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

Percentage Accurate: 88.4% → 99.4%

Time: 11.1s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.4% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.25 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right) \cdot x\_m}}{y\_m \cdot \mathsf{hypot}\left(1, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right) \cdot \left(y\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 1.25e+113)
     (/ (/ 1.0 (* (hypot 1.0 z) x_m)) (* y_m (hypot 1.0 z)))
     (/ (/ 1.0 (hypot 1.0 z)) (* (hypot 1.0 z) (* y_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1.25e+113) {
		tmp = (1.0 / (hypot(1.0, z) * x_m)) / (y_m * hypot(1.0, z));
	} else {
		tmp = (1.0 / hypot(1.0, z)) / (hypot(1.0, z) * (y_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1.25e+113) {
		tmp = (1.0 / (Math.hypot(1.0, z) * x_m)) / (y_m * Math.hypot(1.0, z));
	} else {
		tmp = (1.0 / Math.hypot(1.0, z)) / (Math.hypot(1.0, z) * (y_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if y_m <= 1.25e+113:
		tmp = (1.0 / (math.hypot(1.0, z) * x_m)) / (y_m * math.hypot(1.0, z))
	else:
		tmp = (1.0 / math.hypot(1.0, z)) / (math.hypot(1.0, z) * (y_m * x_m))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 1.25e+113)
		tmp = Float64(Float64(1.0 / Float64(hypot(1.0, z) * x_m)) / Float64(y_m * hypot(1.0, z)));
	else
		tmp = Float64(Float64(1.0 / hypot(1.0, z)) / Float64(hypot(1.0, z) * Float64(y_m * x_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (y_m <= 1.25e+113)
		tmp = (1.0 / (hypot(1.0, z) * x_m)) / (y_m * hypot(1.0, z));
	else
		tmp = (1.0 / hypot(1.0, z)) / (hypot(1.0, z) * (y_m * x_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1.25e+113], N[(N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.25e113

   1. Initial program 90.5%

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

      1. associate-/l/ 90.2%

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

      2. associate-*l* 88.1%

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

      3. *-commutative 88.1%

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

      4. sqr-neg 88.1%

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

      5. +-commutative 88.1%

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

      6. sqr-neg 88.1%

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

      7. 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. clear-num 88.1%

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

      2. associate-*r* 90.1%

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

      3. *-commutative 90.1%

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

      4. *-commutative 90.1%

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

      5. associate-/r/ 90.1%

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

      6. associate-/r* 90.5%

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

   6. Applied egg-rr 90.5%

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

      1. add-sqr-sqrt 90.4%

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

      2. associate-/l* 90.4%

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

      3. sqrt-div 90.4%

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

      4. metadata-eval 90.4%

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

      5. fma-undefine 90.4%

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

      6. unpow2 90.4%

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

      7. +-commutative 90.4%

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

      8. metadata-eval 90.4%

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

      9. unpow2 90.4%

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

      10. hypot-undefine 90.4%

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

      11. sqrt-div 90.4%

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

      12. metadata-eval 90.4%

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

      13. fma-undefine 90.4%

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

      14. unpow2 90.4%

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

      15. +-commutative 90.4%

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

      16. metadata-eval 90.4%

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

      17. unpow2 90.4%

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

      18. hypot-undefine 95.6%

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

   8. Applied egg-rr 95.6%

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

      1. *-commutative 95.6%

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

      2. div-inv 95.7%

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

      3. associate-/r* 97.9%

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

      4. associate-/l/ 97.3%

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

      5. associate-/l/ 97.3%

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

   10. Applied egg-rr 97.3%

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

   if 1.25e113 < y

   1. Initial program 81.8%

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

      1. associate-/l/ 81.8%

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

      2. associate-*l* 97.1%

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

      3. *-commutative 97.1%

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

      4. sqr-neg 97.1%

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

      5. +-commutative 97.1%

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

      6. sqr-neg 97.1%

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

      7. fma-define 97.1%

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

   3. Simplified 97.1%

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

      1. clear-num 97.1%

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

      2. associate-*r* 97.0%

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

      3. *-commutative 97.0%

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

      4. *-commutative 97.0%

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

      5. associate-/r/ 97.0%

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

      6. associate-/r* 97.0%

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

   6. Applied egg-rr 97.0%

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

      1. add-sqr-sqrt 97.0%

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

      2. associate-/l* 97.0%

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

      3. sqrt-div 97.1%

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

      4. metadata-eval 97.1%

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

      5. fma-undefine 97.1%

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

      6. unpow2 97.1%

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

      7. +-commutative 97.1%

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

      8. metadata-eval 97.1%

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

      9. unpow2 97.1%

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

      10. hypot-undefine 97.1%

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

      11. sqrt-div 97.1%

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

      12. metadata-eval 97.1%

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

      13. fma-undefine 97.1%

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

      14. unpow2 97.1%

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

      15. +-commutative 97.1%

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

      16. metadata-eval 97.1%

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

      17. unpow2 97.1%

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

      18. hypot-undefine 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. frac-times 99.7%

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

      2. *-un-lft-identity 99.7%

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

   10. Applied egg-rr 99.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right) \cdot x}}{y \cdot \mathsf{hypot}\left(1, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right) \cdot \left(y \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

   1. associate-/l/ 89.1%

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

   2. associate-*l* 89.3%

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

   3. *-commutative 89.3%

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

   4. sqr-neg 89.3%

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

   5. +-commutative 89.3%

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

   6. sqr-neg 89.3%

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

   7. fma-define 89.3%

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

3. Simplified 89.3%

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

   1. associate-*r* 91.0%

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

   2. *-commutative 91.0%

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

   3. associate-/r* 91.3%

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

   4. *-commutative 91.3%

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

   5. associate-/l/ 91.4%

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

   6. fma-undefine 91.4%

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

   7. +-commutative 91.4%

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

   8. associate-/r* 89.3%

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

   9. *-un-lft-identity 89.3%

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

   10. add-sqr-sqrt 43.5%

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

   11. times-frac 43.5%

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

   12. +-commutative 43.5%

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

   13. fma-undefine 43.5%

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

   14. *-commutative 43.5%

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

   15. sqrt-prod 43.5%

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

   16. fma-undefine 43.5%

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

   17. +-commutative 43.5%

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

   18. hypot-1-def 43.5%

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

   19. +-commutative 43.5%

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

6. Applied egg-rr 48.9%

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

   1. associate-/l/ 48.9%

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

   2. associate-*r/ 48.9%

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

   3. *-rgt-identity 48.9%

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

   4. *-commutative 48.9%

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

   5. associate-/r* 48.9%

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

   6. *-commutative 48.9%

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

8. Simplified 48.9%

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

9. Final simplification 48.9%

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

Alternative 3: 98.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = math.sqrt(y_m) * math.hypot(1.0, z)
	return y_s * (x_s * (1.0 / (t_0 * (x_m * t_0))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(sqrt(y_m) * hypot(1.0, z))
	return Float64(y_s * Float64(x_s * Float64(1.0 / Float64(t_0 * Float64(x_m * t_0)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	t_0 = sqrt(y_m) * hypot(1.0, z);
	tmp = y_s * (x_s * (1.0 / (t_0 * (x_m * t_0))));
end

Wolfram

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

Derivation

1. Initial program 89.3%

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

   1. associate-/l/ 89.1%

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

   2. associate-*l* 89.3%

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

   3. *-commutative 89.3%

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

   4. sqr-neg 89.3%

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

   5. +-commutative 89.3%

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

   6. sqr-neg 89.3%

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

   7. fma-define 89.3%

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

3. Simplified 89.3%

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

   1. associate-*r* 91.0%

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

   2. *-commutative 91.0%

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

   3. associate-/r* 91.3%

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

   4. *-commutative 91.3%

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

   5. associate-/l/ 91.4%

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

   6. fma-undefine 91.4%

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

   7. +-commutative 91.4%

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

   8. associate-/r* 89.3%

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

   9. *-un-lft-identity 89.3%

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

   10. add-sqr-sqrt 43.5%

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

   11. times-frac 43.5%

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

   12. +-commutative 43.5%

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

   13. fma-undefine 43.5%

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

   14. *-commutative 43.5%

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

   15. sqrt-prod 43.5%

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

   16. fma-undefine 43.5%

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

   17. +-commutative 43.5%

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

   18. hypot-1-def 43.5%

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

   19. +-commutative 43.5%

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

6. Applied egg-rr 48.9%

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

   1. associate-/l/ 48.9%

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

   2. associate-*r/ 48.9%

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

   3. *-rgt-identity 48.9%

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

   4. *-commutative 48.9%

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

   5. associate-/r* 48.9%

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

   6. *-commutative 48.9%

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

8. Simplified 48.9%

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

   1. frac-2neg 48.9%

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

   2. div-inv 48.9%

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

   3. associate-/l/ 48.9%

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

   4. distribute-neg-frac 48.9%

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

   5. metadata-eval 48.9%

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

   6. *-commutative 48.9%

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

   7. distribute-rgt-neg-in 48.9%

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

10. Applied egg-rr 48.9%

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

    1. associate-*r/ 48.9%

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

    2. *-rgt-identity 48.9%

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

    3. distribute-rgt-neg-out 48.9%

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

    4. *-commutative 48.9%

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

    5. distribute-neg-frac2 48.9%

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

    6. associate-/l/ 48.9%

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

    7. distribute-neg-frac 48.9%

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

    8. metadata-eval 48.9%

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

    9. *-commutative 48.9%

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

    10. *-commutative 48.9%

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

12. Simplified 48.9%

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

13. Final simplification 48.9%

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

Alternative 4: 90.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x\_m \cdot {z}^{2}\right)}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z z)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 2e+301)
       (/ (/ 1.0 x_m) t_0)
       (/ 1.0 (* y_m (* x_m (pow z 2.0)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+301) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = 1.0 / (y_m * (x_m * pow(z, 2.0)));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z * z))
    if (t_0 <= 2d+301) then
        tmp = (1.0d0 / x_m) / t_0
    else
        tmp = 1.0d0 / (y_m * (x_m * (z ** 2.0d0)))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+301) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = 1.0 / (y_m * (x_m * Math.pow(z, 2.0)));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = y_m * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 2e+301:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = 1.0 / (y_m * (x_m * math.pow(z, 2.0)))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 2e+301)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * (z ^ 2.0))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = y_m * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 2e+301)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = 1.0 / (y_m * (x_m * (z ^ 2.0)));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 2e+301], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(1.0 / N[(y$95$m * N[(x$95$m * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 2.00000000000000011e301

   1. Initial program 93.5%

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

   if 2.00000000000000011e301 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 72.0%

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

      1. associate-/l/ 72.0%

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

      2. associate-*l* 84.6%

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

      3. *-commutative 84.6%

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

      4. sqr-neg 84.6%

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

      5. +-commutative 84.6%

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

      6. sqr-neg 84.6%

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

      7. fma-define 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in z around inf 84.6%

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

Alternative 5: 93.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+240}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m}}{z}}{y\_m \cdot \mathsf{hypot}\left(1, z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e+240)
     (/ (/ 1.0 y_m) (* x_m (fma z z 1.0)))
     (/ (/ (/ 1.0 x_m) z) (* y_m (hypot 1.0 z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+240) {
		tmp = (1.0 / y_m) / (x_m * fma(z, z, 1.0));
	} else {
		tmp = ((1.0 / x_m) / z) / (y_m * hypot(1.0, z));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+240)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / x_m) / z) / Float64(y_m * hypot(1.0, z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+240], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / z), $MachinePrecision] / N[(y$95$m * N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000003e240

   1. Initial program 95.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. associate-*l* 95.9%

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

      3. *-commutative 95.9%

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

      4. sqr-neg 95.9%

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

      5. +-commutative 95.9%

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

      6. sqr-neg 95.9%

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

      7. fma-define 95.9%

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

   3. Simplified 95.9%

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

      1. associate-*r* 98.0%

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

      2. *-commutative 98.0%

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

      3. *-commutative 98.0%

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

      4. add-sqr-sqrt 55.6%

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

      5. sqrt-div 48.5%

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

      6. metadata-eval 48.5%

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

      7. sqrt-prod 48.5%

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

      8. fma-undefine 48.5%

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

      9. +-commutative 48.5%

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

      10. hypot-1-def 48.5%

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

      11. sqrt-div 48.5%

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

      12. metadata-eval 48.5%

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

      13. sqrt-prod 48.7%

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

      14. fma-undefine 48.7%

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

      15. +-commutative 48.7%

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

      16. hypot-1-def 48.7%

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

   6. Applied egg-rr 48.7%

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

      1. unpow-1 48.7%

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

      2. unpow-1 48.7%

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

      3. pow-sqr 48.9%

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

      4. metadata-eval 48.9%

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

   8. Simplified 48.9%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{-2}} \]
   9. Step-by-step derivation

      1. unpow-prod-down 48.8%

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

      2. sqrt-pow2 98.5%

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

      3. *-commutative 98.5%

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

      4. metadata-eval 98.5%

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

      5. inv-pow 98.5%

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

      6. hypot-undefine 98.5%

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

      7. metadata-eval 98.5%

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

      8. unpow2 98.5%

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

      9. sqrt-pow2 98.4%

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

      10. +-commutative 98.4%

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

      11. unpow2 98.4%

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

      12. fma-undefine 98.4%

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

      13. metadata-eval 98.4%

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

      14. inv-pow 98.4%

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

      15. associate-/r* 98.5%

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

      16. frac-times 96.4%

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

      17. *-un-lft-identity 96.4%

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

   10. Applied egg-rr 96.4%

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

   if 5.0000000000000003e240 < (*.f64 z z)

   1. Initial program 75.8%

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

      1. associate-/l/ 75.8%

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

      2. associate-*l* 74.6%

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

      3. *-commutative 74.6%

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

      4. sqr-neg 74.6%

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

      5. +-commutative 74.6%

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

      6. sqr-neg 74.6%

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

      7. fma-define 74.6%

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

   3. Simplified 74.6%

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

      1. clear-num 74.6%

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

      2. associate-*r* 75.5%

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

      3. *-commutative 75.5%

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

      4. *-commutative 75.5%

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

      5. associate-/r/ 75.5%

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

      6. associate-/r* 75.5%

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

   6. Applied egg-rr 75.5%

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

      1. add-sqr-sqrt 75.4%

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

      2. associate-/l* 75.4%

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

      3. sqrt-div 75.5%

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

      4. metadata-eval 75.5%

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

      5. fma-undefine 75.5%

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

      6. unpow2 75.5%

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

      7. +-commutative 75.5%

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

      8. metadata-eval 75.5%

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

      9. unpow2 75.5%

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

      10. hypot-undefine 75.5%

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

      11. sqrt-div 75.5%

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

      12. metadata-eval 75.5%

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

      13. fma-undefine 75.5%

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

      14. unpow2 75.5%

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

      15. +-commutative 75.5%

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

      16. metadata-eval 75.5%

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

      17. unpow2 75.5%

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

      18. hypot-undefine 91.1%

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

   8. Applied egg-rr 91.1%

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

      1. *-commutative 91.1%

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

      2. div-inv 91.1%

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

      3. associate-/r* 98.5%

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

      4. associate-/l/ 96.2%

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

      5. associate-/l/ 96.2%

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

   10. Applied egg-rr 96.2%

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

   11. Taylor expanded in z around inf 84.0%

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

       1. associate-/r* 84.1%

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

   13. Simplified 84.1%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+240}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z}}{y \cdot \mathsf{hypot}\left(1, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right) \cdot x\_m}}{y\_m \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e+260)
     (/ (/ 1.0 y_m) (* x_m (fma z z 1.0)))
     (/ (/ 1.0 (* (hypot 1.0 z) x_m)) (* y_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+260) {
		tmp = (1.0 / y_m) / (x_m * fma(z, z, 1.0));
	} else {
		tmp = (1.0 / (hypot(1.0, z) * x_m)) / (y_m * z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+260)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(hypot(1.0, z) * x_m)) / Float64(y_m * z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+260], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999996e260

   1. Initial program 95.4%

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

      1. associate-/l/ 95.1%

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

      2. associate-*l* 95.4%

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

      3. *-commutative 95.4%

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

      4. sqr-neg 95.4%

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

      5. +-commutative 95.4%

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

      6. sqr-neg 95.4%

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

      7. fma-define 95.4%

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

   3. Simplified 95.4%

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

      1. associate-*r* 98.0%

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

      2. *-commutative 98.0%

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

      3. *-commutative 98.0%

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

      4. add-sqr-sqrt 55.8%

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

      5. sqrt-div 48.8%

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

      6. metadata-eval 48.8%

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

      7. sqrt-prod 48.8%

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

      8. fma-undefine 48.8%

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

      9. +-commutative 48.8%

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

      10. hypot-1-def 48.8%

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

      11. sqrt-div 48.8%

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

      12. metadata-eval 48.8%

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

      13. sqrt-prod 49.0%

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

      14. fma-undefine 49.0%

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

      15. +-commutative 49.0%

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

      16. hypot-1-def 49.0%

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

   6. Applied egg-rr 49.0%

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

      1. unpow-1 49.0%

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

      2. unpow-1 49.0%

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

      3. pow-sqr 49.2%

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

      4. metadata-eval 49.2%

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

   8. Simplified 49.2%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{-2}} \]
   9. Step-by-step derivation

      1. unpow-prod-down 49.1%

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

      2. sqrt-pow2 98.5%

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

      3. *-commutative 98.5%

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

      4. metadata-eval 98.5%

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

      5. inv-pow 98.5%

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

      6. hypot-undefine 98.5%

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

      7. metadata-eval 98.5%

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

      8. unpow2 98.5%

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

      9. sqrt-pow2 98.5%

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

      10. +-commutative 98.5%

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

      11. unpow2 98.5%

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

      12. fma-undefine 98.5%

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

      13. metadata-eval 98.5%

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

      14. inv-pow 98.5%

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

      15. associate-/r* 98.5%

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

      16. frac-times 95.9%

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

      17. *-un-lft-identity 95.9%

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

   10. Applied egg-rr 95.9%

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

   if 4.9999999999999996e260 < (*.f64 z z)

   1. Initial program 74.8%

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

      1. associate-/l/ 74.8%

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

      2. associate-*l* 74.8%

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

      3. *-commutative 74.8%

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

      4. sqr-neg 74.8%

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

      5. +-commutative 74.8%

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

      6. sqr-neg 74.8%

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

      7. fma-define 74.8%

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

   3. Simplified 74.8%

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

      1. clear-num 74.8%

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

      2. associate-*r* 74.5%

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

      3. *-commutative 74.5%

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

      4. *-commutative 74.5%

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

      5. associate-/r/ 74.5%

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

      6. associate-/r* 74.5%

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

   6. Applied egg-rr 74.5%

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

      1. add-sqr-sqrt 74.5%

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

      2. associate-/l* 74.5%

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

      3. sqrt-div 74.5%

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

      4. metadata-eval 74.5%

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

      5. fma-undefine 74.5%

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

      6. unpow2 74.5%

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

      7. +-commutative 74.5%

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

      8. metadata-eval 74.5%

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

      9. unpow2 74.5%

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

      10. hypot-undefine 74.5%

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

      11. sqrt-div 74.5%

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

      12. metadata-eval 74.5%

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

      13. fma-undefine 74.5%

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

      14. unpow2 74.5%

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

      15. +-commutative 74.5%

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

      16. metadata-eval 74.5%

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

      17. unpow2 74.5%

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

      18. hypot-undefine 90.8%

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

   8. Applied egg-rr 90.8%

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

      1. *-commutative 90.8%

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

      2. div-inv 90.8%

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

      3. associate-/r* 98.5%

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

      4. associate-/l/ 96.0%

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

      5. associate-/l/ 96.0%

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

   10. Applied egg-rr 96.0%

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

   11. Taylor expanded in z around inf 84.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right) \cdot x}}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 430000000000:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot {z}^{2}}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 430000000000.0)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z z))))
     (/ (/ 1.0 y_m) (* x_m (pow z 2.0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 430000000000.0) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / y_m) / (x_m * pow(z, 2.0));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 430000000000.0d0) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z * z)))
    else
        tmp = (1.0d0 / y_m) / (x_m * (z ** 2.0d0))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 430000000000.0) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / y_m) / (x_m * Math.pow(z, 2.0));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 430000000000.0:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)))
	else:
		tmp = (1.0 / y_m) / (x_m * math.pow(z, 2.0))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 430000000000.0)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * (z ^ 2.0)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 430000000000.0)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	else
		tmp = (1.0 / y_m) / (x_m * (z ^ 2.0));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 430000000000.0], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.3e11

   1. Initial program 93.2%

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

   if 4.3e11 < z

   1. Initial program 78.4%

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

      1. associate-/l/ 78.4%

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

      2. associate-*l* 80.9%

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

      3. *-commutative 80.9%

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

      4. sqr-neg 80.9%

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

      5. +-commutative 80.9%

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

      6. sqr-neg 80.9%

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

      7. fma-define 80.9%

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

   3. Simplified 80.9%

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

      1. associate-*r* 79.3%

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

      2. *-commutative 79.3%

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

      3. *-commutative 79.3%

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

      4. add-sqr-sqrt 64.2%

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

      5. sqrt-div 39.0%

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

      6. metadata-eval 39.0%

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

      7. sqrt-prod 39.0%

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

      8. fma-undefine 39.0%

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

      9. +-commutative 39.0%

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

      10. hypot-1-def 39.0%

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

      11. sqrt-div 39.0%

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

      12. metadata-eval 39.0%

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

      13. sqrt-prod 38.9%

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

      14. fma-undefine 38.9%

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

      15. +-commutative 38.9%

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

      16. hypot-1-def 41.8%

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

   6. Applied egg-rr 41.8%

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

      1. unpow-1 41.8%

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

      2. unpow-1 41.8%

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

      3. pow-sqr 41.9%

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

      4. metadata-eval 41.9%

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

   8. Simplified 41.9%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{-2}} \]
   9. Step-by-step derivation

      1. unpow-prod-down 40.2%

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

      2. sqrt-pow2 81.1%

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

      3. *-commutative 81.1%

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

      4. metadata-eval 81.1%

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

      5. inv-pow 81.1%

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

      6. hypot-undefine 79.9%

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

      7. metadata-eval 79.9%

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

      8. unpow2 79.9%

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

      9. sqrt-pow2 79.8%

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

      10. +-commutative 79.8%

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

      11. unpow2 79.8%

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

      12. fma-undefine 79.8%

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

      13. metadata-eval 79.8%

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

      14. inv-pow 79.8%

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

      15. associate-/r* 79.8%

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

      16. frac-times 81.4%

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

      17. *-un-lft-identity 81.4%

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

   10. Applied egg-rr 81.4%

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

   11. Taylor expanded in z around inf 81.4%

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

Alternative 8: 92.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

   1. associate-/l/ 89.1%

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

   2. associate-*l* 89.3%

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

   3. *-commutative 89.3%

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

   4. sqr-neg 89.3%

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

   5. +-commutative 89.3%

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

   6. sqr-neg 89.3%

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

   7. fma-define 89.3%

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

3. Simplified 89.3%

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

   1. associate-*r* 91.0%

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

   2. *-commutative 91.0%

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

   3. *-commutative 91.0%

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

   4. add-sqr-sqrt 60.6%

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

   5. sqrt-div 45.8%

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

   6. metadata-eval 45.8%

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

   7. sqrt-prod 45.8%

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

   8. fma-undefine 45.8%

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

   9. +-commutative 45.8%

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

   10. hypot-1-def 45.8%

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

   11. sqrt-div 45.8%

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

   12. metadata-eval 45.8%

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

   13. sqrt-prod 45.9%

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

   14. fma-undefine 45.9%

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

   15. +-commutative 45.9%

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

   16. hypot-1-def 48.2%

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

6. Applied egg-rr 48.2%

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

   1. unpow-1 48.2%

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

   2. unpow-1 48.2%

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

   3. pow-sqr 48.3%

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

   4. metadata-eval 48.3%

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

8. Simplified 48.3%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{-2}} \]
9. Step-by-step derivation

   1. unpow-prod-down 46.7%

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

   2. sqrt-pow2 92.0%

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

   3. *-commutative 92.0%

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

   4. metadata-eval 92.0%

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

   5. inv-pow 92.0%

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

   6. hypot-undefine 91.4%

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

   7. metadata-eval 91.4%

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

   8. unpow2 91.4%

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

   9. sqrt-pow2 91.3%

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

   10. +-commutative 91.3%

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

   11. unpow2 91.3%

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

   12. fma-undefine 91.3%

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

   13. metadata-eval 91.3%

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

   14. inv-pow 91.3%

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

   15. associate-/r* 91.4%

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

   16. frac-times 89.6%

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

   17. *-un-lft-identity 89.6%

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

10. Applied egg-rr 89.6%

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

11. Final simplification 89.6%

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

Alternative 9: 91.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

   1. associate-/l/ 89.1%

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

   2. associate-*l* 89.3%

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

   3. *-commutative 89.3%

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

   4. sqr-neg 89.3%

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

   5. +-commutative 89.3%

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

   6. sqr-neg 89.3%

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

   7. fma-define 89.3%

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

3. Simplified 89.3%

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

Alternative 10: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

Alternative 11: 59.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

   1. associate-/l/ 89.1%

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

   2. associate-*l* 89.3%

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

   3. *-commutative 89.3%

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

   4. sqr-neg 89.3%

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

   5. +-commutative 89.3%

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

   6. sqr-neg 89.3%

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

   7. fma-define 89.3%

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

3. Simplified 89.3%

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

5. Taylor expanded in z around 0 56.4%

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

Developer target: 92.1% 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 2024100 
(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)))))