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

Percentage Accurate: 89.7% → 99.6%

Time: 10.2s

Alternatives: 9

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.7% 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: 99.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m] = \mathsf{sort}([x_m, y, z_m])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z\_m \cdot z\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{1}{y}}}{z\_m}}{x\_m \cdot \left(\sqrt{y} \cdot \mathsf{hypot}\left(1, z\_m\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z_m)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z_m z_m)))))
   (*
    x_s
    (if (<= t_0 5e+307)
      (/ (/ 1.0 x_m) t_0)
      (/ (/ (sqrt (/ 1.0 y)) z_m) (* x_m (* (sqrt y) (hypot 1.0 z_m))))))))

C

z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z_m);
double code(double x_s, double x_m, double y, double z_m) {
	double t_0 = y * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 5e+307) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (sqrt((1.0 / y)) / z_m) / (x_m * (sqrt(y) * hypot(1.0, z_m)));
	}
	return x_s * tmp;
}

Java

z_m = Math.abs(z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z_m;
public static double code(double x_s, double x_m, double y, double z_m) {
	double t_0 = y * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 5e+307) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (Math.sqrt((1.0 / y)) / z_m) / (x_m * (Math.sqrt(y) * Math.hypot(1.0, z_m)));
	}
	return x_s * tmp;
}

Python

z_m = math.fabs(z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z_m] = sort([x_m, y, z_m])
def code(x_s, x_m, y, z_m):
	t_0 = y * (1.0 + (z_m * z_m))
	tmp = 0
	if t_0 <= 5e+307:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = (math.sqrt((1.0 / y)) / z_m) / (x_m * (math.sqrt(y) * math.hypot(1.0, z_m)))
	return x_s * tmp

Julia

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z_m = sort([x_m, y, z_m])
function code(x_s, x_m, y, z_m)
	t_0 = Float64(y * Float64(1.0 + Float64(z_m * z_m)))
	tmp = 0.0
	if (t_0 <= 5e+307)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / y)) / z_m) / Float64(x_m * Float64(sqrt(y) * hypot(1.0, z_m))));
	end
	return Float64(x_s * tmp)
end

MATLAB

z_m = abs(z);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z_m = num2cell(sort([x_m, y, z_m])){:}
function tmp_2 = code(x_s, x_m, y, z_m)
	t_0 = y * (1.0 + (z_m * z_m));
	tmp = 0.0;
	if (t_0 <= 5e+307)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = (sqrt((1.0 / y)) / z_m) / (x_m * (sqrt(y) * hypot(1.0, z_m)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z$95$m_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 5e+307], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] / z$95$m), $MachinePrecision] / N[(x$95$m * N[(N[Sqrt[y], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 5e307

   1. Initial program 94.3%

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

   if 5e307 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 61.4%

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

      1. associate-/l/ 61.4%

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

      2. associate-*l* 74.1%

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

      3. *-commutative 74.1%

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

      4. sqr-neg 74.1%

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

      5. +-commutative 74.1%

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

      6. sqr-neg 74.1%

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

      7. fma-define 74.1%

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

   3. Simplified 74.1%

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

      1. associate-*r* 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-/r* 73.3%

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

      4. *-commutative 73.3%

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

      5. associate-/l/ 73.2%

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

      6. fma-undefine 73.2%

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

      7. +-commutative 73.2%

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

      8. associate-/r* 61.4%

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

      9. *-un-lft-identity 61.4%

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

      10. add-sqr-sqrt 61.4%

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

      11. times-frac 61.4%

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

      12. +-commutative 61.4%

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

      13. fma-undefine 61.4%

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

      14. *-commutative 61.4%

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

      15. sqrt-prod 61.4%

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

      16. fma-undefine 61.4%

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

      17. +-commutative 61.4%

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

      18. hypot-1-def 61.4%

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

      19. +-commutative 61.4%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/l/ 99.5%

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

      2. associate-*r/ 99.5%

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

      3. *-rgt-identity 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.5%

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

      6. *-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in z around inf 81.5%

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

       1. associate-*r/ 81.6%

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

       2. *-rgt-identity 81.6%

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

   11. Simplified 81.6%

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

4. Final simplification 92.5%

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

Alternative 2: 99.4% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

   1. associate-/l/ 89.6%

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

   2. associate-*l* 89.5%

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

   3. *-commutative 89.5%

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

   4. sqr-neg 89.5%

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

   5. +-commutative 89.5%

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

   6. sqr-neg 89.5%

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

   7. fma-define 89.5%

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

3. Simplified 89.5%

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

   1. associate-*r* 91.6%

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

   2. *-commutative 91.6%

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

   3. associate-/r* 91.3%

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

   4. *-commutative 91.3%

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

   5. associate-/l/ 91.2%

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

   6. fma-undefine 91.2%

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

   7. +-commutative 91.2%

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

   8. associate-/r* 89.6%

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

   9. *-un-lft-identity 89.6%

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

   10. add-sqr-sqrt 43.4%

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

   11. times-frac 43.4%

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

   12. +-commutative 43.4%

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

   13. fma-undefine 43.4%

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

   14. *-commutative 43.4%

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

   15. sqrt-prod 43.4%

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

   16. fma-undefine 43.4%

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

   17. +-commutative 43.4%

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

   18. hypot-1-def 43.4%

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

   19. +-commutative 43.4%

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

6. Applied egg-rr 48.9%

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

   1. associate-/l/ 48.9%

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

   2. associate-*r/ 48.9%

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

   3. *-rgt-identity 48.9%

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

   4. *-commutative 48.9%

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

   5. associate-/r* 48.9%

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

   6. *-commutative 48.9%

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

8. Simplified 48.9%

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

9. Final simplification 48.9%

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

Alternative 3: 83.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m] = \mathsf{sort}([x_m, y, z_m])\\ \\ \begin{array}{l} t_0 := z\_m \cdot \sqrt{x\_m}\\ t_1 := y \cdot \left(1 + z\_m \cdot z\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{t\_0}}{t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z_m)
 :precision binary64
 (let* ((t_0 (* z_m (sqrt x_m))) (t_1 (* y (+ 1.0 (* z_m z_m)))))
   (* x_s (if (<= t_1 5e+307) (/ (/ 1.0 x_m) t_1) (/ (/ (/ 1.0 y) t_0) t_0)))))

C

z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z_m);
double code(double x_s, double x_m, double y, double z_m) {
	double t_0 = z_m * sqrt(x_m);
	double t_1 = y * (1.0 + (z_m * z_m));
	double tmp;
	if (t_1 <= 5e+307) {
		tmp = (1.0 / x_m) / t_1;
	} else {
		tmp = ((1.0 / y) / t_0) / t_0;
	}
	return x_s * tmp;
}

Fortran

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z_m should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z_m * sqrt(x_m)
    t_1 = y * (1.0d0 + (z_m * z_m))
    if (t_1 <= 5d+307) then
        tmp = (1.0d0 / x_m) / t_1
    else
        tmp = ((1.0d0 / y) / t_0) / t_0
    end if
    code = x_s * tmp
end function

Java

z_m = Math.abs(z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z_m;
public static double code(double x_s, double x_m, double y, double z_m) {
	double t_0 = z_m * Math.sqrt(x_m);
	double t_1 = y * (1.0 + (z_m * z_m));
	double tmp;
	if (t_1 <= 5e+307) {
		tmp = (1.0 / x_m) / t_1;
	} else {
		tmp = ((1.0 / y) / t_0) / t_0;
	}
	return x_s * tmp;
}

Python

z_m = math.fabs(z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z_m] = sort([x_m, y, z_m])
def code(x_s, x_m, y, z_m):
	t_0 = z_m * math.sqrt(x_m)
	t_1 = y * (1.0 + (z_m * z_m))
	tmp = 0
	if t_1 <= 5e+307:
		tmp = (1.0 / x_m) / t_1
	else:
		tmp = ((1.0 / y) / t_0) / t_0
	return x_s * tmp

Julia

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z_m = sort([x_m, y, z_m])
function code(x_s, x_m, y, z_m)
	t_0 = Float64(z_m * sqrt(x_m))
	t_1 = Float64(y * Float64(1.0 + Float64(z_m * z_m)))
	tmp = 0.0
	if (t_1 <= 5e+307)
		tmp = Float64(Float64(1.0 / x_m) / t_1);
	else
		tmp = Float64(Float64(Float64(1.0 / y) / t_0) / t_0);
	end
	return Float64(x_s * tmp)
end

MATLAB

z_m = abs(z);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z_m = num2cell(sort([x_m, y, z_m])){:}
function tmp_2 = code(x_s, x_m, y, z_m)
	t_0 = z_m * sqrt(x_m);
	t_1 = y * (1.0 + (z_m * z_m));
	tmp = 0.0;
	if (t_1 <= 5e+307)
		tmp = (1.0 / x_m) / t_1;
	else
		tmp = ((1.0 / y) / t_0) / t_0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z$95$m_] := Block[{t$95$0 = N[(z$95$m * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, 5e+307], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 5e307

   1. Initial program 94.3%

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

   if 5e307 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 61.4%

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

      1. associate-/l/ 61.4%

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

      2. associate-*l* 74.1%

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

      3. *-commutative 74.1%

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

      4. sqr-neg 74.1%

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

      5. +-commutative 74.1%

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

      6. sqr-neg 74.1%

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

      7. fma-define 74.1%

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

   3. Simplified 74.1%

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

      1. associate-*r* 73.4%

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

      2. *-commutative 73.4%

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

      3. *-commutative 73.4%

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

      4. add-sqr-sqrt 70.8%

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

      5. pow2 70.8%

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

      6. sqrt-div 48.8%

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

      7. metadata-eval 48.8%

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

      8. sqrt-prod 48.9%

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

      9. fma-undefine 48.9%

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

      10. +-commutative 48.9%

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

      11. hypot-1-def 54.3%

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

   6. Applied egg-rr 54.3%

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

   7. Taylor expanded in z around inf 54.3%

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

      1. metadata-eval 54.3%

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

      2. pow1 54.3%

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

      3. metadata-eval 54.3%

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

      4. sqrt-pow1 48.9%

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

      5. sqrt-prod 48.8%

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

      6. *-commutative 48.8%

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

      7. associate-*r* 49.5%

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

      8. sqrt-div 71.5%

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

      9. pow2 71.5%

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

      10. add-sqr-sqrt 74.1%

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

      11. associate-/r* 74.3%

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

      12. add-sqr-sqrt 49.6%

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

      13. associate-/r* 49.6%

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

      14. *-commutative 49.6%

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

      15. sqrt-prod 49.6%

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

      16. sqrt-pow1 41.8%

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

      17. metadata-eval 41.8%

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

      18. pow1 41.8%

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

   9. Applied egg-rr 70.0%

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

Alternative 4: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m] = \mathsf{sort}([x_m, y, z_m])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 4 \cdot 10^{+298}:\\ \;\;\;\;\frac{1}{y \cdot \left(x\_m \cdot \mathsf{fma}\left(z\_m, z\_m, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot z\_m} \cdot \frac{\frac{1}{\mathsf{hypot}\left(1, z\_m\right)}}{x\_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z_m)
 :precision binary64
 (*
  x_s
  (if (<= (* z_m z_m) 4e+298)
    (/ 1.0 (* y (* x_m (fma z_m z_m 1.0))))
    (* (/ 1.0 (* y z_m)) (/ (/ 1.0 (hypot 1.0 z_m)) x_m)))))

C

z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z_m);
double code(double x_s, double x_m, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 4e+298) {
		tmp = 1.0 / (y * (x_m * fma(z_m, z_m, 1.0)));
	} else {
		tmp = (1.0 / (y * z_m)) * ((1.0 / hypot(1.0, z_m)) / x_m);
	}
	return x_s * tmp;
}

Julia

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z_m = sort([x_m, y, z_m])
function code(x_s, x_m, y, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 4e+298)
		tmp = Float64(1.0 / Float64(y * Float64(x_m * fma(z_m, z_m, 1.0))));
	else
		tmp = Float64(Float64(1.0 / Float64(y * z_m)) * Float64(Float64(1.0 / hypot(1.0, z_m)) / x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z$95$m_] := N[(x$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 4e+298], N[(1.0 / N[(y * N[(x$95$m * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.9999999999999998e298

   1. Initial program 95.7%

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

      1. associate-/l/ 95.7%

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

      2. associate-*l* 95.6%

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

      3. *-commutative 95.6%

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

      4. sqr-neg 95.6%

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

      5. +-commutative 95.6%

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

      6. sqr-neg 95.6%

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

      7. fma-define 95.6%

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

   3. Simplified 95.6%

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

   if 3.9999999999999998e298 < (*.f64 z z)

   1. Initial program 70.9%

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

      1. associate-/l/ 70.9%

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

      2. associate-*l* 70.9%

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

      3. *-commutative 70.9%

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

      4. sqr-neg 70.9%

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

      5. +-commutative 70.9%

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

      6. sqr-neg 70.9%

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

      7. fma-define 70.9%

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

   3. Simplified 70.9%

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

      1. add-sqr-sqrt 41.7%

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

      2. pow2 41.7%

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

      3. *-commutative 41.7%

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

      4. sqrt-prod 41.7%

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

      5. fma-undefine 41.7%

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

      6. +-commutative 41.7%

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

      7. hypot-1-def 49.2%

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

   6. Applied egg-rr 49.2%

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

      1. *-commutative 49.2%

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

      2. unpow2 49.2%

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

      3. swap-sqr 41.7%

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

      4. pow2 41.7%

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

      5. add-sqr-sqrt 70.9%

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

      6. associate-*l* 70.3%

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

      7. pow2 70.3%

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

      8. hypot-undefine 70.3%

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

      9. metadata-eval 70.3%

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

      10. unpow2 70.3%

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

      11. hypot-undefine 70.3%

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

      12. metadata-eval 70.3%

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

      13. unpow2 70.3%

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

      14. add-sqr-sqrt 70.3%

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

      15. +-commutative 70.3%

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

      16. unpow2 70.3%

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

      17. fma-undefine 70.3%

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

      18. *-commutative 70.3%

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

   8. Applied egg-rr 97.9%

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

   9. Taylor expanded in z around inf 81.2%

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

Alternative 5: 96.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m] = \mathsf{sort}([x_m, y, z_m])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 10^{+30}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y \cdot \left(1 + z\_m \cdot z\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z\_m} \cdot \frac{\frac{1}{x\_m}}{z\_m}}{y}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z_m)
 :precision binary64
 (*
  x_s
  (if (<= (* z_m z_m) 1e+30)
    (/ (/ 1.0 x_m) (* y (+ 1.0 (* z_m z_m))))
    (/ (* (/ 1.0 z_m) (/ (/ 1.0 x_m) z_m)) y))))

C

z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z_m);
double code(double x_s, double x_m, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1e+30) {
		tmp = (1.0 / x_m) / (y * (1.0 + (z_m * z_m)));
	} else {
		tmp = ((1.0 / z_m) * ((1.0 / x_m) / z_m)) / y;
	}
	return x_s * tmp;
}

Fortran

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z_m should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((z_m * z_m) <= 1d+30) then
        tmp = (1.0d0 / x_m) / (y * (1.0d0 + (z_m * z_m)))
    else
        tmp = ((1.0d0 / z_m) * ((1.0d0 / x_m) / z_m)) / y
    end if
    code = x_s * tmp
end function

Java

z_m = Math.abs(z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z_m;
public static double code(double x_s, double x_m, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1e+30) {
		tmp = (1.0 / x_m) / (y * (1.0 + (z_m * z_m)));
	} else {
		tmp = ((1.0 / z_m) * ((1.0 / x_m) / z_m)) / y;
	}
	return x_s * tmp;
}

Python

z_m = math.fabs(z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z_m] = sort([x_m, y, z_m])
def code(x_s, x_m, y, z_m):
	tmp = 0
	if (z_m * z_m) <= 1e+30:
		tmp = (1.0 / x_m) / (y * (1.0 + (z_m * z_m)))
	else:
		tmp = ((1.0 / z_m) * ((1.0 / x_m) / z_m)) / y
	return x_s * tmp

Julia

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z_m = sort([x_m, y, z_m])
function code(x_s, x_m, y, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 1e+30)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y * Float64(1.0 + Float64(z_m * z_m))));
	else
		tmp = Float64(Float64(Float64(1.0 / z_m) * Float64(Float64(1.0 / x_m) / z_m)) / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

z_m = abs(z);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z_m = num2cell(sort([x_m, y, z_m])){:}
function tmp_2 = code(x_s, x_m, y, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 1e+30)
		tmp = (1.0 / x_m) / (y * (1.0 + (z_m * z_m)));
	else
		tmp = ((1.0 / z_m) * ((1.0 / x_m) / z_m)) / y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z$95$m_] := N[(x$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1e+30], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e30

   1. Initial program 99.7%

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

   if 1e30 < (*.f64 z z)

   1. Initial program 79.0%

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

      1. associate-/l/ 79.0%

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

      2. associate-*l* 78.9%

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

      3. *-commutative 78.9%

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

      4. sqr-neg 78.9%

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

      5. +-commutative 78.9%

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

      6. sqr-neg 78.9%

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

      7. fma-define 78.9%

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

   3. Simplified 78.9%

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

      1. associate-*r* 83.0%

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

      2. *-commutative 83.0%

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

      3. *-commutative 83.0%

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

      4. add-sqr-sqrt 66.3%

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

      5. pow2 66.3%

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

      6. sqrt-div 40.9%

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

      7. metadata-eval 40.9%

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

      8. sqrt-prod 40.9%

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

      9. fma-undefine 40.9%

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

      10. +-commutative 40.9%

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

      11. hypot-1-def 44.2%

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

   6. Applied egg-rr 44.2%

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

   7. Taylor expanded in z around inf 44.2%

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

      1. metadata-eval 44.2%

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

      2. pow1 44.2%

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

      3. metadata-eval 44.2%

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

      4. sqrt-pow1 40.9%

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

      5. sqrt-prod 40.9%

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

      6. *-commutative 40.9%

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

      7. associate-*r* 37.4%

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

      8. sqrt-div 63.0%

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

      9. pow2 63.0%

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

      10. add-sqr-sqrt 78.9%

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

      11. *-commutative 78.9%

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

      12. associate-/r* 79.1%

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

   9. Applied egg-rr 79.1%

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

       1. associate-/r* 79.5%

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

       2. *-un-lft-identity 79.5%

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

       3. unpow2 79.5%

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

       4. times-frac 87.3%

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

   11. Applied egg-rr 87.3%

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

Alternative 6: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m] = \mathsf{sort}([x_m, y, z_m])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z\_m} \cdot \frac{\frac{1}{x\_m}}{z\_m}}{y}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z_m)
 :precision binary64
 (*
  x_s
  (if (<= (* z_m z_m) 4e-6)
    (/ (/ 1.0 x_m) y)
    (/ (* (/ 1.0 z_m) (/ (/ 1.0 x_m) z_m)) y))))

C

z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z_m);
double code(double x_s, double x_m, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 4e-6) {
		tmp = (1.0 / x_m) / y;
	} else {
		tmp = ((1.0 / z_m) * ((1.0 / x_m) / z_m)) / y;
	}
	return x_s * tmp;
}

Fortran

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z_m should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((z_m * z_m) <= 4d-6) then
        tmp = (1.0d0 / x_m) / y
    else
        tmp = ((1.0d0 / z_m) * ((1.0d0 / x_m) / z_m)) / y
    end if
    code = x_s * tmp
end function

Java

z_m = Math.abs(z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z_m;
public static double code(double x_s, double x_m, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 4e-6) {
		tmp = (1.0 / x_m) / y;
	} else {
		tmp = ((1.0 / z_m) * ((1.0 / x_m) / z_m)) / y;
	}
	return x_s * tmp;
}

Python

z_m = math.fabs(z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z_m] = sort([x_m, y, z_m])
def code(x_s, x_m, y, z_m):
	tmp = 0
	if (z_m * z_m) <= 4e-6:
		tmp = (1.0 / x_m) / y
	else:
		tmp = ((1.0 / z_m) * ((1.0 / x_m) / z_m)) / y
	return x_s * tmp

Julia

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z_m = sort([x_m, y, z_m])
function code(x_s, x_m, y, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 4e-6)
		tmp = Float64(Float64(1.0 / x_m) / y);
	else
		tmp = Float64(Float64(Float64(1.0 / z_m) * Float64(Float64(1.0 / x_m) / z_m)) / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

z_m = abs(z);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z_m = num2cell(sort([x_m, y, z_m])){:}
function tmp_2 = code(x_s, x_m, y, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 4e-6)
		tmp = (1.0 / x_m) / y;
	else
		tmp = ((1.0 / z_m) * ((1.0 / x_m) / z_m)) / y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z_m should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z$95$m_] := N[(x$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 4e-6], N[(N[(1.0 / x$95$m), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.99999999999999982e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0 99.3%

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

   if 3.99999999999999982e-6 < (*.f64 z z)

   1. Initial program 79.9%

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

      1. associate-/l/ 79.9%

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

      2. associate-*l* 79.8%

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

      3. *-commutative 79.8%

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

      4. sqr-neg 79.8%

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

      5. +-commutative 79.8%

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

      6. sqr-neg 79.8%

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

      7. fma-define 79.8%

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

   3. Simplified 79.8%

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

      1. associate-*r* 83.8%

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

      2. *-commutative 83.8%

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

      3. *-commutative 83.8%

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

      4. add-sqr-sqrt 67.8%

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

      5. pow2 67.8%

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

      6. sqrt-div 43.6%

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

      7. metadata-eval 43.6%

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

      8. sqrt-prod 43.6%

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

      9. fma-undefine 43.6%

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

      10. +-commutative 43.6%

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

      11. hypot-1-def 46.8%

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

   6. Applied egg-rr 46.8%

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

   7. Taylor expanded in z around inf 46.8%

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

      1. metadata-eval 46.8%

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

      2. pow1 46.8%

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

      3. metadata-eval 46.8%

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

      4. sqrt-pow1 43.6%

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

      5. sqrt-prod 43.6%

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

      6. *-commutative 43.6%

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

      7. associate-*r* 40.2%

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

      8. sqrt-div 64.6%

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

      9. pow2 64.6%

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

      10. add-sqr-sqrt 79.8%

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

      11. *-commutative 79.8%

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

      12. associate-/r* 80.0%

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

   9. Applied egg-rr 80.0%

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

       1. associate-/r* 80.4%

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

       2. *-un-lft-identity 80.4%

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

       3. unpow2 80.4%

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

       4. times-frac 87.9%

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

   11. Applied egg-rr 87.9%

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

Alternative 7: 91.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0 99.3%

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

   if 1 < (*.f64 z z)

   1. Initial program 79.9%

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

      1. associate-/l/ 79.9%

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

      2. associate-*l* 79.8%

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

      3. *-commutative 79.8%

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

      4. sqr-neg 79.8%

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

      5. +-commutative 79.8%

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

      6. sqr-neg 79.8%

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

      7. fma-define 79.8%

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

   3. Simplified 79.8%

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

      1. associate-*r* 83.8%

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

      2. *-commutative 83.8%

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

      3. *-commutative 83.8%

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

      4. add-sqr-sqrt 67.8%

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

      5. pow2 67.8%

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

      6. sqrt-div 43.6%

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

      7. metadata-eval 43.6%

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

      8. sqrt-prod 43.6%

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

      9. fma-undefine 43.6%

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

      10. +-commutative 43.6%

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

      11. hypot-1-def 46.8%

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

   6. Applied egg-rr 46.8%

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

   7. Taylor expanded in z around inf 46.8%

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

      1. metadata-eval 46.8%

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

      2. pow1 46.8%

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

      3. metadata-eval 46.8%

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

      4. sqrt-pow1 43.6%

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

      5. sqrt-prod 43.6%

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

      6. *-commutative 43.6%

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

      7. associate-*r* 40.2%

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

      8. sqrt-div 64.6%

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

      9. pow2 64.6%

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

      10. add-sqr-sqrt 79.8%

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

      11. *-commutative 79.8%

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

      12. associate-/r* 80.0%

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

   9. Applied egg-rr 80.0%

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

       1. unpow2 80.0%

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

   11. Applied egg-rr 80.0%

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

Alternative 8: 90.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0 99.3%

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

   if 1 < (*.f64 z z)

   1. Initial program 79.9%

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

      1. associate-/l/ 79.9%

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

      2. associate-*l* 79.8%

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

      3. *-commutative 79.8%

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

      4. sqr-neg 79.8%

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

      5. +-commutative 79.8%

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

      6. sqr-neg 79.8%

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

      7. fma-define 79.8%

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

   3. Simplified 79.8%

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

   5. Taylor expanded in z around inf 79.8%

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

      1. unpow2 80.0%

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

   7. Applied egg-rr 79.8%

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

Alternative 9: 58.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

   1. associate-/l/ 89.6%

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

   2. associate-*l* 89.5%

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

   3. *-commutative 89.5%

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

   4. sqr-neg 89.5%

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

   5. +-commutative 89.5%

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

   6. sqr-neg 89.5%

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

   7. fma-define 89.5%

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

3. Simplified 89.5%

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

5. Taylor expanded in z around 0 56.8%

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

Developer target: 92.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + z \cdot z\\ t_1 := y \cdot t\_0\\ t_2 := \frac{\frac{1}{y}}{t\_0 \cdot x}\\ \mathbf{if}\;t\_1 < -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{t\_0 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z z))) (t_1 (* y t_0)) (t_2 (/ (/ 1.0 y) (* t_0 x))))
   (if (< t_1 (- INFINITY))
     t_2
     (if (< t_1 8.680743250567252e+305) (/ (/ 1.0 x) (* t_0 y)) t_2))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z * z)
	t_1 = y * t_0
	t_2 = (1.0 / y) / (t_0 * x)
	tmp = 0
	if t_1 < -math.inf:
		tmp = t_2
	elif t_1 < 8.680743250567252e+305:
		tmp = (1.0 / x) / (t_0 * y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z * z))
	t_1 = Float64(y * t_0)
	t_2 = Float64(Float64(1.0 / y) / Float64(t_0 * x))
	tmp = 0.0
	if (t_1 < Float64(-Inf))
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = Float64(Float64(1.0 / x) / Float64(t_0 * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z * z);
	t_1 = y * t_0;
	t_2 = (1.0 / y) / (t_0 * x);
	tmp = 0.0;
	if (t_1 < -Inf)
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = (1.0 / x) / (t_0 * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / y), $MachinePrecision] / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$1, (-Infinity)], t$95$2, If[Less[t$95$1, 8.680743250567252e+305], N[(N[(1.0 / x), $MachinePrecision] / N[(t$95$0 * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Reproduce

herbie shell --seed 2024118 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

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

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