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

Percentage Accurate: 89.3% → 99.4%

Time: 11.2s

Alternatives: 11

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: 89.3% 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: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot \left(1 + z\_m \cdot z\_m\right) \leq 1.2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(y\_m \cdot z\_m, z\_m, y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{x\_m}}{\mathsf{hypot}\left(1, z\_m\right)}}{y\_m}}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* y_m (+ 1.0 (* z_m z_m))) 1.2e+307)
     (/ (/ 1.0 x_m) (fma (* y_m z_m) z_m y_m))
     (/ (/ (/ (/ 1.0 x_m) (hypot 1.0 z_m)) y_m) z_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((y_m * (1.0 + (z_m * z_m))) <= 1.2e+307) {
		tmp = (1.0 / x_m) / fma((y_m * z_m), z_m, y_m);
	} else {
		tmp = (((1.0 / x_m) / hypot(1.0, z_m)) / y_m) / z_m;
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(y_m * Float64(1.0 + Float64(z_m * z_m))) <= 1.2e+307)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(y_m * z_m), z_m, y_m));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / x_m) / hypot(1.0, z_m)) / y_m) / z_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.2e+307], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(y$95$m * z$95$m), $MachinePrecision] * z$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 1.20000000000000008e307

   1. Initial program 95.9%

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

      1. +-commutative 95.9%

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

      2. distribute-lft-in 95.9%

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

      3. associate-*r* 97.2%

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

      4. *-rgt-identity 97.2%

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

      5. fma-define 97.2%

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

   4. Applied egg-rr 97.2%

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

   if 1.20000000000000008e307 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 91.2%

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

      1. associate-/l/ 91.2%

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

      2. remove-double-neg 91.2%

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

      3. distribute-rgt-neg-out 91.2%

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

      4. distribute-rgt-neg-out 91.2%

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

      5. remove-double-neg 91.2%

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

      6. associate-*l* 91.2%

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

      7. *-commutative 91.2%

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

      8. sqr-neg 91.2%

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

      9. +-commutative 91.2%

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

      10. sqr-neg 91.2%

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

      11. fma-define 91.2%

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

   3. Simplified 91.2%

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

      1. add-sqr-sqrt 57.6%

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

      2. pow2 57.6%

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

      3. *-commutative 57.6%

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

      4. sqrt-prod 57.6%

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

      5. fma-undefine 57.6%

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

      6. +-commutative 57.6%

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

      7. hypot-1-def 57.6%

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

   6. Applied egg-rr 57.6%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in z around inf 93.9%

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

       1. associate-*l/ 93.9%

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

       2. *-un-lft-identity 93.9%

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

       3. associate-/r* 93.9%

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

       4. associate-/l/ 93.9%

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

       5. associate-/r* 93.9%

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

   11. Applied egg-rr 93.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 1.2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* y_m (pow (* (hypot 1.0 z_m) (sqrt x_m)) 2.0))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (1.0 / (y_m * pow((hypot(1.0, z_m) * sqrt(x_m)), 2.0))));
}

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (1.0 / (y_m * Math.pow((Math.hypot(1.0, z_m) * Math.sqrt(x_m)), 2.0))));
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	return y_s * (x_s * (1.0 / (y_m * math.pow((math.hypot(1.0, z_m) * math.sqrt(x_m)), 2.0))))

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	return Float64(y_s * Float64(x_s * Float64(1.0 / Float64(y_m * (Float64(hypot(1.0, z_m) * sqrt(x_m)) ^ 2.0)))))
end

MATLAB

z_m = abs(z);
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_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp = code(y_s, x_s, x_m, y_m, z_m)
	tmp = y_s * (x_s * (1.0 / (y_m * ((hypot(1.0, z_m) * sqrt(x_m)) ^ 2.0))));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * N[(1.0 / N[(y$95$m * N[Power[N[(N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision] * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.3%

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

   1. associate-/l/ 95.0%

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

   2. remove-double-neg 95.0%

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

   3. distribute-rgt-neg-out 95.0%

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

   4. distribute-rgt-neg-out 95.0%

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

   5. remove-double-neg 95.0%

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

   6. associate-*l* 93.9%

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

   7. *-commutative 93.9%

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

   8. sqr-neg 93.9%

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

   9. +-commutative 93.9%

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

   10. sqr-neg 93.9%

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

   11. fma-define 93.9%

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

3. Simplified 93.9%

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

   1. add-sqr-sqrt 48.3%

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

   2. pow2 48.3%

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

   3. *-commutative 48.3%

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

   4. sqrt-prod 48.3%

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

   5. fma-undefine 48.3%

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

   6. +-commutative 48.3%

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

   7. hypot-1-def 49.4%

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

6. Applied egg-rr 49.4%

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

7. Final simplification 49.4%

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

Alternative 3: 98.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 10^{+36}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(y\_m \cdot z\_m, z\_m, y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m} \cdot \left(\frac{1}{z\_m} \cdot \frac{\frac{1}{z\_m}}{x\_m}\right)\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z_m z_m) 1e+36)
     (/ (/ 1.0 x_m) (fma (* y_m z_m) z_m y_m))
     (* (/ 1.0 y_m) (* (/ 1.0 z_m) (/ (/ 1.0 z_m) x_m)))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1e+36) {
		tmp = (1.0 / x_m) / fma((y_m * z_m), z_m, y_m);
	} else {
		tmp = (1.0 / y_m) * ((1.0 / z_m) * ((1.0 / z_m) / x_m));
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 1e+36)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(y_m * z_m), z_m, y_m));
	else
		tmp = Float64(Float64(1.0 / y_m) * Float64(Float64(1.0 / z_m) * Float64(Float64(1.0 / z_m) / x_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1e+36], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(y$95$m * z$95$m), $MachinePrecision] * z$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$95$m), $MachinePrecision] * N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(N[(1.0 / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000004e36

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. associate-*r* 99.7%

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

      4. *-rgt-identity 99.7%

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

      5. fma-define 99.7%

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

   4. Applied egg-rr 99.7%

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

   if 1.00000000000000004e36 < (*.f64 z z)

   1. Initial program 88.8%

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

      1. associate-/l/ 88.8%

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

      2. remove-double-neg 88.8%

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

      3. distribute-rgt-neg-out 88.8%

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

      4. distribute-rgt-neg-out 88.8%

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

      5. remove-double-neg 88.8%

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

      6. associate-*l* 86.2%

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

      7. *-commutative 86.2%

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

      8. sqr-neg 86.2%

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

      9. +-commutative 86.2%

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

      10. sqr-neg 86.2%

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

      11. fma-define 86.2%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in z around inf 88.8%

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

      1. *-commutative 88.8%

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

   7. Simplified 88.8%

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

      1. inv-pow 88.8%

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

      2. *-commutative 88.8%

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

      3. associate-*l* 84.5%

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

      4. unpow2 84.5%

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

      5. *-commutative 84.5%

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

      6. add-sqr-sqrt 43.7%

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

      7. swap-sqr 48.2%

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

      8. pow-prod-down 48.2%

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

      9. pow-sqr 48.2%

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

      10. *-commutative 48.2%

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

      11. metadata-eval 48.2%

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

      12. unpow-prod-down 44.2%

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

      13. sqrt-pow2 84.9%

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

      14. metadata-eval 84.9%

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

      15. inv-pow 84.9%

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

      16. *-commutative 84.9%

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

      17. associate-/r* 84.9%

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

   9. Applied egg-rr 84.9%

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

       1. associate-*l/ 89.1%

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

       2. associate-/l* 86.9%

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

   11. Simplified 86.9%

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

       1. sqr-pow 86.9%

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

       2. associate-/l* 93.8%

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

       3. metadata-eval 93.8%

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

       4. unpow-1 93.8%

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

       5. metadata-eval 93.8%

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

       6. unpow-1 93.8%

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

   13. Applied egg-rr 93.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+36}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(\frac{1}{z} \cdot \frac{\frac{1}{z}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z\_m \cdot z\_m\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 1.2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z\_m} \cdot \frac{\frac{1}{z\_m}}{y\_m \cdot x\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z_m z_m)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 1.2e+307)
       (/ (/ 1.0 x_m) t_0)
       (* (/ 1.0 z_m) (/ (/ 1.0 z_m) (* y_m x_m))))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 1.2e+307) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (1.0 / z_m) * ((1.0 / z_m) / (y_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m = abs(z)
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_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z_m * z_m))
    if (t_0 <= 1.2d+307) then
        tmp = (1.0d0 / x_m) / t_0
    else
        tmp = (1.0d0 / z_m) * ((1.0d0 / z_m) / (y_m * x_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 1.2e+307) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (1.0 / z_m) * ((1.0 / z_m) / (y_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	t_0 = y_m * (1.0 + (z_m * z_m))
	tmp = 0
	if t_0 <= 1.2e+307:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = (1.0 / z_m) * ((1.0 / z_m) / (y_m * x_m))
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z_m * z_m)))
	tmp = 0.0
	if (t_0 <= 1.2e+307)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(Float64(1.0 / z_m) * Float64(Float64(1.0 / z_m) / Float64(y_m * x_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
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_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	t_0 = y_m * (1.0 + (z_m * z_m));
	tmp = 0.0;
	if (t_0 <= 1.2e+307)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = (1.0 / z_m) * ((1.0 / z_m) / (y_m * x_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 1.2e+307], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(N[(1.0 / z$95$m), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 1.20000000000000008e307

   1. Initial program 95.9%

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

   if 1.20000000000000008e307 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 91.2%

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

      1. associate-/l/ 91.2%

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

      2. remove-double-neg 91.2%

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

      3. distribute-rgt-neg-out 91.2%

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

      4. distribute-rgt-neg-out 91.2%

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

      5. remove-double-neg 91.2%

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

      6. associate-*l* 91.2%

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

      7. *-commutative 91.2%

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

      8. sqr-neg 91.2%

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

      9. +-commutative 91.2%

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

      10. sqr-neg 91.2%

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

      11. fma-define 91.2%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in z around inf 88.4%

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

      1. add-sqr-sqrt 88.4%

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

      2. sqrt-div 54.8%

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

      3. metadata-eval 54.8%

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

      4. associate-*r* 57.6%

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

      5. *-commutative 57.6%

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

      6. sqrt-prod 57.6%

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

      7. *-commutative 57.6%

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

      8. sqrt-pow1 57.6%

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

      9. metadata-eval 57.6%

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

      10. pow1 57.6%

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

      11. sqrt-div 57.6%

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

      12. metadata-eval 57.6%

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

      13. associate-*r* 57.6%

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

      14. *-commutative 57.6%

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

      15. sqrt-prod 57.6%

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

      16. *-commutative 57.6%

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

      17. sqrt-pow1 60.6%

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

      18. metadata-eval 60.6%

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

      19. pow1 60.6%

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

   7. Applied egg-rr 60.6%

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

      1. unpow-1 60.6%

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

      2. unpow-1 60.6%

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

      3. pow-sqr 60.6%

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

      4. *-commutative 60.6%

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

      5. *-commutative 60.6%

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

      6. metadata-eval 60.6%

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

   9. Simplified 60.6%

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

       1. metadata-eval 60.6%

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

       2. pow-prod-up 60.6%

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

       3. pow-prod-down 60.5%

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

       4. swap-sqr 57.6%

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

       5. unpow2 57.6%

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

       6. add-sqr-sqrt 91.2%

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

       7. *-commutative 91.2%

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

       8. associate-*l* 88.4%

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

       9. *-commutative 88.4%

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

       10. inv-pow 88.4%

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

       11. associate-*r* 88.4%

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

       12. associate-/r* 88.4%

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

   11. Applied egg-rr 88.4%

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

       1. inv-pow 88.4%

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

       2. unpow-prod-down 88.4%

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

       3. inv-pow 88.4%

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

       4. unpow2 88.4%

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

       5. pow-prod-down 88.5%

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

       6. pow-sqr 88.5%

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

       7. metadata-eval 88.5%

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

       8. *-commutative 88.5%

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

       9. div-inv 88.5%

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

       10. associate-/l/ 91.3%

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

       11. sqr-pow 91.3%

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

       12. associate-/l* 97.0%

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

       13. metadata-eval 97.0%

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

       14. unpow-1 97.0%

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

       15. metadata-eval 97.0%

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

       16. unpow-1 97.0%

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

       17. *-commutative 97.0%

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

   13. Applied egg-rr 97.0%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 1.2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{z}}{y \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m \cdot \left(1 + z\_m \cdot z\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m} \cdot \left(\frac{1}{z\_m} \cdot \frac{\frac{1}{z\_m}}{x\_m}\right)\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z_m z_m) 2e+28)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z_m z_m))))
     (* (/ 1.0 y_m) (* (/ 1.0 z_m) (/ (/ 1.0 z_m) x_m)))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e+28) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	} else {
		tmp = (1.0 / y_m) * ((1.0 / z_m) * ((1.0 / z_m) / x_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m = abs(z)
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_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    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_m
    real(8) :: tmp
    if ((z_m * z_m) <= 2d+28) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z_m * z_m)))
    else
        tmp = (1.0d0 / y_m) * ((1.0d0 / z_m) * ((1.0d0 / z_m) / x_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e+28) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	} else {
		tmp = (1.0 / y_m) * ((1.0 / z_m) * ((1.0 / z_m) / x_m));
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if (z_m * z_m) <= 2e+28:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)))
	else:
		tmp = (1.0 / y_m) * ((1.0 / z_m) * ((1.0 / z_m) / x_m))
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 2e+28)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z_m * z_m))));
	else
		tmp = Float64(Float64(1.0 / y_m) * Float64(Float64(1.0 / z_m) * Float64(Float64(1.0 / z_m) / x_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
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_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 2e+28)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	else
		tmp = (1.0 / y_m) * ((1.0 / z_m) * ((1.0 / z_m) / x_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+28], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$95$m), $MachinePrecision] * N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(N[(1.0 / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999992e28

   1. Initial program 99.7%

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

   if 1.99999999999999992e28 < (*.f64 z z)

   1. Initial program 89.2%

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

      1. associate-/l/ 89.2%

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

      2. remove-double-neg 89.2%

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

      3. distribute-rgt-neg-out 89.2%

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

      4. distribute-rgt-neg-out 89.2%

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

      5. remove-double-neg 89.2%

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

      6. associate-*l* 86.6%

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

      7. *-commutative 86.6%

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

      8. sqr-neg 86.6%

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

      9. +-commutative 86.6%

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

      10. sqr-neg 86.6%

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

      11. fma-define 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in z around inf 89.2%

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

      1. *-commutative 89.2%

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

   7. Simplified 89.2%

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

      1. inv-pow 89.2%

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

      2. *-commutative 89.2%

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

      3. associate-*l* 85.0%

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

      4. unpow2 85.0%

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

      5. *-commutative 85.0%

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

      6. add-sqr-sqrt 43.0%

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

      7. swap-sqr 47.3%

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

      8. pow-prod-down 47.3%

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

      9. pow-sqr 47.4%

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

      10. *-commutative 47.4%

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

      11. metadata-eval 47.4%

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

      12. unpow-prod-down 43.5%

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

      13. sqrt-pow2 85.5%

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

      14. metadata-eval 85.5%

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

      15. inv-pow 85.5%

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

      16. *-commutative 85.5%

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

      17. associate-/r* 85.5%

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

   9. Applied egg-rr 85.5%

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

       1. associate-*l/ 89.5%

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

       2. associate-/l* 87.4%

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

   11. Simplified 87.4%

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

       1. sqr-pow 87.4%

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

       2. associate-/l* 94.1%

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

       3. metadata-eval 94.1%

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

       4. unpow-1 94.1%

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

       5. metadata-eval 94.1%

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

       6. unpow-1 94.1%

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

   13. Applied egg-rr 94.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(\frac{1}{z} \cdot \frac{\frac{1}{z}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.5% accurate, 0.7× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 1.0)
     (/ (/ 1.0 x_m) y_m)
     (* (/ 1.0 z_m) (/ (/ 1.0 z_m) (* y_m x_m)))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / z_m) * ((1.0 / z_m) / (y_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

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

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / z_m) * ((1.0 / z_m) / (y_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0 77.4%

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

   if 1 < z

   1. Initial program 91.5%

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

      1. associate-/l/ 91.5%

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

      2. remove-double-neg 91.5%

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

      3. distribute-rgt-neg-out 91.5%

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

      4. distribute-rgt-neg-out 91.5%

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

      5. remove-double-neg 91.5%

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

      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 87.1%

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

      1. add-sqr-sqrt 66.2%

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

      2. sqrt-div 43.4%

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

      3. metadata-eval 43.4%

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

      4. associate-*r* 43.2%

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

      5. *-commutative 43.2%

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

      6. sqrt-prod 43.3%

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

      7. *-commutative 43.3%

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

      8. sqrt-pow1 43.3%

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

      9. metadata-eval 43.3%

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

      10. pow1 43.3%

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

      11. sqrt-div 43.3%

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

      12. metadata-eval 43.3%

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

      13. associate-*r* 46.6%

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

      14. *-commutative 46.6%

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

      15. sqrt-prod 46.6%

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

      16. *-commutative 46.6%

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

      17. sqrt-pow1 51.5%

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

      18. metadata-eval 51.5%

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

      19. pow1 51.5%

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

   7. Applied egg-rr 51.5%

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

      1. unpow-1 51.5%

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

      2. unpow-1 51.5%

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

      3. pow-sqr 51.5%

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

      4. *-commutative 51.5%

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

      5. *-commutative 51.5%

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

      6. metadata-eval 51.5%

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

   9. Simplified 51.5%

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

       1. metadata-eval 51.5%

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

       2. pow-prod-up 51.5%

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

       3. pow-prod-down 51.5%

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

       4. swap-sqr 46.5%

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

       5. unpow2 46.5%

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

       6. add-sqr-sqrt 88.7%

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

       7. *-commutative 88.7%

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

       8. associate-*l* 90.5%

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

       9. *-commutative 90.5%

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

       10. inv-pow 90.5%

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

       11. associate-*r* 87.1%

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

       12. associate-/r* 87.2%

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

   11. Applied egg-rr 87.2%

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

       1. inv-pow 87.2%

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

       2. unpow-prod-down 87.2%

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

       3. inv-pow 87.2%

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

       4. unpow2 87.2%

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

       5. pow-prod-down 88.6%

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

       6. pow-sqr 88.5%

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

       7. metadata-eval 88.5%

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

       8. *-commutative 88.5%

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

       9. div-inv 88.6%

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

       10. associate-/l/ 90.1%

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

       11. sqr-pow 90.0%

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

       12. associate-/l* 95.3%

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

       13. metadata-eval 95.3%

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

       14. unpow-1 95.3%

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

       15. metadata-eval 95.3%

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

       16. unpow-1 95.3%

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

       17. *-commutative 95.3%

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

   13. Applied egg-rr 95.3%

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

4. Final simplification 81.4%

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

Alternative 7: 97.3% accurate, 0.7× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 1.0)
     (/ (/ 1.0 x_m) y_m)
     (* (/ 1.0 (* y_m z_m)) (/ 1.0 (* z_m x_m)))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / (y_m * z_m)) * (1.0 / (z_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

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

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / (y_m * z_m)) * (1.0 / (z_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0 77.4%

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

   if 1 < z

   1. Initial program 91.5%

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

      1. associate-/l/ 91.5%

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

      2. remove-double-neg 91.5%

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

      3. distribute-rgt-neg-out 91.5%

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

      4. distribute-rgt-neg-out 91.5%

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

      5. remove-double-neg 91.5%

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

      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. Step-by-step derivation

      1. add-sqr-sqrt 38.7%

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

      2. pow2 38.7%

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

      3. *-commutative 38.7%

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

      4. sqrt-prod 38.7%

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

      5. fma-undefine 38.7%

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

      6. +-commutative 38.7%

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

      7. hypot-1-def 38.7%

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

   6. Applied egg-rr 38.7%

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

   8. Applied egg-rr 94.8%

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

   9. Taylor expanded in z around inf 93.9%

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

   10. Taylor expanded in z around inf 93.8%

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

4. Final simplification 81.1%

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

Alternative 8: 97.3% accurate, 0.7× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 1.0)
     (/ (/ 1.0 x_m) y_m)
     (* (/ 1.0 (* y_m z_m)) (/ (/ 1.0 x_m) z_m))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / (y_m * z_m)) * ((1.0 / x_m) / z_m);
	}
	return y_s * (x_s * tmp);
}

Fortran

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

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / (y_m * z_m)) * ((1.0 / x_m) / z_m);
	}
	return y_s * (x_s * tmp);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0 77.4%

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

   if 1 < z

   1. Initial program 91.5%

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

      1. associate-/l/ 91.5%

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

      2. remove-double-neg 91.5%

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

      3. distribute-rgt-neg-out 91.5%

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

      4. distribute-rgt-neg-out 91.5%

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

      5. remove-double-neg 91.5%

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

      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. Step-by-step derivation

      1. add-sqr-sqrt 38.7%

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

      2. pow2 38.7%

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

      3. *-commutative 38.7%

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

      4. sqrt-prod 38.7%

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

      5. fma-undefine 38.7%

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

      6. +-commutative 38.7%

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

      7. hypot-1-def 38.7%

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

   6. Applied egg-rr 38.7%

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

   8. Applied egg-rr 94.8%

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

   9. Taylor expanded in z around inf 93.9%

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

   10. Taylor expanded in z around inf 93.8%

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

       1. associate-/r* 93.7%

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

   12. Simplified 93.7%

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

4. Final simplification 81.1%

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

Alternative 9: 97.3% accurate, 0.7× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 1.0)
     (/ (/ 1.0 x_m) y_m)
     (* (/ (/ 1.0 z_m) x_m) (/ 1.0 (* y_m z_m)))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = ((1.0 / z_m) / x_m) * (1.0 / (y_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

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

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = ((1.0 / z_m) / x_m) * (1.0 / (y_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0 77.4%

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

   if 1 < z

   1. Initial program 91.5%

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

      1. associate-/l/ 91.5%

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

      2. remove-double-neg 91.5%

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

      3. distribute-rgt-neg-out 91.5%

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

      4. distribute-rgt-neg-out 91.5%

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

      5. remove-double-neg 91.5%

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

      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. Step-by-step derivation

      1. add-sqr-sqrt 38.7%

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

      2. pow2 38.7%

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

      3. *-commutative 38.7%

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

      4. sqrt-prod 38.7%

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

      5. fma-undefine 38.7%

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

      6. +-commutative 38.7%

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

      7. hypot-1-def 38.7%

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

   6. Applied egg-rr 38.7%

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

   8. Applied egg-rr 94.8%

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

   9. Taylor expanded in z around inf 93.9%

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

   10. Taylor expanded in z around inf 93.8%

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

4. Final simplification 81.1%

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

Alternative 10: 73.3% accurate, 0.9× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (* x_s (if (<= z_m 1.0) (/ (/ 1.0 x_m) y_m) (/ 1.0 (* x_m (* y_m z_m)))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (x_m * (y_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

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

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (x_m * (y_m * z_m));
	}
	return y_s * (x_s * tmp);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0 77.4%

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

   if 1 < z

   1. Initial program 91.5%

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

      1. associate-/l/ 91.5%

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

      2. remove-double-neg 91.5%

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

      3. distribute-rgt-neg-out 91.5%

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

      4. distribute-rgt-neg-out 91.5%

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

      5. remove-double-neg 91.5%

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

      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. Step-by-step derivation

      1. add-sqr-sqrt 38.7%

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

      2. pow2 38.7%

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

      3. *-commutative 38.7%

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

      4. sqrt-prod 38.7%

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

      5. fma-undefine 38.7%

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

      6. +-commutative 38.7%

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

      7. hypot-1-def 38.7%

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

   6. Applied egg-rr 38.7%

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

   8. Applied egg-rr 94.8%

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

   9. Taylor expanded in z around inf 93.9%

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

   10. Taylor expanded in z around 0 42.9%

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

4. Final simplification 69.8%

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

Alternative 11: 58.5% accurate, 2.2× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* y_m x_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (1.0 / (y_m * x_m)));
}

Fortran

z_m = abs(z)
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_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    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_m
    code = y_s * (x_s * (1.0d0 / (y_m * x_m)))
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (1.0 / (y_m * x_m)));
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	return y_s * (x_s * (1.0 / (y_m * x_m)))

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	return Float64(y_s * Float64(x_s * Float64(1.0 / Float64(y_m * x_m))))
end

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := 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 95.3%

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

   1. associate-/l/ 95.0%

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

   2. remove-double-neg 95.0%

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

   3. distribute-rgt-neg-out 95.0%

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

   4. distribute-rgt-neg-out 95.0%

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

   5. remove-double-neg 95.0%

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

   6. associate-*l* 93.9%

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

   7. *-commutative 93.9%

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

   8. sqr-neg 93.9%

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

   9. +-commutative 93.9%

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

   10. sqr-neg 93.9%

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

   11. fma-define 93.9%

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

3. Simplified 93.9%

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

5. Taylor expanded in z around 0 65.1%

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

6. Final simplification 65.1%

   \[\leadsto \frac{1}{y \cdot x} \]
7. 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 2024089 
(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)))))