un-lft-identity 87.3%

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

   10. add-sqr-sqrt 46.3%

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

   11. times-frac 46.3%

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

   12. +-commutative 46.3%

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

   13. fma-undefine 46.3%

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

   14. *-commutative 46.3%

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

   15. sqrt-prod 46.3%

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

   16. fma-undefine 46.3%

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

   17. +-commutative 46.3%

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

   18. hypot-1-def 46.3%

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

   19. +-commutative 46.3%

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

6. Applied egg-rr 52.7%

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

   1. associate-*r/ 52.7%

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

   2. associate-*r/ 52.7%

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

   3. *-rgt-identity 52.7%

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

   4. *-commutative 52.7%

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

   5. associate-/r* 52.8%

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

8. Simplified 52.8%

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

   1. frac-2neg 52.8%

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

   2. div-inv 52.7%

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

   3. associate-/l/ 52.7%

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

   4. distribute-neg-frac2 52.7%

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

   5. inv-pow 52.7%

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

   6. sqrt-pow2 52.8%

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

   7. metadata-eval 52.8%

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

   8. distribute-neg-frac2 52.8%

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

   9. associate-/l/ 52.8%

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

   10. distribute-neg-frac2 52.8%

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

   11. inv-pow 52.8%

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

   12. sqrt-pow2 52.8%

       \[\leadsto \frac{{y}^{-0.5}}{-x \cdot \mathsf{hypot}\left(1, z\right)} \cdot \frac{\color{blue}{{y}^{\left(\frac{-1}{2}\right)}}}{-\mathsf{hypot}\left(1, z\right)} \]

   13. metadata-eval 52.8%

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

10. Applied egg-rr 52.8%

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

    1. associate-*r/ 52.8%

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

    2. associate-*l/ 52.7%

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

    3. pow-sqr 98.9%

       \[\leadsto \frac{\frac{\color{blue}{{y}^{\left(2 \cdot -0.5\right)}}}{-x \cdot \mathsf{hypot}\left(1, z\right)}}{-\mathsf{hypot}\left(1, z\right)} \]

    4. metadata-eval 98.9%

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

    5. distribute-rgt-neg-in 98.9%

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

12. Simplified 98.9%

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

13. Final simplification 98.9%

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

Alternative 2: 91.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+134}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x\_m} \cdot \frac{1}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{y\_m}^{-1}}{x\_m \cdot z}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 1e+134)
     (* (/ (/ 1.0 (fma z z 1.0)) x_m) (/ 1.0 y_m))
     (/ (/ (pow y_m -1.0) (* x_m z)) (hypot 1.0 z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1e+134) {
		tmp = ((1.0 / fma(z, z, 1.0)) / x_m) * (1.0 / y_m);
	} else {
		tmp = (pow(y_m, -1.0) / (x_m * z)) / hypot(1.0, z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+134)
		tmp = Float64(Float64(Float64(1.0 / fma(z, z, 1.0)) / x_m) * Float64(1.0 / y_m));
	else
		tmp = Float64(Float64((y_m ^ -1.0) / Float64(x_m * z)) / hypot(1.0, z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1e+134], N[(N[(N[(1.0 / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[y$95$m, -1.0], $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999921e133

   1. Initial program 98.4%

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

      1. associate-/l/ 98.1%

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

      2. remove-double-neg 98.1%

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

      3. distribute-rgt-neg-out 98.1%

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

      4. distribute-rgt-neg-out 98.1%

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

      5. remove-double-neg 98.1%

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

      6. associate-*l* 98.2%

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

      7. *-commutative 98.2%

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

      8. sqr-neg 98.2%

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

      9. +-commutative 98.2%

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

      10. sqr-neg 98.2%

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

      11. fma-define 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in y around 0 98.1%

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

      1. associate-*r* 99.3%

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

      2. +-commutative 99.3%

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

      3. unpow2 99.3%

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

      4. fma-undefine 99.3%

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

      5. associate-/l/ 99.3%

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

   7. Simplified 99.3%

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

      1. associate-/r* 98.4%

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

      2. div-inv 98.4%

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

   9. Applied egg-rr 98.4%

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

   if 9.99999999999999921e133 < (*.f64 z z)

   1. Initial program 70.4%

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

      1. associate-/l/ 69.9%

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

      2. remove-double-neg 69.9%

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

      3. distribute-rgt-neg-out 69.9%

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

      4. distribute-rgt-neg-out 69.9%

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

      5. remove-double-neg 69.9%

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

      6. associate-*l* 70.0%

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

      7. *-commutative 70.0%

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

      8. sqr-neg 70.0%

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

      9. +-commutative 70.0%

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

      10. sqr-neg 70.0%

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

      11. fma-define 70.0%

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

   3. Simplified 70.0%

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

      1. associate-*r* 69.6%

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

      2. *-commutative 69.6%

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

      3. associate-/r* 69.9%

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

      4. *-commutative 69.9%

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

      5. associate-/l/ 69.9%

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

      6. fma-undefine 69.9%

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

      7. +-commutative 69.9%

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

      8. associate-/r* 70.4%

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

      9. *-un-lft-identity 70.4%

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

      10. add-sqr-sqrt 29.4%

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

      11. times-frac 29.4%

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

      12. +-commutative 29.4%

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

      13. fma-undefine 29.4%

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

      14. *-commutative 29.4%

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

      15. sqrt-prod 29.4%

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

      16. fma-undefine 29.4%

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

      17. +-commutative 29.4%

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

      18. hypot-1-def 29.4%

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

      19. +-commutative 29.4%

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

   6. Applied egg-rr 43.8%

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

      1. associate-*r/ 43.8%

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

      2. associate-*r/ 43.8%

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

      3. *-rgt-identity 43.8%

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

      4. *-commutative 43.8%

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

      5. associate-/r* 43.8%

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

   8. Simplified 43.8%

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

      1. frac-2neg 43.8%

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

      2. div-inv 43.8%

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

      3. associate-/l/ 43.9%

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

      4. distribute-neg-frac2 43.9%

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

      5. inv-pow 43.9%

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

      6. sqrt-pow2 43.9%

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

      7. metadata-eval 43.9%

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

      8. distribute-neg-frac2 43.9%

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

      9. associate-/l/ 43.9%

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

      10. distribute-neg-frac2 43.9%

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

      11. inv-pow 43.9%

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

      12. sqrt-pow2 44.0%

          \[\leadsto \frac{{y}^{-0.5}}{-x \cdot \mathsf{hypot}\left(1, z\right)} \cdot \frac{\color{blue}{{y}^{\left(\frac{-1}{2}\right)}}}{-\mathsf{hypot}\left(1, z\right)} \]

      13. metadata-eval 44.0%

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

   10. Applied egg-rr 44.0%

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

       1. associate-*r/ 43.9%

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

       2. associate-*l/ 43.9%

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

       3. pow-sqr 98.9%

          \[\leadsto \frac{\frac{\color{blue}{{y}^{\left(2 \cdot -0.5\right)}}}{-x \cdot \mathsf{hypot}\left(1, z\right)}}{-\mathsf{hypot}\left(1, z\right)} \]

       4. metadata-eval 98.9%

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

       5. distribute-rgt-neg-in 98.9%

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

   12. Simplified 98.9%

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

   13. Taylor expanded in z around inf 71.3%

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

       1. associate-*r* 71.3%

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

       2. neg-mul-1 71.3%

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

   15. Simplified 71.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+134}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x} \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{y}^{-1}}{x \cdot z}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[(x$95$m * N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.3%

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

   1. associate-/l/ 86.9%

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

   2. remove-double-neg 86.9%

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

   3. distribute-rgt-neg-out 86.9%

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

   4. distribute-rgt-neg-out 86.9%

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

   5. remove-double-neg 86.9%

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

   6. associate-*l* 87.0%

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

   7. *-commutative 87.0%

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

   8. sqr-neg 87.0%

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

   9. +-commutative 87.0%

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

   10. sqr-neg 87.0%

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

   11. fma-define 87.0%

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

3. Simplified 87.0%

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

   1. associate-*r* 87.5%

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

   2. *-commutative 87.5%

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

   3. associate-/r* 87.6%

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

   4. *-commutative 87.6%

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

   5. associate-/l/ 87.8%

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

   6. fma-undefine 87.8%

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

   7. +-commutative 87.8%

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

   8. associate-/r* 87.3%

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

   9. *-un-lft-identity 87.3%

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

   10. add-sqr-sqrt 46.3%

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

   11. times-frac 46.3%

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

   12. +-commutative 46.3%

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

   13. fma-undefine 46.3%

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

   14. *-commutative 46.3%

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

   15. sqrt-prod 46.3%

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

   16. fma-undefine 46.3%

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

   17. +-commutative 46.3%

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

   18. hypot-1-def 46.3%

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

   19. +-commutative 46.3%

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

6. Applied egg-rr 52.7%

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

   1. associate-*r/ 52.7%

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

   2. associate-*r/ 52.7%

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

   3. *-rgt-identity 52.7%

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

   4. *-commutative 52.7%

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

   5. associate-/r* 52.8%

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

8. Simplified 52.8%

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

   1. *-un-lft-identity 52.8%

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

   2. div-inv 52.7%

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

   3. associate-/l/ 52.7%

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

   4. associate-/l/ 52.7%

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

   5. frac-times 50.0%

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

   6. frac-times 50.0%

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

   7. metadata-eval 50.0%

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

   8. add-sqr-sqrt 93.7%

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

10. Applied egg-rr 93.7%

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

11. Final simplification 93.7%

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

Alternative 4: 94.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x\_m} \cdot \frac{1}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m}}{y\_m \cdot z}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 2e+258)
     (* (/ (/ 1.0 (fma z z 1.0)) x_m) (/ 1.0 y_m))
     (/ (/ (/ 1.0 x_m) (* y_m z)) (hypot 1.0 z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+258) {
		tmp = ((1.0 / fma(z, z, 1.0)) / x_m) * (1.0 / y_m);
	} else {
		tmp = ((1.0 / x_m) / (y_m * z)) / hypot(1.0, z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+258)
		tmp = Float64(Float64(Float64(1.0 / fma(z, z, 1.0)) / x_m) * Float64(1.0 / y_m));
	else
		tmp = Float64(Float64(Float64(1.0 / x_m) / Float64(y_m * z)) / hypot(1.0, z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+258], N[(N[(N[(1.0 / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * z), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000011e258

   1. Initial program 98.1%

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

      1. associate-/l/ 97.5%

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

      2. remove-double-neg 97.5%

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

      3. distribute-rgt-neg-out 97.5%

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

      4. distribute-rgt-neg-out 97.5%

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

      5. remove-double-neg 97.5%

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

      6. associate-*l* 96.5%

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

      7. *-commutative 96.5%

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

      8. sqr-neg 96.5%

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

      9. +-commutative 96.5%

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

      10. sqr-neg 96.5%

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

      11. fma-define 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in y around 0 97.5%

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

      1. associate-*r* 96.9%

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

      2. +-commutative 96.9%

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

      3. unpow2 96.9%

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

      4. fma-undefine 96.9%

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

      5. associate-/l/ 97.2%

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

   7. Simplified 97.2%

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

      1. associate-/r* 97.6%

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

      2. div-inv 97.5%

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

   9. Applied egg-rr 97.5%

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

   if 2.00000000000000011e258 < (*.f64 z z)

   1. Initial program 63.1%

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

      1. associate-/l/ 63.1%

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

      2. remove-double-neg 63.1%

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

      3. distribute-rgt-neg-out 63.1%

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

      4. distribute-rgt-neg-out 63.1%

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

      5. remove-double-neg 63.1%

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

      6. associate-*l* 65.6%

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

      7. *-commutative 65.6%

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

      8. sqr-neg 65.6%

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

      9. +-commutative 65.6%

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

      10. sqr-neg 65.6%

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

      11. fma-define 65.6%

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

   3. Simplified 65.6%

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

      1. associate-*r* 66.3%

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

      2. *-commutative 66.3%

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

      3. associate-/r* 66.1%

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

      4. *-commutative 66.1%

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

      5. associate-/l/ 66.1%

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

      6. fma-undefine 66.1%

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

      7. +-commutative 66.1%

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

      8. associate-/r* 63.1%

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

      9. *-un-lft-identity 63.1%

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

      10. add-sqr-sqrt 24.0%

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

      11. times-frac 24.0%

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

      12. +-commutative 24.0%

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

      13. fma-undefine 24.0%

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

      14. *-commutative 24.0%

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

      15. sqrt-prod 24.0%

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

      16. fma-undefine 24.0%

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

      17. +-commutative 24.0%

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

      18. hypot-1-def 24.0%

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

      19. +-commutative 24.0%

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

   6. Applied egg-rr 41.6%

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

      1. associate-*r/ 41.6%

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

      2. associate-*r/ 41.6%

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

      3. *-rgt-identity 41.6%

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

      4. *-commutative 41.6%

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

      5. associate-/r* 41.6%

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

   8. Simplified 41.6%

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

      1. frac-2neg 41.6%

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

      2. div-inv 41.5%

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

      3. associate-/l/ 41.6%

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

      4. distribute-neg-frac2 41.6%

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

      5. inv-pow 41.6%

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

      6. sqrt-pow2 41.6%

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

      7. metadata-eval 41.6%

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

      8. distribute-neg-frac2 41.6%

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

      9. associate-/l/ 41.7%

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

      10. distribute-neg-frac2 41.7%

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

      11. inv-pow 41.7%

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

      12. sqrt-pow2 41.7%

          \[\leadsto \frac{{y}^{-0.5}}{-x \cdot \mathsf{hypot}\left(1, z\right)} \cdot \frac{\color{blue}{{y}^{\left(\frac{-1}{2}\right)}}}{-\mathsf{hypot}\left(1, z\right)} \]

      13. metadata-eval 41.7%

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

   10. Applied egg-rr 41.7%

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

       1. associate-*r/ 41.6%

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

       2. associate-*l/ 41.6%

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

       3. pow-sqr 99.8%

          \[\leadsto \frac{\frac{\color{blue}{{y}^{\left(2 \cdot -0.5\right)}}}{-x \cdot \mathsf{hypot}\left(1, z\right)}}{-\mathsf{hypot}\left(1, z\right)} \]

       4. metadata-eval 99.8%

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

       5. distribute-rgt-neg-in 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in z around inf 74.7%

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

       1. associate-/r* 74.6%

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

   15. Simplified 74.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x} \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y \cdot z}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x\_m} \cdot \frac{1}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot \left(y\_m \cdot z\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 2e+258)
     (* (/ (/ 1.0 (fma z z 1.0)) x_m) (/ 1.0 y_m))
     (/ (/ 1.0 (* x_m (* y_m z))) (hypot 1.0 z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+258) {
		tmp = ((1.0 / fma(z, z, 1.0)) / x_m) * (1.0 / y_m);
	} else {
		tmp = (1.0 / (x_m * (y_m * z))) / hypot(1.0, z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+258)
		tmp = Float64(Float64(Float64(1.0 / fma(z, z, 1.0)) / x_m) * Float64(1.0 / y_m));
	else
		tmp = Float64(Float64(1.0 / Float64(x_m * Float64(y_m * z))) / hypot(1.0, z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+258], N[(N[(N[(1.0 / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x$95$m * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000011e258

   1. Initial program 98.1%

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

      1. associate-/l/ 97.5%

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

      2. remove-double-neg 97.5%

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

      3. distribute-rgt-neg-out 97.5%

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

      4. distribute-rgt-neg-out 97.5%

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

      5. remove-double-neg 97.5%

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

      6. associate-*l* 96.5%

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

      7. *-commutative 96.5%

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

      8. sqr-neg 96.5%

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

      9. +-commutative 96.5%

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

      10. sqr-neg 96.5%

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

      11. fma-define 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in y around 0 97.5%

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

      1. associate-*r* 96.9%

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

      2. +-commutative 96.9%

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

      3. unpow2 96.9%

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

      4. fma-undefine 96.9%

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

      5. associate-/l/ 97.2%

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

   7. Simplified 97.2%

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

      1. associate-/r* 97.6%

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

      2. div-inv 97.5%

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

   9. Applied egg-rr 97.5%

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

   if 2.00000000000000011e258 < (*.f64 z z)

   1. Initial program 63.1%

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

      1. associate-/l/ 63.1%

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

      2. remove-double-neg 63.1%

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

      3. distribute-rgt-neg-out 63.1%

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

      4. distribute-rgt-neg-out 63.1%

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

      5. remove-double-neg 63.1%

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

      6. associate-*l* 65.6%

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

      7. *-commutative 65.6%

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

      8. sqr-neg 65.6%

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

      9. +-commutative 65.6%

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

      10. sqr-neg 65.6%

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

      11. fma-define 65.6%

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

   3. Simplified 65.6%

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

      1. associate-*r* 66.3%

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

      2. *-commutative 66.3%

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

      3. associate-/r* 66.1%

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

      4. *-commutative 66.1%

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

      5. associate-/l/ 66.1%

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

      6. fma-undefine 66.1%

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

      7. +-commutative 66.1%

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

      8. associate-/r* 63.1%

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

      9. *-un-lft-identity 63.1%

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

      10. add-sqr-sqrt 24.0%

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

      11. times-frac 24.0%

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

      12. +-commutative 24.0%

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

      13. fma-undefine 24.0%

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

      14. *-commutative 24.0%

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

      15. sqrt-prod 24.0%

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

      16. fma-undefine 24.0%

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

      17. +-commutative 24.0%

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

      18. hypot-1-def 24.0%

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

      19. +-commutative 24.0%

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

   6. Applied egg-rr 41.6%

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

      1. associate-*r/ 41.6%

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

      2. associate-*r/ 41.6%

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

      3. *-rgt-identity 41.6%

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

      4. *-commutative 41.6%

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

      5. associate-/r* 41.6%

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

   8. Simplified 41.6%

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

      1. frac-2neg 41.6%

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

      2. div-inv 41.5%

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

      3. associate-/l/ 41.6%

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

      4. distribute-neg-frac2 41.6%

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

      5. inv-pow 41.6%

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

      6. sqrt-pow2 41.6%

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

      7. metadata-eval 41.6%

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

      8. distribute-neg-frac2 41.6%

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

      9. associate-/l/ 41.7%

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

      10. distribute-neg-frac2 41.7%

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

      11. inv-pow 41.7%

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

      12. sqrt-pow2 41.7%

          \[\leadsto \frac{{y}^{-0.5}}{-x \cdot \mathsf{hypot}\left(1, z\right)} \cdot \frac{\color{blue}{{y}^{\left(\frac{-1}{2}\right)}}}{-\mathsf{hypot}\left(1, z\right)} \]

      13. metadata-eval 41.7%

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

   10. Applied egg-rr 41.7%

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

       1. associate-*r/ 41.6%

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

       2. associate-*l/ 41.6%

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

       3. pow-sqr 99.8%

          \[\leadsto \frac{\frac{\color{blue}{{y}^{\left(2 \cdot -0.5\right)}}}{-x \cdot \mathsf{hypot}\left(1, z\right)}}{-\mathsf{hypot}\left(1, z\right)} \]

       4. metadata-eval 99.8%

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

       5. distribute-rgt-neg-in 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in z around inf 74.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x} \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(y \cdot z\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := y\_m \cdot \left(z \cdot z + 1\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{{z}^{-2}}{y\_m \cdot x\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* y_m (+ (* z z) 1.0))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 INFINITY)
       (/ (/ 1.0 x_m) t_0)
       (/ (pow z -2.0) (* y_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m * ((z * z) + 1.0);
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = pow(z, -2.0) / (y_m * x_m);
	}
	return y_s * (x_s * tmp);
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m * ((z * z) + 1.0);
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = Math.pow(z, -2.0) / (y_m * x_m);
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = y_m * ((z * z) + 1.0)
	tmp = 0
	if t_0 <= math.inf:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = math.pow(z, -2.0) / (y_m * x_m)
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(y_m * Float64(Float64(z * z) + 1.0))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64((z ^ -2.0) / Float64(y_m * x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = y_m * ((z * z) + 1.0);
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = (z ^ -2.0) / (y_m * x_m);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(y$95$m * N[(N[(z * z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, Infinity], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Power[z, -2.0], $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < +inf.0

   1. Initial program 87.3%

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

   if +inf.0 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 87.3%

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

      1. remove-double-neg 87.3%

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

      2. distribute-lft-neg-out 87.3%

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

      3. distribute-rgt-neg-in 87.3%

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

      4. associate-/r* 87.8%

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

      5. associate-/l/ 87.6%

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

      6. associate-/l/ 87.5%

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

      7. distribute-lft-neg-out 87.5%

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

      8. distribute-rgt-neg-in 87.5%

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

      9. distribute-lft-neg-in 87.5%

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

      10. remove-double-neg 87.5%

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

      11. sqr-neg 87.5%

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

      12. +-commutative 87.5%

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

      13. sqr-neg 87.5%

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

      14. fma-define 87.5%

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

      15. *-commutative 87.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in z around inf 47.9%

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

      1. *-un-lft-identity 47.9%

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

      2. pow2 47.9%

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

      3. associate-/r* 48.1%

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

      4. pow2 48.1%

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

      5. pow-flip 48.3%

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

      6. metadata-eval 48.3%

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

      7. *-commutative 48.3%

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

   7. Applied egg-rr 48.3%

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

      1. *-lft-identity 48.3%

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

      2. *-commutative 48.3%

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

   9. Simplified 48.3%

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

4. Final simplification 87.3%

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

Alternative 7: 94.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(y\_m \cdot z, z, y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m} \cdot \frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 5e-54)
     (/ (/ 1.0 x_m) (fma (* y_m z) z y_m))
     (* (/ 1.0 y_m) (/ (/ 1.0 (fma z z 1.0)) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 5e-54) {
		tmp = (1.0 / x_m) / fma((y_m * z), z, y_m);
	} else {
		tmp = (1.0 / y_m) * ((1.0 / fma(z, z, 1.0)) / x_m);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 5e-54)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(y_m * z), z, y_m));
	else
		tmp = Float64(Float64(1.0 / y_m) * Float64(Float64(1.0 / fma(z, z, 1.0)) / x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 5e-54], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(y$95$m * z), $MachinePrecision] * z + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$95$m), $MachinePrecision] * N[(N[(1.0 / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.00000000000000015e-54

   1. Initial program 86.4%

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

      1. +-commutative 86.4%

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

      2. distribute-lft-in 86.4%

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

      3. associate-*r* 93.5%

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

      4. *-rgt-identity 93.5%

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

      5. fma-define 93.5%

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

   4. Applied egg-rr 93.5%

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

   if 5.00000000000000015e-54 < y

   1. Initial program 89.5%

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

      1. associate-/l/ 89.3%

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

      2. remove-double-neg 89.3%

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

      3. distribute-rgt-neg-out 89.3%

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

      4. distribute-rgt-neg-out 89.3%

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

      5. remove-double-neg 89.3%

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

      6. associate-*l* 94.5%

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

      7. *-commutative 94.5%

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

      8. sqr-neg 94.5%

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

      9. +-commutative 94.5%

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

      10. sqr-neg 94.5%

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

      11. fma-define 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in y around 0 89.3%

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

      1. associate-*r* 94.5%

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

      2. +-commutative 94.5%

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

      3. unpow2 94.5%

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

      4. fma-undefine 94.5%

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

      5. associate-/l/ 94.5%

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

   7. Simplified 94.5%

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

      1. associate-/r* 94.6%

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

      2. div-inv 94.5%

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

   9. Applied egg-rr 94.5%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 10000000000000:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(y\_m \cdot z, z, y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{y\_m \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 10000000000000.0)
     (/ (/ 1.0 x_m) (fma (* y_m z) z y_m))
     (/ (/ 1.0 (fma z z 1.0)) (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 10000000000000.0) {
		tmp = (1.0 / x_m) / fma((y_m * z), z, y_m);
	} else {
		tmp = (1.0 / fma(z, z, 1.0)) / (y_m * x_m);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 10000000000000.0)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(y_m * z), z, y_m));
	else
		tmp = Float64(Float64(1.0 / fma(z, z, 1.0)) / Float64(y_m * x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 10000000000000.0], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(y$95$m * z), $MachinePrecision] * z + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1e13

   1. Initial program 86.2%

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

      1. +-commutative 86.2%

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

      2. distribute-lft-in 86.2%

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

      3. associate-*r* 92.7%

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

      4. *-rgt-identity 92.7%

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

      5. fma-define 92.7%

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

   4. Applied egg-rr 92.7%

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

   if 1e13 < y

   1. Initial program 91.2%

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

      1. associate-/l/ 91.1%

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

      2. remove-double-neg 91.1%

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

      3. distribute-rgt-neg-out 91.1%

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

      4. distribute-rgt-neg-out 91.1%

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

      5. remove-double-neg 91.1%

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

      6. associate-*l* 98.0%

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

      7. *-commutative 98.0%

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

      8. sqr-neg 98.0%

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

      9. +-commutative 98.0%

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

      10. sqr-neg 98.0%

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

      11. fma-define 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in y around 0 91.1%

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

      1. associate-*r* 98.0%

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

      2. +-commutative 98.0%

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

      3. unpow2 98.0%

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

      4. fma-undefine 98.0%

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

      5. associate-/l/ 98.0%

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

   7. Simplified 98.0%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10000000000000:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{y \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 0.019:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(y\_m \cdot z, z, y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(y\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 0.019)
     (/ (/ 1.0 x_m) (fma (* y_m z) z y_m))
     (/ 1.0 (* (fma z z 1.0) (* y_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 0.019) {
		tmp = (1.0 / x_m) / fma((y_m * z), z, y_m);
	} else {
		tmp = 1.0 / (fma(z, z, 1.0) * (y_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 0.019)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(y_m * z), z, y_m));
	else
		tmp = Float64(1.0 / Float64(fma(z, z, 1.0) * Float64(y_m * x_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 0.019], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(y$95$m * z), $MachinePrecision] * z + y$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(z * z + 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 < 0.0189999999999999995

   1. Initial program 86.1%

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

      1. +-commutative 86.1%

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

      2. distribute-lft-in 86.1%

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

      3. associate-*r* 92.7%

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

      4. *-rgt-identity 92.7%

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

      5. fma-define 92.7%

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

   4. Applied egg-rr 92.7%

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

   if 0.0189999999999999995 < y

   1. Initial program 91.4%

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

      1. remove-double-neg 91.4%

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

      2. distribute-lft-neg-out 91.4%

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

      3. distribute-rgt-neg-in 91.4%

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

      4. associate-/r* 98.2%

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

      5. associate-/l/ 98.0%

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

      6. associate-/l/ 98.0%

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

      7. distribute-lft-neg-out 98.0%

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

      8. distribute-rgt-neg-in 98.0%

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

      9. distribute-lft-neg-in 98.0%

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

      10. remove-double-neg 98.0%

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

      11. sqr-neg 98.0%

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

      12. +-commutative 98.0%

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

      13. sqr-neg 98.0%

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

      14. fma-define 98.0%

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

      15. *-commutative 98.0%

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

   3. Simplified 98.0%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.019:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(y \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. associate-/l/ 86.9%

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

   2. remove-double-neg 86.9%

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

   3. distribute-rgt-neg-out 86.9%

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

   4. distribute-rgt-neg-out 86.9%

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

   5. remove-double-neg 86.9%

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

   6. associate-*l* 87.0%

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

   7. *-commutative 87.0%

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

   8. sqr-neg 87.0%

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

   9. +-commutative 87.0%

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

   10. sqr-neg 87.0%

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

   11. fma-define 87.0%

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

3. Simplified 87.0%

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

Alternative 11: 88.7% accurate, 0.4× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* y_m (+ (* z z) 1.0))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 INFINITY)
       (/ (/ 1.0 x_m) t_0)
       (/ 1.0 (* (* z z) (* x_m (+ y_m (/ y_m (* z z)))))))))))

C

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

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m * ((z * z) + 1.0);
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = 1.0 / ((z * z) * (x_m * (y_m + (y_m / (z * z)))));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = y_m * ((z * z) + 1.0)
	tmp = 0
	if t_0 <= math.inf:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = 1.0 / ((z * z) * (x_m * (y_m + (y_m / (z * z)))))
	return y_s * (x_s * tmp)

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = y_m * ((z * z) + 1.0);
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = 1.0 / ((z * z) * (x_m * (y_m + (y_m / (z * z)))));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < +inf.0

   1. Initial program 87.3%

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

   if +inf.0 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 87.3%

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

      1. associate-/l/ 86.9%

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

      2. remove-double-neg 86.9%

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

      3. distribute-rgt-neg-out 86.9%

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

      4. distribute-rgt-neg-out 86.9%

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

      5. remove-double-neg 86.9%

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

      6. associate-*l* 87.0%

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

      7. *-commutative 87.0%

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

      8. sqr-neg 87.0%

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

      9. +-commutative 87.0%

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

      10. sqr-neg 87.0%

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

      11. fma-define 87.0%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in z around inf 57.9%

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

      1. associate-/l* 59.2%

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

      2. distribute-lft-out 59.2%

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

   7. Simplified 59.2%

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

      1. unpow2 59.2%

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

   9. Applied egg-rr 59.2%

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

       1. unpow2 59.2%

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

   11. Applied egg-rr 59.2%

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

4. Final simplification 87.3%

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

Alternative 12: 88.7% accurate, 0.5× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* y_m (+ (* z z) 1.0))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 INFINITY)
       (/ (/ 1.0 x_m) t_0)
       (/ 1.0 (* (* z z) (* y_m x_m))))))))

C

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

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m * ((z * z) + 1.0);
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = 1.0 / ((z * z) * (y_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = y_m * ((z * z) + 1.0)
	tmp = 0
	if t_0 <= math.inf:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = 1.0 / ((z * z) * (y_m * x_m))
	return y_s * (x_s * tmp)

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = y_m * ((z * z) + 1.0);
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = 1.0 / ((z * z) * (y_m * x_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

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

   1. Initial program 87.3%

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

   if +inf.0 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 87.3%

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

      1. remove-double-neg 87.3%

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

      2. distribute-lft-neg-out 87.3%

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

      3. distribute-rgt-neg-in 87.3%

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

      4. associate-/r* 87.8%

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

      5. associate-/l/ 87.6%

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

      6. associate-/l/ 87.5%

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

      7. distribute-lft-neg-out 87.5%

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

      8. distribute-rgt-neg-in 87.5%

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

      9. distribute-lft-neg-in 87.5%

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

      10. remove-double-neg 87.5%

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

      11. sqr-neg 87.5%

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

      12. +-commutative 87.5%

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

      13. sqr-neg 87.5%

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

      14. fma-define 87.5%

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

      15. *-commutative 87.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in z around inf 47.9%

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

      1. unpow2 59.2%

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

   7. Applied egg-rr 47.9%

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

4. Final simplification 87.3%

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

Alternative 13: 74.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 89.1%

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

      1. associate-/l/ 88.5%

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

      2. remove-double-neg 88.5%

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

      3. distribute-rgt-neg-out 88.5%

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

      4. distribute-rgt-neg-out 88.5%

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

      5. remove-double-neg 88.5%

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

      6. associate-*l* 87.6%

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

      7. *-commutative 87.6%

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

      8. sqr-neg 87.6%

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

      9. +-commutative 87.6%

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

      10. sqr-neg 87.6%

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

      11. fma-define 87.6%

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

   3. Simplified 87.6%

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

   5. Taylor expanded in z around 0 69.5%

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

      1. *-un-lft-identity 69.5%

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

      2. associate-/r* 69.7%

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

   7. Applied egg-rr 69.7%

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

      1. *-un-lft-identity 69.7%

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

      2. frac-2neg 69.7%

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

      3. distribute-neg-frac 69.7%

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

      4. metadata-eval 69.7%

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

   9. Applied egg-rr 69.7%

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

   if 1 < z

   1. Initial program 81.8%

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

      1. remove-double-neg 81.8%

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

      2. distribute-lft-neg-out 81.8%

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

      3. distribute-rgt-neg-in 81.8%

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

      4. associate-/r* 84.5%

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

      5. associate-/l/ 84.4%

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

      6. associate-/l/ 84.6%

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

      7. distribute-lft-neg-out 84.6%

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

      8. distribute-rgt-neg-in 84.6%

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

      9. distribute-lft-neg-in 84.6%

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

      10. remove-double-neg 84.6%

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

      11. sqr-neg 84.6%

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

      12. +-commutative 84.6%

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

      13. sqr-neg 84.6%

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

      14. fma-define 84.6%

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

      15. *-commutative 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in z around inf 83.6%

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

      1. unpow2 84.6%

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

   7. Applied egg-rr 83.6%

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

4. Final simplification 73.2%

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

Alternative 14: 58.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. associate-/l/ 86.9%

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

   2. remove-double-neg 86.9%

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

   3. distribute-rgt-neg-out 86.9%

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

   4. distribute-rgt-neg-out 86.9%

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

   5. remove-double-neg 86.9%

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

   6. associate-*l* 87.0%

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

   7. *-commutative 87.0%

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

   8. sqr-neg 87.0%

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

   9. +-commutative 87.0%

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

   10. sqr-neg 87.0%

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

   11. fma-define 87.0%

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

3. Simplified 87.0%

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

5. Taylor expanded in z around 0 58.3%

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

   1. *-un-lft-identity 58.3%

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

   2. associate-/r* 58.5%

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

7. Applied egg-rr 58.5%

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

   1. *-un-lft-identity 58.5%

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

   2. frac-2neg 58.5%

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

   3. distribute-neg-frac 58.5%

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

   4. metadata-eval 58.5%

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

9. Applied egg-rr 58.5%

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

10. Final simplification 58.5%

    \[\leadsto \frac{\frac{1}{y}}{x} \]
11. Add Preprocessing

Alternative 15: 58.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in z around 0 58.5%

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

Alternative 16: 58.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. associate-/l/ 86.9%

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

   2. remove-double-neg 86.9%

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

   3. distribute-rgt-neg-out 86.9%

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

   4. distribute-rgt-neg-out 86.9%

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

   5. remove-double-neg 86.9%

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

   6. associate-*l* 87.0%

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

   7. *-commutative 87.0%

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

   8. sqr-neg 87.0%

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

   9. +-commutative 87.0%

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

   10. sqr-neg 87.0%

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

   11. fma-define 87.0%

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

3. Simplified 87.0%

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

5. Taylor expanded in z around 0 58.3%

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

Developer Target 1: 92.6% accurate, 0.3× 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 2024113 
(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)))))