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

Percentage Accurate: 90.6% → 99.1%

Time: 17.4s

Alternatives: 12

Speedup: 0.6×

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: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ [x, y, z_m] = \mathsf{sort}([x, y, z_m])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z\_m \cdot z\_m\right) \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(z\_m \cdot y, z\_m, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\_m\right)} \cdot \frac{1}{y \cdot \left(z\_m \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x y z_m)
 :precision binary64
 (if (<= (* y (+ 1.0 (* z_m z_m))) 4e+307)
   (/ (/ 1.0 x) (fma (* z_m y) z_m y))
   (* (/ 1.0 (hypot 1.0 z_m)) (/ 1.0 (* y (* z_m x))))))

C

z_m = fabs(z);
assert(x < y && y < z_m);
double code(double x, double y, double z_m) {
	double tmp;
	if ((y * (1.0 + (z_m * z_m))) <= 4e+307) {
		tmp = (1.0 / x) / fma((z_m * y), z_m, y);
	} else {
		tmp = (1.0 / hypot(1.0, z_m)) * (1.0 / (y * (z_m * x)));
	}
	return tmp;
}

Julia

z_m = abs(z)
x, y, z_m = sort([x, y, z_m])
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z_m * z_m))) <= 4e+307)
		tmp = Float64(Float64(1.0 / x) / fma(Float64(z_m * y), z_m, y));
	else
		tmp = Float64(Float64(1.0 / hypot(1.0, z_m)) * Float64(1.0 / Float64(y * Float64(z_m * x))));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
code[x_, y_, z$95$m_] := If[LessEqual[N[(y * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+307], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(z$95$m * y), $MachinePrecision] * z$95$m + y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y * N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 3.99999999999999994e307

   1. Initial program 96.6%

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

      1. +-commutative 96.6%

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

      2. distribute-lft-in 96.6%

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

      3. associate-*r* 97.1%

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

      4. *-rgt-identity 97.1%

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

      5. fma-define 97.1%

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

   4. Applied egg-rr 97.1%

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

   if 3.99999999999999994e307 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 78.4%

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

      1. associate-/l/ 78.4%

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

      2. remove-double-neg 78.4%

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

      3. distribute-rgt-neg-out 78.4%

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

      4. distribute-rgt-neg-out 78.4%

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

      5. remove-double-neg 78.4%

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

      6. associate-*l* 85.5%

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

      7. *-commutative 85.5%

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

      8. sqr-neg 85.5%

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

      9. +-commutative 85.5%

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

      10. sqr-neg 85.5%

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

      11. fma-define 85.5%

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

   3. Simplified 85.5%

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

      1. associate-*r* 85.3%

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

      2. *-commutative 85.3%

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

      3. associate-/r* 85.3%

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

      4. *-commutative 85.3%

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

      5. associate-/l/ 85.3%

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

      6. fma-undefine 85.3%

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

      7. +-commutative 85.3%

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

      8. associate-/r* 78.4%

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

      9. *-un-lft-identity 78.4%

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

      10. add-sqr-sqrt 78.4%

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

      11. times-frac 78.4%

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

      12. +-commutative 78.4%

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

      13. fma-undefine 78.4%

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

      14. *-commutative 78.4%

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

      15. sqrt-prod 78.4%

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

      16. fma-undefine 78.4%

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

      17. +-commutative 78.4%

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

      18. hypot-1-def 78.4%

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

      19. +-commutative 78.4%

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

   6. Applied egg-rr 99.6%

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

      1. associate-/l/ 99.7%

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

      2. associate-*r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

      4. associate-/r* 99.7%

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

      5. *-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-/l/ 99.7%

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

      2. div-inv 99.8%

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

      3. associate-*r* 99.7%

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

      4. associate-*l* 99.8%

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

      5. add-sqr-sqrt 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in z around inf 87.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(z \cdot y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{y \cdot \left(z \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.7% accurate, 0.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x y z_m)
 :precision binary64
 (/ (/ (/ 1.0 (hypot 1.0 z_m)) (sqrt y)) (* x (* (hypot 1.0 z_m) (sqrt y)))))

C

z_m = fabs(z);
assert(x < y && y < z_m);
double code(double x, double y, double z_m) {
	return ((1.0 / hypot(1.0, z_m)) / sqrt(y)) / (x * (hypot(1.0, z_m) * sqrt(y)));
}

Java

z_m = Math.abs(z);
assert x < y && y < z_m;
public static double code(double x, double y, double z_m) {
	return ((1.0 / Math.hypot(1.0, z_m)) / Math.sqrt(y)) / (x * (Math.hypot(1.0, z_m) * Math.sqrt(y)));
}

Python

z_m = math.fabs(z)
[x, y, z_m] = sort([x, y, z_m])
def code(x, y, z_m):
	return ((1.0 / math.hypot(1.0, z_m)) / math.sqrt(y)) / (x * (math.hypot(1.0, z_m) * math.sqrt(y)))

Julia

z_m = abs(z)
x, y, z_m = sort([x, y, z_m])
function code(x, y, z_m)
	return Float64(Float64(Float64(1.0 / hypot(1.0, z_m)) / sqrt(y)) / Float64(x * Float64(hypot(1.0, z_m) * sqrt(y))))
end

MATLAB

z_m = abs(z);
x, y, z_m = num2cell(sort([x, y, z_m])){:}
function tmp = code(x, y, z_m)
	tmp = ((1.0 / hypot(1.0, z_m)) / sqrt(y)) / (x * (hypot(1.0, z_m) * sqrt(y)));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
code[x_, y_, z$95$m_] := N[(N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] / N[(x * N[(N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.9%

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

   1. associate-/l/ 93.3%

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

   2. remove-double-neg 93.3%

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

   3. distribute-rgt-neg-out 93.3%

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

   4. distribute-rgt-neg-out 93.3%

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

   5. remove-double-neg 93.3%

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

   6. associate-*l* 94.0%

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

   7. *-commutative 94.0%

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

   8. sqr-neg 94.0%

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

   9. +-commutative 94.0%

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

   10. sqr-neg 94.0%

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

   11. fma-define 94.0%

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

3. Simplified 94.0%

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

   1. associate-*r* 93.4%

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

   2. *-commutative 93.4%

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

   3. associate-/r* 93.4%

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

   4. *-commutative 93.4%

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

   5. associate-/l/ 93.8%

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

   6. fma-undefine 93.8%

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

   7. +-commutative 93.8%

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

   8. associate-/r* 93.9%

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

   9. *-un-lft-identity 93.9%

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

   10. add-sqr-sqrt 50.0%

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

   11. times-frac 49.9%

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

   12. +-commutative 49.9%

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

   13. fma-undefine 49.9%

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

   14. *-commutative 49.9%

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

   15. sqrt-prod 49.9%

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

   16. fma-undefine 49.9%

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

   17. +-commutative 49.9%

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

   18. hypot-1-def 49.9%

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

   19. +-commutative 49.9%

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

6. Applied egg-rr 53.1%

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

   1. associate-/l/ 53.2%

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

   2. associate-*r/ 53.2%

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

   3. *-rgt-identity 53.2%

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

   4. associate-/r* 53.2%

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

   5. *-commutative 53.2%

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

8. Simplified 53.2%

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

Alternative 3: 73.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ [x, y, z_m] = \mathsf{sort}([x, y, z_m])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(1, z\_m\right) \cdot \sqrt{y}\\ \frac{\frac{1}{x \cdot t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (* (hypot 1.0 z_m) (sqrt y)))) (/ (/ 1.0 (* x t_0)) t_0)))

C

z_m = fabs(z);
assert(x < y && y < z_m);
double code(double x, double y, double z_m) {
	double t_0 = hypot(1.0, z_m) * sqrt(y);
	return (1.0 / (x * t_0)) / t_0;
}

Java

z_m = Math.abs(z);
assert x < y && y < z_m;
public static double code(double x, double y, double z_m) {
	double t_0 = Math.hypot(1.0, z_m) * Math.sqrt(y);
	return (1.0 / (x * t_0)) / t_0;
}

Python

z_m = math.fabs(z)
[x, y, z_m] = sort([x, y, z_m])
def code(x, y, z_m):
	t_0 = math.hypot(1.0, z_m) * math.sqrt(y)
	return (1.0 / (x * t_0)) / t_0

Julia

z_m = abs(z)
x, y, z_m = sort([x, y, z_m])
function code(x, y, z_m)
	t_0 = Float64(hypot(1.0, z_m) * sqrt(y))
	return Float64(Float64(1.0 / Float64(x * t_0)) / t_0)
end

MATLAB

z_m = abs(z);
x, y, z_m = num2cell(sort([x, y, z_m])){:}
function tmp = code(x, y, z_m)
	t_0 = hypot(1.0, z_m) * sqrt(y);
	tmp = (1.0 / (x * t_0)) / t_0;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 93.9%

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

   1. associate-/l/ 93.3%

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

   2. remove-double-neg 93.3%

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

   3. distribute-rgt-neg-out 93.3%

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

   4. distribute-rgt-neg-out 93.3%

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

   5. remove-double-neg 93.3%

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

   6. associate-*l* 94.0%

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

   7. *-commutative 94.0%

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

   8. sqr-neg 94.0%

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

   9. +-commutative 94.0%

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

   10. sqr-neg 94.0%

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

   11. fma-define 94.0%

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

3. Simplified 94.0%

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

   1. associate-*r* 93.4%

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

   2. *-commutative 93.4%

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

   3. associate-/r* 93.4%

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

   4. *-commutative 93.4%

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

   5. associate-/l/ 93.8%

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

   6. fma-undefine 93.8%

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

   7. +-commutative 93.8%

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

   8. associate-/r* 93.9%

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

   9. *-un-lft-identity 93.9%

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

   10. add-sqr-sqrt 50.0%

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

   11. times-frac 49.9%

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

   12. +-commutative 49.9%

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

   13. fma-undefine 49.9%

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

   14. *-commutative 49.9%

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

   15. sqrt-prod 49.9%

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

   16. fma-undefine 49.9%

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

   17. +-commutative 49.9%

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

   18. hypot-1-def 49.9%

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

   19. +-commutative 49.9%

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

6. Applied egg-rr 53.1%

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

   1. associate-*l/ 53.2%

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

   2. *-lft-identity 53.2%

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

   3. associate-/l/ 53.2%

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

   4. *-commutative 53.2%

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

8. Simplified 53.2%

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

Alternative 4: 24.9% accurate, 0.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x y z_m)
 :precision binary64
 (/ 1.0 (* y (pow (* (hypot 1.0 z_m) (sqrt x)) 2.0))))

C

z_m = fabs(z);
assert(x < y && y < z_m);
double code(double x, double y, double z_m) {
	return 1.0 / (y * pow((hypot(1.0, z_m) * sqrt(x)), 2.0));
}

Java

z_m = Math.abs(z);
assert x < y && y < z_m;
public static double code(double x, double y, double z_m) {
	return 1.0 / (y * Math.pow((Math.hypot(1.0, z_m) * Math.sqrt(x)), 2.0));
}

Python

z_m = math.fabs(z)
[x, y, z_m] = sort([x, y, z_m])
def code(x, y, z_m):
	return 1.0 / (y * math.pow((math.hypot(1.0, z_m) * math.sqrt(x)), 2.0))

Julia

z_m = abs(z)
x, y, z_m = sort([x, y, z_m])
function code(x, y, z_m)
	return Float64(1.0 / Float64(y * (Float64(hypot(1.0, z_m) * sqrt(x)) ^ 2.0)))
end

MATLAB

z_m = abs(z);
x, y, z_m = num2cell(sort([x, y, z_m])){:}
function tmp = code(x, y, z_m)
	tmp = 1.0 / (y * ((hypot(1.0, z_m) * sqrt(x)) ^ 2.0));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
code[x_, y_, z$95$m_] := N[(1.0 / N[(y * N[Power[N[(N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.9%

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

   1. associate-/l/ 93.3%

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

   2. remove-double-neg 93.3%

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

   3. distribute-rgt-neg-out 93.3%

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

   4. distribute-rgt-neg-out 93.3%

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

   5. remove-double-neg 93.3%

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

   6. associate-*l* 94.0%

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

   7. *-commutative 94.0%

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

   8. sqr-neg 94.0%

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

   9. +-commutative 94.0%

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

   10. sqr-neg 94.0%

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

   11. fma-define 94.0%

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

3. Simplified 94.0%

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

   1. add-sqr-sqrt 47.6%

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

   2. pow2 47.6%

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

   3. *-commutative 47.6%

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

   4. sqrt-prod 47.5%

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

   5. fma-undefine 47.5%

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

   6. +-commutative 47.5%

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

   7. hypot-1-def 48.6%

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

6. Applied egg-rr 48.6%

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

Alternative 5: 97.4% accurate, 0.1× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x y z_m)
 :precision binary64
 (* (/ 1.0 (hypot 1.0 z_m)) (/ 1.0 (* y (* (hypot 1.0 z_m) x)))))

C

z_m = fabs(z);
assert(x < y && y < z_m);
double code(double x, double y, double z_m) {
	return (1.0 / hypot(1.0, z_m)) * (1.0 / (y * (hypot(1.0, z_m) * x)));
}

Java

z_m = Math.abs(z);
assert x < y && y < z_m;
public static double code(double x, double y, double z_m) {
	return (1.0 / Math.hypot(1.0, z_m)) * (1.0 / (y * (Math.hypot(1.0, z_m) * x)));
}

Python

z_m = math.fabs(z)
[x, y, z_m] = sort([x, y, z_m])
def code(x, y, z_m):
	return (1.0 / math.hypot(1.0, z_m)) * (1.0 / (y * (math.hypot(1.0, z_m) * x)))

Julia

z_m = abs(z)
x, y, z_m = sort([x, y, z_m])
function code(x, y, z_m)
	return Float64(Float64(1.0 / hypot(1.0, z_m)) * Float64(1.0 / Float64(y * Float64(hypot(1.0, z_m) * x))))
end

MATLAB

z_m = abs(z);
x, y, z_m = num2cell(sort([x, y, z_m])){:}
function tmp = code(x, y, z_m)
	tmp = (1.0 / hypot(1.0, z_m)) * (1.0 / (y * (hypot(1.0, z_m) * x)));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
code[x_, y_, z$95$m_] := N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y * N[(N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.9%

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

   1. associate-/l/ 93.3%

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

   2. remove-double-neg 93.3%

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

   3. distribute-rgt-neg-out 93.3%

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

   4. distribute-rgt-neg-out 93.3%

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

   5. remove-double-neg 93.3%

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

   6. associate-*l* 94.0%

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

   7. *-commutative 94.0%

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

   8. sqr-neg 94.0%

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

   9. +-commutative 94.0%

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

   10. sqr-neg 94.0%

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

   11. fma-define 94.0%

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

3. Simplified 94.0%

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

   1. associate-*r* 93.4%

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

   2. *-commutative 93.4%

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

   3. associate-/r* 93.4%

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

   4. *-commutative 93.4%

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

   5. associate-/l/ 93.8%

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

   6. fma-undefine 93.8%

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

   7. +-commutative 93.8%

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

   8. associate-/r* 93.9%

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

   9. *-un-lft-identity 93.9%

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

   10. add-sqr-sqrt 50.0%

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

   11. times-frac 49.9%

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

   12. +-commutative 49.9%

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

   13. fma-undefine 49.9%

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

   14. *-commutative 49.9%

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

   15. sqrt-prod 49.9%

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

   16. fma-undefine 49.9%

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

   17. +-commutative 49.9%

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

   18. hypot-1-def 49.9%

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

   19. +-commutative 49.9%

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

6. Applied egg-rr 53.1%

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

   1. associate-/l/ 53.2%

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

   2. associate-*r/ 53.2%

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

   3. *-rgt-identity 53.2%

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

   4. associate-/r* 53.2%

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

   5. *-commutative 53.2%

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

8. Simplified 53.2%

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

   1. associate-/l/ 52.7%

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

   2. div-inv 52.7%

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

   3. associate-*r* 52.6%

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

   4. associate-*l* 52.6%

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

   5. add-sqr-sqrt 97.8%

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

10. Applied egg-rr 97.8%

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

11. Final simplification 97.8%

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

Alternative 6: 97.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ [x, y, z_m] = \mathsf{sort}([x, y, z_m])\\ \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 2 \cdot 10^{+241}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \mathsf{fma}\left(z\_m, z\_m, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z\_m}}{y} \cdot \frac{\frac{1}{z\_m}}{x}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x y z_m)
 :precision binary64
 (if (<= (* z_m z_m) 2e+241)
   (/ 1.0 (* y (* x (fma z_m z_m 1.0))))
   (* (/ (/ 1.0 z_m) y) (/ (/ 1.0 z_m) x))))

C

z_m = fabs(z);
assert(x < y && y < z_m);
double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e+241) {
		tmp = 1.0 / (y * (x * fma(z_m, z_m, 1.0)));
	} else {
		tmp = ((1.0 / z_m) / y) * ((1.0 / z_m) / x);
	}
	return tmp;
}

Julia

z_m = abs(z)
x, y, z_m = sort([x, y, z_m])
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 2e+241)
		tmp = Float64(1.0 / Float64(y * Float64(x * fma(z_m, z_m, 1.0))));
	else
		tmp = Float64(Float64(Float64(1.0 / z_m) / y) * Float64(Float64(1.0 / z_m) / x));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
code[x_, y_, z$95$m_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+241], N[(1.0 / N[(y * N[(x * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z$95$m), $MachinePrecision] / y), $MachinePrecision] * N[(N[(1.0 / z$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e241

   1. Initial program 97.7%

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

      1. associate-/l/ 96.9%

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

      2. remove-double-neg 96.9%

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

      3. distribute-rgt-neg-out 96.9%

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

      4. distribute-rgt-neg-out 96.9%

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

      5. remove-double-neg 96.9%

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

      6. associate-*l* 96.9%

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

      7. *-commutative 96.9%

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

      8. sqr-neg 96.9%

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

      9. +-commutative 96.9%

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

      10. sqr-neg 96.9%

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

      11. fma-define 96.9%

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

   3. Simplified 96.9%

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

   if 2.0000000000000001e241 < (*.f64 z z)

   1. Initial program 82.8%

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

      1. remove-double-neg 82.8%

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

      2. distribute-lft-neg-out 82.8%

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

      3. distribute-rgt-neg-in 82.8%

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

      4. associate-/r* 84.0%

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

      5. associate-/l/ 84.0%

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

      6. associate-/l/ 84.0%

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

      7. distribute-lft-neg-out 84.0%

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

      8. distribute-rgt-neg-in 84.0%

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

      9. distribute-lft-neg-in 84.0%

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

      10. remove-double-neg 84.0%

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

      11. sqr-neg 84.0%

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

      12. +-commutative 84.0%

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

      13. sqr-neg 84.0%

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

      14. fma-define 84.0%

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

      15. *-commutative 84.0%

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

   3. Simplified 84.0%

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

   5. Taylor expanded in z around inf 84.0%

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

      1. associate-/r* 84.0%

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

      2. add-sqr-sqrt 84.0%

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

      3. *-commutative 84.0%

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

      4. times-frac 85.7%

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

      5. sqrt-div 85.7%

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

      6. metadata-eval 85.7%

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

      7. sqrt-pow1 79.8%

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

      8. metadata-eval 79.8%

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

      9. pow1 79.8%

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

      10. sqrt-div 79.8%

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

      11. metadata-eval 79.8%

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

      12. sqrt-pow1 98.5%

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

      13. metadata-eval 98.5%

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

      14. pow1 98.5%

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

   7. Applied egg-rr 98.5%

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

Alternative 7: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ [x, y, z_m] = \mathsf{sort}([x, y, z_m])\\ \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{elif}\;z\_m \cdot z\_m \leq 10^{+243}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \left(z\_m \cdot z\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y \cdot x}}{z\_m}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x y z_m)
 :precision binary64
 (if (<= (* z_m z_m) 1e-9)
   (/ (/ 1.0 x) y)
   (if (<= (* z_m z_m) 1e+243)
     (/ 1.0 (* y (* x (* z_m z_m))))
     (/ (/ (/ 1.0 (* y x)) z_m) z_m))))

C

z_m = fabs(z);
assert(x < y && y < z_m);
double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1e-9) {
		tmp = (1.0 / x) / y;
	} else if ((z_m * z_m) <= 1e+243) {
		tmp = 1.0 / (y * (x * (z_m * z_m)));
	} else {
		tmp = ((1.0 / (y * x)) / z_m) / z_m;
	}
	return tmp;
}

Fortran

z_m = abs(z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((z_m * z_m) <= 1d-9) then
        tmp = (1.0d0 / x) / y
    else if ((z_m * z_m) <= 1d+243) then
        tmp = 1.0d0 / (y * (x * (z_m * z_m)))
    else
        tmp = ((1.0d0 / (y * x)) / z_m) / z_m
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
assert x < y && y < z_m;
public static double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1e-9) {
		tmp = (1.0 / x) / y;
	} else if ((z_m * z_m) <= 1e+243) {
		tmp = 1.0 / (y * (x * (z_m * z_m)));
	} else {
		tmp = ((1.0 / (y * x)) / z_m) / z_m;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
[x, y, z_m] = sort([x, y, z_m])
def code(x, y, z_m):
	tmp = 0
	if (z_m * z_m) <= 1e-9:
		tmp = (1.0 / x) / y
	elif (z_m * z_m) <= 1e+243:
		tmp = 1.0 / (y * (x * (z_m * z_m)))
	else:
		tmp = ((1.0 / (y * x)) / z_m) / z_m
	return tmp

Julia

z_m = abs(z)
x, y, z_m = sort([x, y, z_m])
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 1e-9)
		tmp = Float64(Float64(1.0 / x) / y);
	elseif (Float64(z_m * z_m) <= 1e+243)
		tmp = Float64(1.0 / Float64(y * Float64(x * Float64(z_m * z_m))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(y * x)) / z_m) / z_m);
	end
	return tmp
end

MATLAB

z_m = abs(z);
x, y, z_m = num2cell(sort([x, y, z_m])){:}
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 1e-9)
		tmp = (1.0 / x) / y;
	elseif ((z_m * z_m) <= 1e+243)
		tmp = 1.0 / (y * (x * (z_m * z_m)));
	else
		tmp = ((1.0 / (y * x)) / z_m) / z_m;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
code[x_, y_, z$95$m_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1e-9], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1e+243], N[(1.0 / N[(y * N[(x * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(y * x), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 1.00000000000000006e-9

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0 99.3%

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

   if 1.00000000000000006e-9 < (*.f64 z z) < 1.0000000000000001e243

   1. Initial program 92.4%

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

      1. associate-/l/ 91.7%

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

      2. remove-double-neg 91.7%

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

      3. distribute-rgt-neg-out 91.7%

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

      4. distribute-rgt-neg-out 91.7%

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

      5. remove-double-neg 91.7%

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

      6. associate-*l* 91.5%

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

      7. *-commutative 91.5%

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

      8. sqr-neg 91.5%

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

      9. +-commutative 91.5%

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

      10. sqr-neg 91.5%

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

      11. fma-define 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in z around inf 89.4%

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

      1. pow2 89.4%

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

   7. Applied egg-rr 89.4%

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

   if 1.0000000000000001e243 < (*.f64 z z)

   1. Initial program 82.6%

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

      1. associate-/l/ 82.6%

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

      2. remove-double-neg 82.6%

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

      3. distribute-rgt-neg-out 82.6%

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

      4. distribute-rgt-neg-out 82.6%

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

      5. remove-double-neg 82.6%

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

      6. associate-*l* 85.4%

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

      7. *-commutative 85.4%

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

      8. sqr-neg 85.4%

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

      9. +-commutative 85.4%

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

      10. sqr-neg 85.4%

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

      11. fma-define 85.4%

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

   3. Simplified 85.4%

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

   5. Taylor expanded in z around inf 85.4%

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

      1. associate-/r* 85.4%

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

      2. *-un-lft-identity 85.4%

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

      3. pow2 85.4%

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

      4. *-commutative 85.4%

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

      5. times-frac 85.2%

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

      6. pow2 85.2%

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

      7. associate-/l/ 85.2%

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

   7. Applied egg-rr 85.2%

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

      1. associate-*l/ 85.2%

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

      2. *-un-lft-identity 85.2%

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

      3. associate-/r* 85.2%

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

      4. inv-pow 85.2%

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

      5. metadata-eval 85.2%

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

      6. pow-pow 47.9%

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

      7. unpow2 47.9%

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

      8. associate-/r* 50.8%

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

      9. pow-pow 93.9%

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

      10. metadata-eval 93.9%

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

      11. inv-pow 93.9%

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

      12. associate-/l/ 93.9%

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

   9. Applied egg-rr 93.9%

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

Alternative 8: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ [x, y, z_m] = \mathsf{sort}([x, y, z_m])\\ \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 20000000:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z\_m \cdot z\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z\_m} \cdot \frac{\frac{1}{x}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x y z_m)
 :precision binary64
 (if (<= (* z_m z_m) 20000000.0)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z_m z_m))))
   (* (/ (/ 1.0 y) z_m) (/ (/ 1.0 x) z_m))))

C

z_m = fabs(z);
assert(x < y && y < z_m);
double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 20000000.0) {
		tmp = (1.0 / x) / (y * (1.0 + (z_m * z_m)));
	} else {
		tmp = ((1.0 / y) / z_m) * ((1.0 / x) / z_m);
	}
	return tmp;
}

Fortran

z_m = abs(z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((z_m * z_m) <= 20000000.0d0) then
        tmp = (1.0d0 / x) / (y * (1.0d0 + (z_m * z_m)))
    else
        tmp = ((1.0d0 / y) / z_m) * ((1.0d0 / x) / z_m)
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
assert x < y && y < z_m;
public static double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 20000000.0) {
		tmp = (1.0 / x) / (y * (1.0 + (z_m * z_m)));
	} else {
		tmp = ((1.0 / y) / z_m) * ((1.0 / x) / z_m);
	}
	return tmp;
}

Python

z_m = math.fabs(z)
[x, y, z_m] = sort([x, y, z_m])
def code(x, y, z_m):
	tmp = 0
	if (z_m * z_m) <= 20000000.0:
		tmp = (1.0 / x) / (y * (1.0 + (z_m * z_m)))
	else:
		tmp = ((1.0 / y) / z_m) * ((1.0 / x) / z_m)
	return tmp

Julia

z_m = abs(z)
x, y, z_m = sort([x, y, z_m])
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 20000000.0)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z_m * z_m))));
	else
		tmp = Float64(Float64(Float64(1.0 / y) / z_m) * Float64(Float64(1.0 / x) / z_m));
	end
	return tmp
end

MATLAB

z_m = abs(z);
x, y, z_m = num2cell(sort([x, y, z_m])){:}
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 20000000.0)
		tmp = (1.0 / x) / (y * (1.0 + (z_m * z_m)));
	else
		tmp = ((1.0 / y) / z_m) * ((1.0 / x) / z_m);
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
code[x_, y_, z$95$m_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 20000000.0], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] / z$95$m), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e7

   1. Initial program 99.7%

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

   if 2e7 < (*.f64 z z)

   1. Initial program 86.7%

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

      1. remove-double-neg 86.7%

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

      2. distribute-lft-neg-out 86.7%

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

      3. distribute-rgt-neg-in 86.7%

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

      4. associate-/r* 86.5%

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

      5. associate-/l/ 86.6%

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

      6. associate-/l/ 86.6%

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

      7. distribute-lft-neg-out 86.6%

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

      8. distribute-rgt-neg-in 86.6%

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

      9. distribute-lft-neg-in 86.6%

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

      10. remove-double-neg 86.6%

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

      11. sqr-neg 86.6%

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

      12. +-commutative 86.6%

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

      13. sqr-neg 86.6%

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

      14. fma-define 86.6%

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

      15. *-commutative 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in z around inf 86.6%

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

      1. metadata-eval 86.6%

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

      2. frac-times 86.6%

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

      3. associate-*l/ 86.6%

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

      4. associate-/r* 86.5%

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

      5. inv-pow 86.5%

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

      6. metadata-eval 86.5%

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

      7. pow-pow 45.4%

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

      8. associate-*r/ 45.4%

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

      9. associate-*l/ 45.4%

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

      10. unpow2 45.4%

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

      11. times-frac 50.4%

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

      12. pow-pow 96.5%

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

      13. metadata-eval 96.5%

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

      14. inv-pow 96.5%

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

   7. Applied egg-rr 96.5%

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

Alternative 9: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ [x, y, z_m] = \mathsf{sort}([x, y, z_m])\\ \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z\_m} \cdot \frac{\frac{1}{x}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x y z_m)
 :precision binary64
 (if (<= (* z_m z_m) 1e-9)
   (/ (/ 1.0 x) y)
   (* (/ (/ 1.0 y) z_m) (/ (/ 1.0 x) z_m))))

C

z_m = fabs(z);
assert(x < y && y < z_m);
double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1e-9) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = ((1.0 / y) / z_m) * ((1.0 / x) / z_m);
	}
	return tmp;
}

Fortran

z_m = abs(z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((z_m * z_m) <= 1d-9) then
        tmp = (1.0d0 / x) / y
    else
        tmp = ((1.0d0 / y) / z_m) * ((1.0d0 / x) / z_m)
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
assert x < y && y < z_m;
public static double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1e-9) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = ((1.0 / y) / z_m) * ((1.0 / x) / z_m);
	}
	return tmp;
}

Python

z_m = math.fabs(z)
[x, y, z_m] = sort([x, y, z_m])
def code(x, y, z_m):
	tmp = 0
	if (z_m * z_m) <= 1e-9:
		tmp = (1.0 / x) / y
	else:
		tmp = ((1.0 / y) / z_m) * ((1.0 / x) / z_m)
	return tmp

Julia

z_m = abs(z)
x, y, z_m = sort([x, y, z_m])
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 1e-9)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(Float64(Float64(1.0 / y) / z_m) * Float64(Float64(1.0 / x) / z_m));
	end
	return tmp
end

MATLAB

z_m = abs(z);
x, y, z_m = num2cell(sort([x, y, z_m])){:}
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 1e-9)
		tmp = (1.0 / x) / y;
	else
		tmp = ((1.0 / y) / z_m) * ((1.0 / x) / z_m);
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
code[x_, y_, z$95$m_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1e-9], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] / z$95$m), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000006e-9

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0 99.3%

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

   if 1.00000000000000006e-9 < (*.f64 z z)

   1. Initial program 86.9%

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

      1. remove-double-neg 86.9%

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

      2. distribute-lft-neg-out 86.9%

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

      3. distribute-rgt-neg-in 86.9%

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

      4. associate-/r* 86.7%

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

      5. associate-/l/ 86.8%

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

      6. associate-/l/ 86.8%

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

      7. distribute-lft-neg-out 86.8%

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

      8. distribute-rgt-neg-in 86.8%

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

      9. distribute-lft-neg-in 86.8%

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

      10. remove-double-neg 86.8%

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

      11. sqr-neg 86.8%

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

      12. +-commutative 86.8%

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

      13. sqr-neg 86.8%

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

      14. fma-define 86.8%

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

      15. *-commutative 86.8%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in z around inf 85.9%

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

      1. metadata-eval 85.9%

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

      2. frac-times 85.8%

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

      3. associate-*l/ 85.8%

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

      4. associate-/r* 85.8%

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

      5. inv-pow 85.8%

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

      6. metadata-eval 85.8%

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

      7. pow-pow 44.6%

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

      8. associate-*r/ 44.6%

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

      9. associate-*l/ 44.6%

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

      10. unpow2 44.6%

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

      11. times-frac 49.5%

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

      12. pow-pow 95.6%

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

      13. metadata-eval 95.6%

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

      14. inv-pow 95.6%

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

   7. Applied egg-rr 95.6%

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

Alternative 10: 89.9% accurate, 0.7× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x y z_m)
 :precision binary64
 (if (<= (* z_m z_m) 1.0) (/ (/ 1.0 x) y) (/ 1.0 (* y (* x (* z_m z_m))))))

C

z_m = fabs(z);
assert(x < y && y < z_m);
double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1.0) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (y * (x * (z_m * z_m)));
	}
	return tmp;
}

Fortran

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

Java

z_m = Math.abs(z);
assert x < y && y < z_m;
public static double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1.0) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (y * (x * (z_m * z_m)));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
[x, y, z_m] = sort([x, y, z_m])
def code(x, y, z_m):
	tmp = 0
	if (z_m * z_m) <= 1.0:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (y * (x * (z_m * z_m)))
	return tmp

Julia

z_m = abs(z)
x, y, z_m = sort([x, y, z_m])
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 1.0)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(y * Float64(x * Float64(z_m * z_m))));
	end
	return tmp
end

MATLAB

z_m = abs(z);
x, y, z_m = num2cell(sort([x, y, z_m])){:}
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 1.0)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (y * (x * (z_m * z_m)));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
code[x_, y_, z$95$m_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1.0], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(y * N[(x * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0 99.3%

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

   if 1 < (*.f64 z z)

   1. Initial program 86.9%

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

      1. associate-/l/ 86.7%

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

      2. remove-double-neg 86.7%

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

      3. distribute-rgt-neg-out 86.7%

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

      4. distribute-rgt-neg-out 86.7%

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

      5. remove-double-neg 86.7%

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

      6. associate-*l* 88.1%

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

      7. *-commutative 88.1%

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

      8. sqr-neg 88.1%

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

      9. +-commutative 88.1%

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

      10. sqr-neg 88.1%

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

      11. fma-define 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in z around inf 87.2%

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

      1. pow2 87.2%

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

   7. Applied egg-rr 87.2%

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

Alternative 11: 58.6% accurate, 2.2× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x y z_m) :precision binary64 (/ (/ 1.0 x) y))

C

z_m = fabs(z);
assert(x < y && y < z_m);
double code(double x, double y, double z_m) {
	return (1.0 / x) / y;
}

Fortran

z_m = abs(z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = (1.0d0 / x) / y
end function

Java

z_m = Math.abs(z);
assert x < y && y < z_m;
public static double code(double x, double y, double z_m) {
	return (1.0 / x) / y;
}

Python

z_m = math.fabs(z)
[x, y, z_m] = sort([x, y, z_m])
def code(x, y, z_m):
	return (1.0 / x) / y

Julia

z_m = abs(z)
x, y, z_m = sort([x, y, z_m])
function code(x, y, z_m)
	return Float64(Float64(1.0 / x) / y)
end

MATLAB

z_m = abs(z);
x, y, z_m = num2cell(sort([x, y, z_m])){:}
function tmp = code(x, y, z_m)
	tmp = (1.0 / x) / y;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
code[x_, y_, z$95$m_] := N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision]

Derivation

1. Initial program 93.9%

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

3. Taylor expanded in z around 0 62.5%

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

Alternative 12: 58.6% accurate, 2.2× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x y z_m) :precision binary64 (/ 1.0 (* y x)))

C

z_m = fabs(z);
assert(x < y && y < z_m);
double code(double x, double y, double z_m) {
	return 1.0 / (y * x);
}

Fortran

z_m = abs(z)
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = 1.0d0 / (y * x)
end function

Java

z_m = Math.abs(z);
assert x < y && y < z_m;
public static double code(double x, double y, double z_m) {
	return 1.0 / (y * x);
}

Python

z_m = math.fabs(z)
[x, y, z_m] = sort([x, y, z_m])
def code(x, y, z_m):
	return 1.0 / (y * x)

Julia

z_m = abs(z)
x, y, z_m = sort([x, y, z_m])
function code(x, y, z_m)
	return Float64(1.0 / Float64(y * x))
end

MATLAB

z_m = abs(z);
x, y, z_m = num2cell(sort([x, y, z_m])){:}
function tmp = code(x, y, z_m)
	tmp = 1.0 / (y * x);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
NOTE: x, y, and z_m should be sorted in increasing order before calling this function.
code[x_, y_, z$95$m_] := N[(1.0 / N[(y * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.9%

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

   1. associate-/l/ 93.3%

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

   2. remove-double-neg 93.3%

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

   3. distribute-rgt-neg-out 93.3%

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

   4. distribute-rgt-neg-out 93.3%

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

   5. remove-double-neg 93.3%

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

   6. associate-*l* 94.0%

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

   7. *-commutative 94.0%

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

   8. sqr-neg 94.0%

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

   9. +-commutative 94.0%

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

   10. sqr-neg 94.0%

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

   11. fma-define 94.0%

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

3. Simplified 94.0%

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

5. Taylor expanded in z around 0 62.4%

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

Developer target: 92.8% 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 2024110 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (if (< (* y (+ 1.0 (* z z))) (- INFINITY)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

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