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

Percentage Accurate: 88.4% → 99.5%

Time: 15.9s

Alternatives: 12

Speedup: 0.8×

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: 99.5% accurate, 0.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (/
    (/ 1.0 (/ (* (hypot 1.0 z) x_m) (pow y_m -0.5)))
    (* (hypot 1.0 z) (sqrt y_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((1.0 / ((hypot(1.0, z) * x_m) / pow(y_m, -0.5))) / (hypot(1.0, z) * sqrt(y_m))));
}

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((1.0 / ((Math.hypot(1.0, z) * x_m) / Math.pow(y_m, -0.5))) / (Math.hypot(1.0, z) * Math.sqrt(y_m))));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * ((1.0 / ((math.hypot(1.0, z) * x_m) / math.pow(y_m, -0.5))) / (math.hypot(1.0, z) * math.sqrt(y_m))))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(Float64(1.0 / Float64(Float64(hypot(1.0, z) * x_m) / (y_m ^ -0.5))) / Float64(hypot(1.0, z) * sqrt(y_m)))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * ((1.0 / ((hypot(1.0, z) * x_m) / (y_m ^ -0.5))) / (hypot(1.0, z) * sqrt(y_m))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(1.0 / N[(N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * x$95$m), $MachinePrecision] / N[Power[y$95$m, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.0%

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

   1. /-rgt-identity 91.0%

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

   2. associate-/l* 91.0%

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

   3. *-commutative 91.0%

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

   4. associate-/l* 90.2%

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

   5. associate-/r/ 90.7%

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

   6. *-commutative 90.7%

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

   7. /-rgt-identity 90.7%

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

   8. +-commutative 90.7%

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

   9. fma-def 90.7%

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

3. Simplified 90.7%

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

   1. associate-/r* 91.0%

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

   2. div-inv 91.0%

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

   3. add-sqr-sqrt 45.3%

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

   4. times-frac 45.2%

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

   5. sqrt-prod 45.2%

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

   6. fma-udef 45.2%

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

   7. +-commutative 45.2%

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

   8. hypot-1-def 45.2%

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

   9. associate-/l/ 45.7%

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

   10. sqrt-prod 46.4%

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

   11. fma-udef 46.4%

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

   12. +-commutative 46.4%

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

   13. hypot-1-def 50.1%

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

5. Applied egg-rr 50.1%

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

   1. associate-*l/ 50.1%

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

   2. *-lft-identity 50.1%

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

   3. associate-*l* 49.4%

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

7. Simplified 49.4%

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

   1. /-rgt-identity 49.4%

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

   2. *-commutative 49.4%

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

   3. associate-/l* 49.3%

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

   4. pow1/2 49.3%

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

   5. pow-flip 49.4%

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

   6. metadata-eval 49.4%

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

9. Applied egg-rr 49.4%

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

10. Final simplification 49.4%

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

Alternative 2: 99.4% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

   1. /-rgt-identity 91.0%

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

   2. associate-/l* 91.0%

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

   3. *-commutative 91.0%

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

   4. associate-/l* 90.2%

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

   5. associate-/r/ 90.7%

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

   6. *-commutative 90.7%

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

   7. /-rgt-identity 90.7%

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

   8. +-commutative 90.7%

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

   9. fma-def 90.7%

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

3. Simplified 90.7%

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

   1. associate-/r* 91.0%

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

   2. div-inv 91.0%

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

   3. add-sqr-sqrt 45.3%

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

   4. times-frac 45.2%

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

   5. sqrt-prod 45.2%

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

   6. fma-udef 45.2%

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

   7. +-commutative 45.2%

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

   8. hypot-1-def 45.2%

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

   9. associate-/l/ 45.7%

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

   10. sqrt-prod 46.4%

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

   11. fma-udef 46.4%

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

   12. +-commutative 46.4%

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

   13. hypot-1-def 50.1%

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

5. Applied egg-rr 50.1%

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

   1. associate-*l/ 50.1%

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

   2. *-lft-identity 50.1%

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

   3. associate-*l* 49.4%

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

7. Simplified 49.4%

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

8. Final simplification 49.4%

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

Alternative 3: 98.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

   1. associate-/l/ 89.3%

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

   2. /-rgt-identity 89.3%

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

   3. associate-/l* 89.3%

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

   4. *-rgt-identity 89.3%

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

   5. /-rgt-identity 89.3%

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

   6. associate-/l* 89.3%

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

   7. *-commutative 89.3%

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

   8. associate-/l* 88.7%

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

   9. associate-/r/ 88.7%

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

   10. *-commutative 88.7%

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

   11. /-rgt-identity 88.7%

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

   12. +-commutative 88.7%

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

   13. fma-def 88.7%

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

3. Simplified 88.7%

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

   1. /-rgt-identity 88.7%

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

   2. associate-/l* 88.7%

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

5. Applied egg-rr 88.7%

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

   1. associate-/r/ 88.7%

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

   2. add-sqr-sqrt 88.7%

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

   3. associate-*r* 88.7%

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

   4. /-rgt-identity 88.7%

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

   5. fma-udef 88.7%

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

   6. unpow2 88.7%

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

   7. +-commutative 88.7%

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

   8. metadata-eval 88.7%

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

   9. unpow2 88.7%

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

   10. hypot-udef 88.7%

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

   11. *-commutative 88.7%

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

   12. fma-udef 88.7%

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

   13. unpow2 88.7%

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

   14. +-commutative 88.7%

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

   15. metadata-eval 88.7%

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

   16. unpow2 88.7%

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

   17. hypot-udef 91.7%

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

7. Applied egg-rr 91.7%

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

   1. associate-*l* 91.7%

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

9. Simplified 91.7%

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

10. Final simplification 91.7%

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

Alternative 4: 98.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

   1. associate-/l/ 89.3%

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

   2. /-rgt-identity 89.3%

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

   3. associate-/l* 89.3%

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

   4. *-rgt-identity 89.3%

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

   5. /-rgt-identity 89.3%

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

   6. associate-/l* 89.3%

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

   7. *-commutative 89.3%

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

   8. associate-/l* 88.7%

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

   9. associate-/r/ 88.7%

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

   10. *-commutative 88.7%

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

   11. /-rgt-identity 88.7%

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

   12. +-commutative 88.7%

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

   13. fma-def 88.7%

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

3. Simplified 88.7%

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

   1. associate-/r* 89.3%

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

   2. div-inv 89.3%

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

   3. add-sqr-sqrt 89.3%

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

   4. times-frac 89.3%

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

   5. fma-udef 89.3%

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

   6. +-commutative 89.3%

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

   7. hypot-1-def 89.3%

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

   8. fma-udef 89.3%

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

   9. +-commutative 89.3%

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

   10. hypot-1-def 92.2%

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

5. Applied egg-rr 92.2%

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

   1. associate-*l/ 92.2%

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

   2. *-lft-identity 92.2%

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

   3. associate-/l/ 92.2%

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

   4. *-commutative 92.2%

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

7. Simplified 92.2%

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

8. Final simplification 92.2%

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

Alternative 5: 98.8% accurate, 0.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ (/ (/ 1.0 (* (hypot 1.0 z) x_m)) 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 / (hypot(1.0, z) * x_m)) / 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.hypot(1.0, z) * x_m)) / 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.hypot(1.0, z) * x_m)) / 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 / Float64(hypot(1.0, z) * x_m)) / 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 / (hypot(1.0, z) * x_m)) / 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[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.0%

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

   1. /-rgt-identity 91.0%

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

   2. associate-/l* 91.0%

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

   3. *-commutative 91.0%

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

   4. associate-/l* 90.2%

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

   5. associate-/r/ 90.7%

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

   6. *-commutative 90.7%

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

   7. /-rgt-identity 90.7%

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

   8. +-commutative 90.7%

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

   9. fma-def 90.7%

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

3. Simplified 90.7%

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

   1. associate-/r* 91.0%

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

   2. div-inv 91.0%

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

   3. add-sqr-sqrt 45.3%

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

   4. times-frac 45.2%

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

   5. sqrt-prod 45.2%

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

   6. fma-udef 45.2%

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

   7. +-commutative 45.2%

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

   8. hypot-1-def 45.2%

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

   9. associate-/l/ 45.7%

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

   10. sqrt-prod 46.4%

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

   11. fma-udef 46.4%

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

   12. +-commutative 46.4%

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

   13. hypot-1-def 50.1%

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

5. Applied egg-rr 50.1%

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

   1. associate-*l/ 50.1%

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

   2. *-lft-identity 50.1%

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

   3. associate-*l* 49.4%

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

7. Simplified 49.4%

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

   1. /-rgt-identity 49.4%

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

   2. *-commutative 49.4%

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

   3. associate-/l* 49.3%

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

   4. pow1/2 49.3%

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

   5. pow-flip 49.4%

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

   6. metadata-eval 49.4%

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

9. Applied egg-rr 49.4%

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

    1. *-un-lft-identity 49.4%

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

    2. *-commutative 49.4%

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

    3. associate-/r* 49.4%

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

    4. associate-/r/ 49.0%

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

    5. associate-/l* 49.0%

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

    6. metadata-eval 49.0%

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

    7. metadata-eval 49.0%

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

    8. sqrt-pow1 49.0%

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

    9. metadata-eval 49.0%

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

    10. inv-pow 49.0%

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

    11. sqrt-div 48.9%

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

    12. metadata-eval 48.9%

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

    13. associate-/l* 49.0%

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

    14. add-sqr-sqrt 97.5%

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

    15. /-rgt-identity 97.5%

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

11. Applied egg-rr 97.5%

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

12. Final simplification 97.5%

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

Alternative 6: 95.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{1}{x_m \cdot \mathsf{fma}\left(z, z, 1\right)}}{y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y_m} \cdot \frac{\frac{1}{z}}{x_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 1.35e+154)
     (/ (/ 1.0 (* x_m (fma z z 1.0))) y_m)
     (* (/ (/ 1.0 z) y_m) (/ (/ 1.0 z) x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.35e+154) {
		tmp = (1.0 / (x_m * fma(z, z, 1.0))) / y_m;
	} else {
		tmp = ((1.0 / z) / y_m) * ((1.0 / z) / x_m);
	}
	return y_s * (x_s * tmp);
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 1.35e+154)
		tmp = Float64(Float64(1.0 / Float64(x_m * fma(z, z, 1.0))) / y_m);
	else
		tmp = Float64(Float64(Float64(1.0 / z) / y_m) * Float64(Float64(1.0 / z) / x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 1.35e+154], N[(N[(1.0 / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.35000000000000003e154

   1. Initial program 93.4%

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

      1. associate-/l/ 91.6%

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

      2. /-rgt-identity 91.6%

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

      3. associate-/l* 91.6%

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

      4. *-rgt-identity 91.6%

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

      5. /-rgt-identity 91.6%

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

      6. associate-/l* 91.6%

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

      7. *-commutative 91.6%

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

      8. associate-/l* 90.9%

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

      9. associate-/r/ 91.0%

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

      10. *-commutative 91.0%

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

      11. /-rgt-identity 91.0%

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

      12. +-commutative 91.0%

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

      13. fma-def 91.0%

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

   3. Simplified 91.0%

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

   if 1.35000000000000003e154 < z

   1. Initial program 66.3%

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

      1. associate-/l/ 66.3%

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

      2. /-rgt-identity 66.3%

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

      3. associate-/l* 66.3%

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

      4. *-rgt-identity 66.3%

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

      5. /-rgt-identity 66.3%

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

      6. associate-/l* 66.3%

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

      7. *-commutative 66.3%

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

      8. associate-/l* 66.3%

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

      9. associate-/r/ 66.3%

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

      10. *-commutative 66.3%

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

      11. /-rgt-identity 66.3%

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

      12. +-commutative 66.3%

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

      13. fma-def 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in z around inf 66.3%

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

      1. /-rgt-identity 66.3%

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

      2. *-commutative 66.3%

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

      3. associate-/l* 66.3%

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

   6. Applied egg-rr 66.3%

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

      1. associate-/r/ 66.3%

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

      2. /-rgt-identity 66.3%

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

      3. unpow2 66.3%

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

      4. associate-*l* 79.6%

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

   8. Applied egg-rr 79.6%

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

      1. associate-/r* 79.4%

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

      2. div-inv 79.4%

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

      3. associate-/l/ 79.4%

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

      4. associate-/l* 99.5%

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

      5. associate-/r/ 94.3%

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

      6. associate-/l/ 90.2%

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

      7. associate-/r* 94.2%

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

   10. Applied egg-rr 94.2%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y} \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \]

Alternative 7: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{y_m}}{x_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{x_m}}{z}}{y_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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-11)
     (/ (/ 1.0 y_m) x_m)
     (/ (/ (/ (/ 1.0 z) x_m) z) y_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-11) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = (((1.0 / z) / x_m) / z) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d-11) then
        tmp = (1.0d0 / y_m) / x_m
    else
        tmp = (((1.0d0 / z) / x_m) / z) / y_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-11) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = (((1.0 / z) / x_m) / z) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 5e-11:
		tmp = (1.0 / y_m) / x_m
	else:
		tmp = (((1.0 / z) / x_m) / z) / y_m
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-11)
		tmp = Float64(Float64(1.0 / y_m) / x_m);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / z) / x_m) / z) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 5e-11)
		tmp = (1.0 / y_m) / x_m;
	else
		tmp = (((1.0 / z) / x_m) / z) / y_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e-11], N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(N[(1.0 / z), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000018e-11

   1. Initial program 99.1%

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

      1. /-rgt-identity 99.1%

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

      2. associate-/l* 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-/l* 97.8%

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

      5. associate-/r/ 98.6%

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

      6. *-commutative 98.6%

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

      7. /-rgt-identity 98.6%

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

      8. +-commutative 98.6%

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

      9. fma-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around 0 98.6%

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

      1. associate-/l/ 99.7%

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

   6. Simplified 99.7%

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

   if 5.00000000000000018e-11 < (*.f64 z z)

   1. Initial program 81.4%

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

      1. /-rgt-identity 81.4%

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

      2. associate-/l* 81.4%

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

      3. *-commutative 81.4%

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

      4. associate-/l* 81.4%

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

      5. associate-/r/ 81.4%

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

      6. *-commutative 81.4%

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

      7. /-rgt-identity 81.4%

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

      8. +-commutative 81.4%

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

      9. fma-def 81.4%

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

   3. Simplified 81.4%

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

      1. associate-/r* 81.4%

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

      2. div-inv 81.4%

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

      3. add-sqr-sqrt 40.1%

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

      4. times-frac 40.1%

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

      5. sqrt-prod 40.1%

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

      6. fma-udef 40.1%

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

      7. +-commutative 40.1%

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

      8. hypot-1-def 40.1%

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

      9. associate-/l/ 40.1%

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

      10. sqrt-prod 41.6%

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

      11. fma-udef 41.6%

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

      12. +-commutative 41.6%

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

      13. hypot-1-def 49.7%

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

   5. Applied egg-rr 49.7%

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

      1. associate-*l/ 49.7%

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

      2. *-lft-identity 49.7%

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

      3. associate-*l* 48.1%

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

   7. Simplified 48.1%

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

   8. Taylor expanded in z around inf 80.7%

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

      1. associate-/r* 80.7%

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

      2. associate-/l/ 77.1%

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

   10. Simplified 77.1%

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

       1. div-inv 77.1%

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

       2. unpow2 77.1%

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

       3. times-frac 83.5%

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

   12. Applied egg-rr 83.5%

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

       1. associate-*r/ 83.5%

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

       2. un-div-inv 83.5%

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

   14. Applied egg-rr 83.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{x}}{z}}{y}\\ \end{array} \]

Alternative 8: 78.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.0%

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

      1. /-rgt-identity 93.0%

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

      2. associate-/l* 93.0%

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

      3. *-commutative 93.0%

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

      4. associate-/l* 92.1%

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

      5. associate-/r/ 92.7%

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

      6. *-commutative 92.7%

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

      7. /-rgt-identity 92.7%

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

      8. +-commutative 92.7%

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

      9. fma-def 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in z around 0 73.5%

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

      1. associate-/l/ 74.0%

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

   6. Simplified 74.0%

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

   if 1 < z

   1. Initial program 82.7%

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

      1. associate-/l/ 72.6%

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

      2. /-rgt-identity 72.6%

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

      3. associate-/l* 72.6%

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

      4. *-rgt-identity 72.6%

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

      5. /-rgt-identity 72.6%

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

      6. associate-/l* 72.6%

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

      7. *-commutative 72.6%

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

      8. associate-/l* 69.7%

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

      9. associate-/r/ 69.9%

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

      10. *-commutative 69.9%

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

      11. /-rgt-identity 69.9%

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

      12. +-commutative 69.9%

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

      13. fma-def 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in z around inf 69.9%

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

      1. div-inv 69.8%

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

      2. add-sqr-sqrt 52.2%

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

      3. associate-*l* 52.3%

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

      4. associate-/r* 52.1%

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

      5. sqrt-div 34.6%

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

      6. inv-pow 34.6%

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

      7. metadata-eval 34.6%

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

      8. pow-sqr 34.6%

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

      9. sqrt-unprod 34.6%

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

      10. add-sqr-sqrt 34.6%

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

      11. unpow2 34.6%

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

      12. sqrt-prod 34.6%

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

      13. add-sqr-sqrt 34.6%

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

   6. Applied egg-rr 52.6%

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

      1. associate-*r* 38.6%

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

      2. unpow2 38.6%

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

   8. Simplified 38.6%

      \[\leadsto \color{blue}{{\left(\frac{{x}^{-0.5}}{z}\right)}^{2} \cdot \frac{1}{y}} \]
   9. Step-by-step derivation

      1. unpow2 38.6%

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

      2. frac-times 35.0%

         \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{z \cdot z}} \cdot \frac{1}{y} \]

      3. pow-prod-up 72.6%

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

      4. metadata-eval 72.6%

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

      5. inv-pow 72.6%

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

      6. unpow2 72.6%

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

      7. clear-num 69.7%

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

      8. associate-*l/ 69.7%

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

      9. div-inv 69.7%

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

      10. clear-num 69.7%

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

      11. div-inv 69.7%

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

      12. associate-/r* 69.8%

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

      13. remove-double-div 69.8%

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

      14. pow-flip 69.9%

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

      15. metadata-eval 69.9%

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

   10. Applied egg-rr 69.9%

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

       1. div-inv 69.8%

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

       2. pow-flip 69.8%

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

       3. metadata-eval 69.8%

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

       4. pow2 69.8%

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

       5. associate-*l* 75.7%

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

       6. *-commutative 75.7%

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

   12. Applied egg-rr 75.7%

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

4. Final simplification 74.4%

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

Alternative 9: 78.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.0%

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

      1. /-rgt-identity 93.0%

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

      2. associate-/l* 93.0%

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

      3. *-commutative 93.0%

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

      4. associate-/l* 92.1%

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

      5. associate-/r/ 92.7%

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

      6. *-commutative 92.7%

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

      7. /-rgt-identity 92.7%

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

      8. +-commutative 92.7%

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

      9. fma-def 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in z around 0 73.5%

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

      1. associate-/l/ 74.0%

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

   6. Simplified 74.0%

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

   if 1 < z

   1. Initial program 82.7%

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

      1. associate-/l/ 72.6%

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

      2. /-rgt-identity 72.6%

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

      3. associate-/l* 72.6%

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

      4. *-rgt-identity 72.6%

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

      5. /-rgt-identity 72.6%

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

      6. associate-/l* 72.6%

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

      7. *-commutative 72.6%

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

      8. associate-/l* 69.7%

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

      9. associate-/r/ 69.9%

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

      10. *-commutative 69.9%

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

      11. /-rgt-identity 69.9%

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

      12. +-commutative 69.9%

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

      13. fma-def 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in z around inf 69.9%

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

      1. /-rgt-identity 69.9%

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

      2. *-commutative 69.9%

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

      3. associate-/l* 69.7%

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

   6. Applied egg-rr 69.7%

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

      1. associate-/r/ 69.9%

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

      2. /-rgt-identity 69.9%

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

      3. unpow2 69.9%

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

      4. associate-*l* 75.9%

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

   8. Applied egg-rr 75.9%

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

4. Final simplification 74.4%

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

Alternative 10: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1.15:\\ \;\;\;\;\frac{\frac{1}{y_m}}{x_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z}}{z \cdot x_m}}{y_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 1.15) (/ (/ 1.0 y_m) x_m) (/ (/ (/ 1.0 z) (* z x_m)) y_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.15) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = ((1.0 / z) / (z * x_m)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.15d0) then
        tmp = (1.0d0 / y_m) / x_m
    else
        tmp = ((1.0d0 / z) / (z * x_m)) / y_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.15) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = ((1.0 / z) / (z * x_m)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 1.15:
		tmp = (1.0 / y_m) / x_m
	else:
		tmp = ((1.0 / z) / (z * x_m)) / y_m
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 1.15)
		tmp = Float64(Float64(1.0 / y_m) / x_m);
	else
		tmp = Float64(Float64(Float64(1.0 / z) / Float64(z * x_m)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 1.15)
		tmp = (1.0 / y_m) / x_m;
	else
		tmp = ((1.0 / z) / (z * x_m)) / y_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 1.15], N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.1499999999999999

   1. Initial program 93.0%

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

      1. /-rgt-identity 93.0%

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

      2. associate-/l* 93.0%

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

      3. *-commutative 93.0%

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

      4. associate-/l* 92.1%

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

      5. associate-/r/ 92.7%

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

      6. *-commutative 92.7%

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

      7. /-rgt-identity 92.7%

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

      8. +-commutative 92.7%

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

      9. fma-def 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in z around 0 73.5%

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

      1. associate-/l/ 74.0%

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

   6. Simplified 74.0%

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

   if 1.1499999999999999 < z

   1. Initial program 82.7%

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

      1. /-rgt-identity 82.7%

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

      2. associate-/l* 82.7%

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

      3. *-commutative 82.7%

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

      4. associate-/l* 82.6%

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

      5. associate-/r/ 82.7%

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

      6. *-commutative 82.7%

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

      7. /-rgt-identity 82.7%

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

      8. +-commutative 82.7%

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

      9. fma-def 82.7%

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

   3. Simplified 82.7%

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

      1. associate-/r* 82.7%

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

      2. div-inv 82.7%

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

      3. add-sqr-sqrt 43.1%

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

      4. times-frac 43.2%

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

      5. sqrt-prod 43.2%

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

      6. fma-udef 43.2%

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

      7. +-commutative 43.2%

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

      8. hypot-1-def 43.2%

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

      9. associate-/l/ 43.2%

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

      10. sqrt-prod 43.2%

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

      11. fma-udef 43.2%

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

      12. +-commutative 43.2%

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

      13. hypot-1-def 54.6%

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

   5. Applied egg-rr 54.6%

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

      1. associate-*l/ 54.6%

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

      2. *-lft-identity 54.6%

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

      3. associate-*l* 52.8%

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

   7. Simplified 52.8%

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

   8. Taylor expanded in z around inf 82.7%

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

      1. associate-/r* 82.7%

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

      2. associate-/l/ 72.6%

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

   10. Simplified 72.6%

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

       1. div-inv 72.6%

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

       2. unpow2 72.6%

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

       3. times-frac 78.7%

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

   12. Applied egg-rr 78.7%

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

       1. associate-/l/ 78.6%

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

       2. div-inv 78.7%

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

   14. Applied egg-rr 78.7%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.15:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z}}{z \cdot x}}{y}\\ \end{array} \]

Alternative 11: 58.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

   1. /-rgt-identity 91.0%

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

   2. associate-/l* 91.0%

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

   3. *-commutative 91.0%

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

   4. associate-/l* 90.2%

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

   5. associate-/r/ 90.7%

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

   6. *-commutative 90.7%

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

   7. /-rgt-identity 90.7%

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

   8. +-commutative 90.7%

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

   9. fma-def 90.7%

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

3. Simplified 90.7%

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

4. Taylor expanded in z around 0 61.2%

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

5. Final simplification 61.2%

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

Alternative 12: 58.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

   1. /-rgt-identity 91.0%

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

   2. associate-/l* 91.0%

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

   3. *-commutative 91.0%

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

   4. associate-/l* 90.2%

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

   5. associate-/r/ 90.7%

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

   6. *-commutative 90.7%

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

   7. /-rgt-identity 90.7%

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

   8. +-commutative 90.7%

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

   9. fma-def 90.7%

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

3. Simplified 90.7%

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

4. Taylor expanded in z around 0 61.2%

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

   1. associate-/l/ 61.4%

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

6. Simplified 61.4%

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

7. Final simplification 61.4%

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

Developer target: 92.3% accurate, 0.4× 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 2023326 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (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)))))