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

Percentage Accurate: 89.0% → 99.1%

Time: 10.7s

Alternatives: 9

Speedup: 0.7×

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: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(z, z, 1\right)}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y\_m \cdot \left(z \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

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 z) 2e+203)
     (/ (/ (/ 1.0 x_m) (fma z z 1.0)) y_m)
     (/ (/ 1.0 z) (* y_m (* z 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 * z) <= 2e+203) {
		tmp = ((1.0 / x_m) / fma(z, z, 1.0)) / y_m;
	} else {
		tmp = (1.0 / z) / (y_m * (z * 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 (Float64(z * z) <= 2e+203)
		tmp = Float64(Float64(Float64(1.0 / x_m) / fma(z, z, 1.0)) / y_m);
	else
		tmp = Float64(Float64(1.0 / z) / Float64(y_m * Float64(z * 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[N[(z * z), $MachinePrecision], 2e+203], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e203

   1. Initial program 98.6%

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

      1. associate-/l/ 98.3%

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

      2. associate-*l* 98.8%

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

      3. *-commutative 98.8%

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

      4. sqr-neg 98.8%

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

      5. +-commutative 98.8%

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

      6. sqr-neg 98.8%

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

      7. fma-define 98.8%

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

   3. Simplified 98.8%

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

      1. fma-undefine 98.8%

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

      2. +-commutative 98.8%

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

      3. *-commutative 98.8%

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

      4. associate-*r* 98.3%

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

      5. associate-/l/ 98.6%

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

      6. *-un-lft-identity 98.6%

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

      7. add-sqr-sqrt 48.5%

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

      8. times-frac 48.5%

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

      9. *-commutative 48.5%

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

      10. sqrt-prod 48.5%

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

      11. hypot-1-def 48.5%

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

      12. *-commutative 48.5%

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

      13. sqrt-prod 49.0%

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

      14. hypot-1-def 49.0%

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

   6. Applied egg-rr 49.0%

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

      1. *-commutative 49.0%

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

      2. div-inv 49.1%

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

      3. associate-/r* 48.5%

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

      4. pow2 48.5%

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

      5. *-commutative 48.5%

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

      6. hypot-1-def 48.5%

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

      7. sqrt-prod 48.5%

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

      8. +-commutative 48.5%

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

      9. fma-undefine 48.5%

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

      10. pow2 48.5%

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

      11. add-sqr-sqrt 98.6%

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

      12. fma-undefine 98.6%

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

      13. +-commutative 98.6%

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

      14. *-commutative 98.6%

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

      15. associate-/r* 99.2%

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

      16. +-commutative 99.2%

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

      17. fma-undefine 99.2%

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

   8. Applied egg-rr 99.2%

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

   if 2e203 < (*.f64 z z)

   1. Initial program 78.5%

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

      1. associate-/l/ 78.5%

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

      2. associate-*l* 75.2%

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

      3. *-commutative 75.2%

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

      4. sqr-neg 75.2%

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

      5. +-commutative 75.2%

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

      6. sqr-neg 75.2%

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

      7. fma-define 75.2%

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

   3. Simplified 75.2%

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

   5. Taylor expanded in z around inf 78.5%

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

      1. associate-/r* 78.5%

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

      2. associate-/r* 77.1%

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

      3. associate-/r* 77.1%

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

   7. Simplified 77.1%

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

      1. *-un-lft-identity 77.1%

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

      2. pow2 77.1%

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

      3. times-frac 91.5%

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

      4. *-commutative 91.5%

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

   9. Applied egg-rr 91.5%

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

       1. clear-num 91.5%

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

       2. un-div-inv 91.6%

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

       3. div-inv 91.6%

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

       4. clear-num 92.7%

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

       5. /-rgt-identity 92.7%

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

       6. *-commutative 92.7%

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

       7. associate-*r* 98.7%

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

   11. Applied egg-rr 98.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y \cdot \left(z \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.0× speedup

Math

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

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 (hypot 1.0 z)) (sqrt y_m))
    (* (sqrt y_m) (* (hypot 1.0 z) 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 / hypot(1.0, z)) / sqrt(y_m)) / (sqrt(y_m) * (hypot(1.0, z) * x_m))));
}

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 / Math.hypot(1.0, z)) / Math.sqrt(y_m)) / (Math.sqrt(y_m) * (Math.hypot(1.0, z) * 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 / math.hypot(1.0, z)) / math.sqrt(y_m)) / (math.sqrt(y_m) * (math.hypot(1.0, z) * 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(Float64(1.0 / hypot(1.0, z)) / sqrt(y_m)) / Float64(sqrt(y_m) * Float64(hypot(1.0, z) * 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 / hypot(1.0, z)) / sqrt(y_m)) / (sqrt(y_m) * (hypot(1.0, z) * x_m))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y$95$m], $MachinePrecision] * N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.2%

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

   1. associate-/l/ 92.0%

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

   2. associate-*l* 91.3%

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

   3. *-commutative 91.3%

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

   4. sqr-neg 91.3%

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

   5. +-commutative 91.3%

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

   6. sqr-neg 91.3%

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

   7. fma-define 91.3%

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

3. Simplified 91.3%

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

   1. fma-undefine 91.3%

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

   2. +-commutative 91.3%

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

   3. *-commutative 91.3%

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

   4. associate-*r* 92.0%

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

   5. associate-/l/ 92.2%

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

   6. *-un-lft-identity 92.2%

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

   7. add-sqr-sqrt 48.2%

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

   8. times-frac 48.2%

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

   9. *-commutative 48.2%

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

   10. sqrt-prod 48.2%

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

   11. hypot-1-def 48.2%

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

   12. *-commutative 48.2%

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

   13. sqrt-prod 48.6%

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

   14. hypot-1-def 52.4%

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

6. Applied egg-rr 52.4%

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

   1. associate-/l/ 52.4%

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

   2. un-div-inv 52.5%

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

   3. associate-/r* 52.5%

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

   4. *-commutative 52.5%

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

   5. associate-*l* 52.1%

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

8. Applied egg-rr 52.1%

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

Alternative 3: 99.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot {\left(\frac{{x\_m}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y\_m}}\right)}^{2}\right) \end{array} \]

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 (/ (pow x_m -0.5) (* (hypot 1.0 z) (sqrt y_m))) 2.0))))

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 * pow((pow(x_m, -0.5) / (hypot(1.0, z) * sqrt(y_m))), 2.0));
}

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((Math.pow(x_m, -0.5) / (Math.hypot(1.0, z) * Math.sqrt(y_m))), 2.0));
}

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((math.pow(x_m, -0.5) / (math.hypot(1.0, z) * math.sqrt(y_m))), 2.0))

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((x_m ^ -0.5) / Float64(hypot(1.0, z) * sqrt(y_m))) ^ 2.0)))
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 * (((x_m ^ -0.5) / (hypot(1.0, z) * sqrt(y_m))) ^ 2.0));
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[Power[N[(N[Power[x$95$m, -0.5], $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.2%

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

   1. associate-/l/ 92.0%

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

   2. associate-*l* 91.3%

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

   3. *-commutative 91.3%

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

   4. sqr-neg 91.3%

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

   5. +-commutative 91.3%

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

   6. sqr-neg 91.3%

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

   7. fma-define 91.3%

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

3. Simplified 91.3%

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

   1. fma-undefine 91.3%

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

   2. +-commutative 91.3%

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

   3. *-commutative 91.3%

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

   4. associate-*r* 92.0%

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

   5. associate-/l/ 92.2%

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

   6. add-sqr-sqrt 63.3%

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

   7. sqrt-div 26.7%

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

   8. inv-pow 26.7%

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

   9. sqrt-pow1 26.7%

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

   10. metadata-eval 26.7%

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

   11. *-commutative 26.7%

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

   12. sqrt-prod 26.7%

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

   13. hypot-1-def 26.7%

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

   14. sqrt-div 26.6%

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

   15. inv-pow 26.7%

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

   16. sqrt-pow1 26.7%

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

   17. metadata-eval 26.7%

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

   18. *-commutative 26.7%

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

6. Applied egg-rr 29.5%

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

   1. unpow2 29.5%

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

8. Simplified 29.5%

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

Alternative 4: 98.9% accurate, 0.1× speedup

Math

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

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 (hypot 1.0 z)) (/ (/ 1.0 x_m) (hypot 1.0 z))) 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 / hypot(1.0, z)) * ((1.0 / x_m) / hypot(1.0, z))) / y_m));
}

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 / Math.hypot(1.0, z)) * ((1.0 / x_m) / Math.hypot(1.0, z))) / 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 / math.hypot(1.0, z)) * ((1.0 / x_m) / math.hypot(1.0, z))) / 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(Float64(1.0 / hypot(1.0, z)) * Float64(Float64(1.0 / x_m) / hypot(1.0, z))) / 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 / hypot(1.0, z)) * ((1.0 / x_m) / hypot(1.0, z))) / 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[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.2%

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

   1. associate-/l/ 92.0%

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

   2. associate-*l* 91.3%

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

   3. *-commutative 91.3%

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

   4. sqr-neg 91.3%

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

   5. +-commutative 91.3%

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

   6. sqr-neg 91.3%

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

   7. fma-define 91.3%

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

3. Simplified 91.3%

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

   1. fma-undefine 91.3%

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

   2. +-commutative 91.3%

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

   3. *-commutative 91.3%

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

   4. associate-*r* 92.0%

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

   5. associate-/l/ 92.2%

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

   6. *-un-lft-identity 92.2%

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

   7. add-sqr-sqrt 48.2%

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

   8. times-frac 48.2%

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

   9. *-commutative 48.2%

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

   10. sqrt-prod 48.2%

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

   11. hypot-1-def 48.2%

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

   12. *-commutative 48.2%

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

   13. sqrt-prod 48.6%

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

   14. hypot-1-def 52.4%

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

6. Applied egg-rr 52.4%

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

   1. *-commutative 52.4%

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

   2. associate-/r* 52.2%

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

   3. associate-/r* 52.2%

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

   4. frac-times 50.2%

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

   5. add-sqr-sqrt 95.2%

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

8. Applied egg-rr 95.2%

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

9. Final simplification 95.2%

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

Alternative 5: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(z \cdot x\_m\right)}}{z}\\ \end{array}\right) \end{array} \end{array} \]

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 (+ 1.0 (* z z)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 5e+291)
       (/ (/ 1.0 x_m) t_0)
       (/ (/ 1.0 (* y_m (* z x_m))) 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 * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 5e+291) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z * z))
    if (t_0 <= 5d+291) then
        tmp = (1.0d0 / x_m) / t_0
    else
        tmp = (1.0d0 / (y_m * (z * x_m))) / z
    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 t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 5e+291) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / 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 * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 5e+291:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = (1.0 / (y_m * (z * x_m))) / 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(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 5e+291)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(z * x_m))) / 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 * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 5e+291)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = (1.0 / (y_m * (z * x_m))) / 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[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 5e+291], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 5.0000000000000001e291

   1. Initial program 95.6%

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

   if 5.0000000000000001e291 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 77.7%

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

      1. associate-/l/ 77.7%

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

      2. associate-*l* 79.6%

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

      3. *-commutative 79.6%

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

      4. sqr-neg 79.6%

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

      5. +-commutative 79.6%

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

      6. sqr-neg 79.6%

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

      7. fma-define 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in z around inf 77.7%

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

      1. associate-/r* 77.7%

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

      2. associate-/r* 79.5%

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

      3. associate-/r* 79.5%

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

   7. Simplified 79.5%

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

      1. *-un-lft-identity 79.5%

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

      2. pow2 79.5%

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

      3. times-frac 95.8%

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

      4. *-commutative 95.8%

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

   9. Applied egg-rr 95.8%

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

       1. associate-*r/ 95.9%

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

       2. associate-/l/ 95.9%

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

       3. clear-num 95.9%

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

       4. frac-times 96.6%

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

       5. metadata-eval 96.6%

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

       6. div-inv 96.6%

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

       7. clear-num 96.6%

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

       8. /-rgt-identity 96.6%

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

       9. *-commutative 96.6%

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

       10. associate-*r* 99.8%

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

   11. Applied egg-rr 99.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 3.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y\_m \cdot \left(z \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

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 z) 3.5e-6)
     (/ (/ 1.0 x_m) y_m)
     (/ (/ 1.0 z) (* y_m (* z 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 * z) <= 3.5e-6) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / z) / (y_m * (z * 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 * z) <= 3.5d-6) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = (1.0d0 / z) / (y_m * (z * 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 * z) <= 3.5e-6) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / z) / (y_m * (z * 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 * z) <= 3.5e-6:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = (1.0 / z) / (y_m * (z * 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 (Float64(z * z) <= 3.5e-6)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(Float64(1.0 / z) / Float64(y_m * Float64(z * 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 * z) <= 3.5e-6)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = (1.0 / z) / (y_m * (z * 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[N[(z * z), $MachinePrecision], 3.5e-6], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.49999999999999995e-6

   1. Initial program 99.7%

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

      1. associate-/l/ 99.3%

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

      2. associate-*l* 99.3%

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

      3. *-commutative 99.3%

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

      4. sqr-neg 99.3%

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

      5. +-commutative 99.3%

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

      6. sqr-neg 99.3%

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

      7. fma-define 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in z around 0 98.9%

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

      1. associate-/r* 99.3%

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

   7. Simplified 99.3%

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

   if 3.49999999999999995e-6 < (*.f64 z z)

   1. Initial program 84.8%

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

      1. associate-/l/ 84.8%

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

      2. associate-*l* 83.3%

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

      3. *-commutative 83.3%

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

      4. sqr-neg 83.3%

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

      5. +-commutative 83.3%

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

      6. sqr-neg 83.3%

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

      7. fma-define 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in z around inf 84.3%

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

      1. associate-/r* 84.3%

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

      2. associate-/r* 83.4%

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

      3. associate-/r* 83.4%

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

   7. Simplified 83.4%

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

      1. *-un-lft-identity 83.4%

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

      2. pow2 83.4%

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

      3. times-frac 92.5%

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

      4. *-commutative 92.5%

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

   9. Applied egg-rr 92.5%

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

       1. clear-num 92.5%

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

       2. un-div-inv 92.6%

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

       3. div-inv 92.5%

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

       4. clear-num 93.3%

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

       5. /-rgt-identity 93.3%

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

       6. *-commutative 93.3%

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

       7. associate-*r* 97.1%

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

   11. Applied egg-rr 97.1%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 3.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y \cdot \left(z \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.0019:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y\_m \cdot \left(z \cdot x\_m\right)\right)}\\ \end{array}\right) \end{array} \]

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 0.0019) (/ (/ 1.0 x_m) y_m) (/ 1.0 (* z (* y_m (* z 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 <= 0.0019) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (z * (y_m * (z * 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 <= 0.0019d0) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = 1.0d0 / (z * (y_m * (z * 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 <= 0.0019) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (z * (y_m * (z * 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 <= 0.0019:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = 1.0 / (z * (y_m * (z * 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 <= 0.0019)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(1.0 / Float64(z * Float64(y_m * Float64(z * 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 <= 0.0019)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = 1.0 / (z * (y_m * (z * 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, 0.0019], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(z * N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.0019

   1. Initial program 94.0%

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

      1. associate-/l/ 93.7%

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

      2. associate-*l* 93.3%

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

      3. *-commutative 93.3%

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

      4. sqr-neg 93.3%

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

      5. +-commutative 93.3%

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

      6. sqr-neg 93.3%

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

      7. fma-define 93.3%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in z around 0 68.2%

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

      1. associate-/r* 68.5%

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

   7. Simplified 68.5%

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

   if 0.0019 < z

   1. Initial program 85.7%

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

      1. associate-/l/ 85.8%

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

      2. associate-*l* 84.1%

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

      3. *-commutative 84.1%

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

      4. sqr-neg 84.1%

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

      5. +-commutative 84.1%

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

      6. sqr-neg 84.1%

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

      7. fma-define 84.1%

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

   3. Simplified 84.1%

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

   5. Taylor expanded in z around inf 85.7%

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

      1. associate-/r* 85.6%

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

      2. associate-/r* 83.9%

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

      3. associate-/r* 83.8%

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

   7. Simplified 83.8%

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

      1. *-un-lft-identity 83.8%

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

      2. pow2 83.8%

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

      3. times-frac 91.1%

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

      4. *-commutative 91.1%

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

   9. Applied egg-rr 91.1%

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

       1. *-commutative 91.1%

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

       2. clear-num 91.1%

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

       3. frac-times 90.6%

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

       4. metadata-eval 90.6%

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

       5. div-inv 90.7%

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

       6. clear-num 91.3%

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

       7. /-rgt-identity 91.3%

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

       8. *-commutative 91.3%

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

       9. associate-*r* 97.5%

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

   11. Applied egg-rr 97.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.0019:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.3% 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 92.2%

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

   1. associate-/l/ 92.0%

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

   2. associate-*l* 91.3%

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

   3. *-commutative 91.3%

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

   4. sqr-neg 91.3%

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

   5. +-commutative 91.3%

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

   6. sqr-neg 91.3%

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

   7. fma-define 91.3%

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

3. Simplified 91.3%

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

5. Taylor expanded in z around 0 59.6%

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

   1. associate-/r* 59.8%

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

7. Simplified 59.8%

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

Alternative 9: 58.4% 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 92.2%

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

   1. associate-/l/ 92.0%

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

   2. associate-*l* 91.3%

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

   3. *-commutative 91.3%

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

   4. sqr-neg 91.3%

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

   5. +-commutative 91.3%

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

   6. sqr-neg 91.3%

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

   7. fma-define 91.3%

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

3. Simplified 91.3%

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

5. Taylor expanded in z around 0 59.6%

   \[\leadsto \frac{1}{y \cdot \color{blue}{x}} \]
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 2024133 
(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)))))