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

Percentage Accurate: 89.3% → 99.4%

Time: 11.8s

Alternatives: 11

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. associate-/l/ 90.4%

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

   2. remove-double-neg 90.4%

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

   3. distribute-rgt-neg-out 90.4%

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

   4. distribute-rgt-neg-out 90.4%

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

   5. remove-double-neg 90.4%

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

   6. associate-*l* 89.2%

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

   7. *-commutative 89.2%

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

   8. sqr-neg 89.2%

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

   9. +-commutative 89.2%

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

   10. sqr-neg 89.2%

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

   11. fma-define 89.2%

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

3. Simplified 89.2%

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

   1. associate-*r* 91.3%

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

   2. *-commutative 91.3%

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

   3. associate-/r* 91.0%

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

   4. *-commutative 91.0%

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

   5. associate-/l/ 91.0%

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

   6. fma-undefine 91.0%

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

   7. +-commutative 91.0%

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

   8. associate-/r* 90.4%

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

   9. *-un-lft-identity 90.4%

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

   10. add-sqr-sqrt 42.3%

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

   11. times-frac 42.3%

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

   12. +-commutative 42.3%

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

   13. fma-undefine 42.3%

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

   14. *-commutative 42.3%

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

   15. sqrt-prod 42.3%

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

   16. fma-undefine 42.3%

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

   17. +-commutative 42.3%

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

   18. hypot-1-def 42.3%

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

   19. +-commutative 42.3%

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

6. Applied egg-rr 47.8%

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

   1. associate-/l/ 47.8%

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

   2. associate-*r/ 47.8%

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

   3. *-rgt-identity 47.8%

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

   4. *-commutative 47.8%

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

   5. associate-/r* 47.8%

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

   6. *-commutative 47.8%

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

8. Simplified 47.8%

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

9. Final simplification 47.8%

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

Alternative 2: 98.7% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. associate-/l/ 90.4%

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

   2. remove-double-neg 90.4%

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

   3. distribute-rgt-neg-out 90.4%

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

   4. distribute-rgt-neg-out 90.4%

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

   5. remove-double-neg 90.4%

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

   6. associate-*l* 89.2%

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

   7. *-commutative 89.2%

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

   8. sqr-neg 89.2%

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

   9. +-commutative 89.2%

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

   10. sqr-neg 89.2%

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

   11. fma-define 89.2%

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

3. Simplified 89.2%

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

   1. associate-/r* 89.2%

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

   2. div-inv 89.2%

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

6. Applied egg-rr 89.2%

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

   1. un-div-inv 89.2%

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

   2. *-un-lft-identity 89.2%

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

   3. fma-undefine 89.2%

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

   4. +-commutative 89.2%

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

   5. times-frac 90.8%

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

   6. metadata-eval 90.8%

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

   7. add-sqr-sqrt 42.7%

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

   8. frac-times 42.8%

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

   9. add-sqr-sqrt 42.8%

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

   10. hypot-1-def 42.8%

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

   11. hypot-1-def 42.8%

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

   12. frac-times 44.8%

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

   13. associate-/l/ 44.8%

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

   14. un-div-inv 44.8%

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

   15. times-frac 47.8%

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

   16. *-un-lft-identity 47.8%

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

   17. div-inv 47.8%

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

8. Applied egg-rr 47.7%

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

9. Final simplification 47.7%

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

Alternative 3: 98.1% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(N[Power[y$95$m, -0.5], $MachinePrecision] / N[(N[(N[Sqrt[y$95$m], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.4%

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

   1. associate-/l/ 90.4%

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

   2. remove-double-neg 90.4%

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

   3. distribute-rgt-neg-out 90.4%

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

   4. distribute-rgt-neg-out 90.4%

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

   5. remove-double-neg 90.4%

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

   6. associate-*l* 89.2%

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

   7. *-commutative 89.2%

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

   8. sqr-neg 89.2%

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

   9. +-commutative 89.2%

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

   10. sqr-neg 89.2%

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

   11. fma-define 89.2%

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

3. Simplified 89.2%

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

   1. associate-/r* 89.2%

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

   2. div-inv 89.2%

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

6. Applied egg-rr 89.2%

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

   1. un-div-inv 89.2%

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

   2. *-un-lft-identity 89.2%

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

   3. fma-undefine 89.2%

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

   4. +-commutative 89.2%

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

   5. times-frac 90.8%

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

   6. metadata-eval 90.8%

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

   7. add-sqr-sqrt 42.7%

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

   8. frac-times 42.8%

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

   9. add-sqr-sqrt 42.8%

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

   10. hypot-1-def 42.8%

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

   11. hypot-1-def 42.8%

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

   12. frac-times 44.8%

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

   13. associate-/l/ 44.8%

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

   14. un-div-inv 44.8%

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

   15. times-frac 47.8%

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

   16. *-un-lft-identity 47.8%

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

   17. associate-/l/ 47.4%

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

8. Applied egg-rr 46.7%

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

9. Final simplification 46.7%

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

Alternative 4: 49.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (pow((hypot(1.0, z) * sqrt(x)), -2.0) / y_m);
}

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (Math.pow((Math.hypot(1.0, z) * Math.sqrt(x)), -2.0) / y_m);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	return y_s * (math.pow((math.hypot(1.0, z) * math.sqrt(x)), -2.0) / y_m)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64((Float64(hypot(1.0, z) * sqrt(x)) ^ -2.0) / y_m))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (((hypot(1.0, z) * sqrt(x)) ^ -2.0) / y_m);
end

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. associate-/l/ 90.4%

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

   2. remove-double-neg 90.4%

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

   3. distribute-rgt-neg-out 90.4%

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

   4. distribute-rgt-neg-out 90.4%

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

   5. remove-double-neg 90.4%

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

   6. associate-*l* 89.2%

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

   7. *-commutative 89.2%

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

   8. sqr-neg 89.2%

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

   9. +-commutative 89.2%

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

   10. sqr-neg 89.2%

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

   11. fma-define 89.2%

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

3. Simplified 89.2%

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

   1. associate-*r* 91.3%

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

   2. *-commutative 91.3%

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

   3. associate-/r* 91.0%

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

   4. *-commutative 91.0%

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

   5. associate-/l/ 91.0%

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

   6. fma-undefine 91.0%

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

   7. +-commutative 91.0%

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

   8. associate-/r* 90.4%

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

   9. *-un-lft-identity 90.4%

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

   10. add-sqr-sqrt 42.3%

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

   11. times-frac 42.3%

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

   12. +-commutative 42.3%

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

   13. fma-undefine 42.3%

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

   14. *-commutative 42.3%

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

   15. sqrt-prod 42.3%

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

   16. fma-undefine 42.3%

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

   17. +-commutative 42.3%

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

   18. hypot-1-def 42.3%

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

   19. +-commutative 42.3%

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

6. Applied egg-rr 47.8%

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

   1. associate-/l/ 47.8%

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

   2. associate-*r/ 47.8%

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

   3. *-rgt-identity 47.8%

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

   4. *-commutative 47.8%

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

   5. associate-/r* 47.8%

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

   6. *-commutative 47.8%

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

8. Simplified 47.8%

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

   1. *-un-lft-identity 47.8%

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

   2. times-frac 44.8%

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

   3. un-div-inv 44.8%

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

   4. associate-/l/ 44.8%

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

   5. frac-times 42.8%

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

   6. frac-times 42.7%

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

   7. metadata-eval 42.7%

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

   8. add-sqr-sqrt 90.8%

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

   9. hypot-1-def 90.8%

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

   10. hypot-1-def 90.8%

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

   11. add-sqr-sqrt 90.8%

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

   12. times-frac 89.2%

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

   13. +-commutative 89.2%

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

   14. fma-undefine 89.2%

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

   15. add-sqr-sqrt 42.6%

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

10. Applied egg-rr 46.6%

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

    1. associate-/r* 46.6%

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

    2. associate-/l/ 46.6%

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

    3. associate-*r/ 44.1%

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

    4. unpow-1 44.1%

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

    5. unpow-1 44.1%

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

    6. pow-sqr 44.1%

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

    7. metadata-eval 44.1%

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

12. Simplified 44.1%

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

Alternative 5: 91.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (*.f64 z z)) < +inf.0

   1. Initial program 90.4%

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

      1. associate-/l/ 90.4%

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

      2. remove-double-neg 90.4%

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

      3. distribute-rgt-neg-out 90.4%

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

      4. distribute-rgt-neg-out 90.4%

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

      5. remove-double-neg 90.4%

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

      6. associate-*l* 89.2%

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

      7. *-commutative 89.2%

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

      8. sqr-neg 89.2%

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

      9. +-commutative 89.2%

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

      10. sqr-neg 89.2%

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

      11. fma-define 89.2%

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

   3. Simplified 89.2%

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

      1. associate-*r* 91.3%

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

      2. *-commutative 91.3%

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

      3. associate-/r* 91.0%

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

      4. *-commutative 91.0%

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

      5. associate-/l/ 91.0%

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

      6. fma-undefine 91.0%

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

      7. +-commutative 91.0%

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

      8. associate-/r* 90.4%

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

      9. *-un-lft-identity 90.4%

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

      10. add-sqr-sqrt 42.3%

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

      11. times-frac 42.3%

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

      12. +-commutative 42.3%

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

      13. fma-undefine 42.3%

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

      14. *-commutative 42.3%

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

      15. sqrt-prod 42.3%

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

      16. fma-undefine 42.3%

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

      17. +-commutative 42.3%

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

      18. hypot-1-def 42.3%

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

      19. +-commutative 42.3%

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

   6. Applied egg-rr 47.8%

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

      1. associate-/l/ 47.8%

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

      2. associate-*r/ 47.8%

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

      3. *-rgt-identity 47.8%

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

      4. *-commutative 47.8%

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

      5. associate-/r* 47.8%

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

      6. *-commutative 47.8%

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

   8. Simplified 47.8%

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

      1. *-un-lft-identity 47.8%

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

      2. times-frac 44.8%

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

      3. un-div-inv 44.8%

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

      4. associate-/l/ 44.8%

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

      5. frac-times 42.8%

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

      6. frac-times 42.7%

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

      7. metadata-eval 42.7%

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

      8. add-sqr-sqrt 90.8%

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

      9. hypot-1-def 90.8%

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

      10. hypot-1-def 90.8%

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

      11. add-sqr-sqrt 90.8%

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

      12. times-frac 89.2%

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

      13. +-commutative 89.2%

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

      14. fma-undefine 89.2%

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

      15. add-sqr-sqrt 42.6%

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

   10. Applied egg-rr 46.6%

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

       1. associate-/r* 46.6%

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

       2. associate-/l/ 46.6%

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

       3. associate-*r/ 44.1%

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

       4. unpow-1 44.1%

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

       5. unpow-1 44.1%

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

       6. pow-sqr 44.1%

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

       7. metadata-eval 44.1%

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

   12. Simplified 44.1%

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

   13. Taylor expanded in x around 0 89.2%

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

       1. associate-/r* 89.3%

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

       2. +-commutative 89.3%

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

       3. unpow2 89.3%

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

       4. fma-define 89.3%

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

   15. Simplified 89.3%

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

   if +inf.0 < (+.f64 #s(literal 1 binary64) (*.f64 z z))

   1. Initial program 90.4%

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

      1. remove-double-neg 90.4%

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

      2. distribute-lft-neg-out 90.4%

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

      3. distribute-rgt-neg-in 90.4%

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

      4. associate-/r* 91.0%

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

      5. associate-/l/ 91.0%

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

      6. associate-/l/ 91.3%

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

      7. distribute-lft-neg-out 91.3%

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

      8. distribute-rgt-neg-in 91.3%

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

      9. distribute-lft-neg-in 91.3%

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

      10. remove-double-neg 91.3%

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

      11. sqr-neg 91.3%

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

      12. +-commutative 91.3%

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

      13. sqr-neg 91.3%

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

      14. fma-define 91.3%

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

      15. *-commutative 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in z around inf 59.3%

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

      1. *-un-lft-identity 59.3%

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

      2. pow2 59.3%

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

      3. associate-/r* 59.3%

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

      4. pow2 59.3%

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

      5. pow-flip 59.4%

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

      6. metadata-eval 59.4%

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

   7. Applied egg-rr 59.4%

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

      1. *-lft-identity 59.4%

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

   9. Simplified 59.4%

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

       1. sqr-pow 59.2%

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

       2. *-commutative 59.2%

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

       3. times-frac 63.0%

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

       4. metadata-eval 63.0%

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

       5. unpow-1 63.0%

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

       6. metadata-eval 63.0%

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

       7. unpow-1 63.0%

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

   11. Applied egg-rr 63.0%

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

Alternative 6: 90.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], Infinity], N[(1.0 / N[(y$95$m * N[(x * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < +inf.0

   1. Initial program 90.4%

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

      1. associate-/l/ 90.4%

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

      2. remove-double-neg 90.4%

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

      3. distribute-rgt-neg-out 90.4%

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

      4. distribute-rgt-neg-out 90.4%

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

      5. remove-double-neg 90.4%

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

      6. associate-*l* 89.2%

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

      7. *-commutative 89.2%

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

      8. sqr-neg 89.2%

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

      9. +-commutative 89.2%

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

      10. sqr-neg 89.2%

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

      11. fma-define 89.2%

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

   3. Simplified 89.2%

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

   if +inf.0 < (*.f64 z z)

   1. Initial program 90.4%

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

      1. remove-double-neg 90.4%

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

      2. distribute-lft-neg-out 90.4%

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

      3. distribute-rgt-neg-in 90.4%

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

      4. associate-/r* 91.0%

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

      5. associate-/l/ 91.0%

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

      6. associate-/l/ 91.3%

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

      7. distribute-lft-neg-out 91.3%

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

      8. distribute-rgt-neg-in 91.3%

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

      9. distribute-lft-neg-in 91.3%

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

      10. remove-double-neg 91.3%

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

      11. sqr-neg 91.3%

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

      12. +-commutative 91.3%

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

      13. sqr-neg 91.3%

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

      14. fma-define 91.3%

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

      15. *-commutative 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in z around inf 59.3%

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

      1. *-un-lft-identity 59.3%

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

      2. pow2 59.3%

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

      3. associate-/r* 59.3%

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

      4. pow2 59.3%

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

      5. pow-flip 59.4%

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

      6. metadata-eval 59.4%

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

   7. Applied egg-rr 59.4%

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

      1. *-lft-identity 59.4%

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

   9. Simplified 59.4%

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

       1. sqr-pow 59.2%

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

       2. *-commutative 59.2%

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

       3. times-frac 63.0%

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

       4. metadata-eval 63.0%

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

       5. unpow-1 63.0%

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

       6. metadata-eval 63.0%

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

       7. unpow-1 63.0%

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

   11. Applied egg-rr 63.0%

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

Alternative 7: 77.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{y\_m \cdot \left(1 + z \cdot z\right)}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-273}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{z}}{y\_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) (* y_m (+ 1.0 (* z z))))))
   (* y_s (if (<= t_0 5e-273) (* (/ 1.0 z) (/ (/ 1.0 z) (* y_m x))) t_0))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (1.0 / x) / (y_m * (1.0 + (z * z)));
	double tmp;
	if (t_0 <= 5e-273) {
		tmp = (1.0 / z) * ((1.0 / z) / (y_m * x));
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / x) / (y_m * (1.0d0 + (z * z)))
    if (t_0 <= 5d-273) then
        tmp = (1.0d0 / z) * ((1.0d0 / z) / (y_m * x))
    else
        tmp = t_0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (1.0 / x) / (y_m * (1.0 + (z * z)));
	double tmp;
	if (t_0 <= 5e-273) {
		tmp = (1.0 / z) * ((1.0 / z) / (y_m * x));
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	t_0 = (1.0 / x) / (y_m * (1.0 + (z * z)))
	tmp = 0
	if t_0 <= 5e-273:
		tmp = (1.0 / z) * ((1.0 / z) / (y_m * x))
	else:
		tmp = t_0
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(1.0 / x) / Float64(y_m * Float64(1.0 + Float64(z * z))))
	tmp = 0.0
	if (t_0 <= 5e-273)
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(1.0 / z) / Float64(y_m * x)));
	else
		tmp = t_0;
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (1.0 / x) / (y_m * (1.0 + (z * z)));
	tmp = 0.0;
	if (t_0 <= 5e-273)
		tmp = (1.0 / z) * ((1.0 / z) / (y_m * x));
	else
		tmp = t_0;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 5e-273], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] / N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 1 binary64) x) (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))) < 4.99999999999999965e-273

   1. Initial program 86.3%

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

      1. remove-double-neg 86.3%

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

      2. distribute-lft-neg-out 86.3%

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

      3. distribute-rgt-neg-in 86.3%

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

      4. associate-/r* 88.7%

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

      5. associate-/l/ 88.7%

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

      6. associate-/l/ 89.1%

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

      7. distribute-lft-neg-out 89.1%

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

      8. distribute-rgt-neg-in 89.1%

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

      9. distribute-lft-neg-in 89.1%

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

      10. remove-double-neg 89.1%

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

      11. sqr-neg 89.1%

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

      12. +-commutative 89.1%

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

      13. sqr-neg 89.1%

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

      14. fma-define 89.1%

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

      15. *-commutative 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in z around inf 66.5%

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

      1. *-un-lft-identity 66.5%

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

      2. pow2 66.5%

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

      3. associate-/r* 66.5%

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

      4. pow2 66.5%

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

      5. pow-flip 66.6%

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

      6. metadata-eval 66.6%

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

   7. Applied egg-rr 66.6%

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

      1. *-lft-identity 66.6%

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

   9. Simplified 66.6%

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

       1. sqr-pow 66.5%

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

       2. associate-/l* 76.1%

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

       3. metadata-eval 76.1%

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

       4. unpow-1 76.1%

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

       5. metadata-eval 76.1%

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

       6. unpow-1 76.1%

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

   11. Applied egg-rr 76.1%

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

   if 4.99999999999999965e-273 < (/.f64 (/.f64 #s(literal 1 binary64) x) (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))))

   1. Initial program 99.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \leq 5 \cdot 10^{-273}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{z}}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.01:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x}\\ \mathbf{elif}\;z \cdot z \leq \infty:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y\_m} \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* z z) 0.01)
    (/ (/ 1.0 y_m) x)
    (if (<= (* z z) INFINITY)
      (/ 1.0 (* y_m (* x (* z z))))
      (* (/ (/ 1.0 z) y_m) (/ (/ 1.0 z) x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.01) {
		tmp = (1.0 / y_m) / x;
	} else if ((z * z) <= ((double) INFINITY)) {
		tmp = 1.0 / (y_m * (x * (z * z)));
	} else {
		tmp = ((1.0 / z) / y_m) * ((1.0 / z) / x);
	}
	return y_s * tmp;
}

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.01) {
		tmp = (1.0 / y_m) / x;
	} else if ((z * z) <= Double.POSITIVE_INFINITY) {
		tmp = 1.0 / (y_m * (x * (z * z)));
	} else {
		tmp = ((1.0 / z) / y_m) * ((1.0 / z) / x);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z * z) <= 0.01:
		tmp = (1.0 / y_m) / x
	elif (z * z) <= math.inf:
		tmp = 1.0 / (y_m * (x * (z * z)))
	else:
		tmp = ((1.0 / z) / y_m) * ((1.0 / z) / x)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.01)
		tmp = Float64(Float64(1.0 / y_m) / x);
	elseif (Float64(z * z) <= Inf)
		tmp = Float64(1.0 / Float64(y_m * Float64(x * Float64(z * z))));
	else
		tmp = Float64(Float64(Float64(1.0 / z) / y_m) * Float64(Float64(1.0 / z) / x));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 0.01)
		tmp = (1.0 / y_m) / x;
	elseif ((z * z) <= Inf)
		tmp = 1.0 / (y_m * (x * (z * z)));
	else
		tmp = ((1.0 / z) / y_m) * ((1.0 / z) / x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 0.01], N[(N[(1.0 / y$95$m), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], Infinity], N[(1.0 / N[(y$95$m * N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 0.0100000000000000002

   1. Initial program 99.7%

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

      1. associate-/l/ 99.7%

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

      2. remove-double-neg 99.7%

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

      3. distribute-rgt-neg-out 99.7%

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

      4. distribute-rgt-neg-out 99.7%

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

      5. remove-double-neg 99.7%

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

      6. associate-*l* 99.7%

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

      7. *-commutative 99.7%

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

      8. sqr-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. sqr-neg 99.7%

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

      11. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 99.0%

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

      1. add-sqr-sqrt 59.4%

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

      2. pow2 59.4%

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

      3. inv-pow 59.4%

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

      4. sqrt-pow1 59.4%

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

      5. *-commutative 59.4%

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

      6. metadata-eval 59.4%

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

   7. Applied egg-rr 59.4%

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

      1. pow-pow 99.0%

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

      2. metadata-eval 99.0%

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

      3. inv-pow 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-/r* 98.9%

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

   9. Applied egg-rr 98.9%

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

   if 0.0100000000000000002 < (*.f64 z z) < +inf.0

   1. Initial program 82.6%

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

      1. associate-/l/ 82.6%

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

      2. remove-double-neg 82.6%

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

      3. distribute-rgt-neg-out 82.6%

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

      4. distribute-rgt-neg-out 82.6%

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

      5. remove-double-neg 82.6%

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

      6. associate-*l* 80.6%

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

      7. *-commutative 80.6%

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

      8. sqr-neg 80.6%

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

      9. +-commutative 80.6%

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

      10. sqr-neg 80.6%

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

      11. fma-define 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in z around inf 80.6%

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

      1. pow2 80.6%

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

   7. Applied egg-rr 80.6%

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

   if +inf.0 < (*.f64 z z)

   1. Initial program 90.4%

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

      1. remove-double-neg 90.4%

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

      2. distribute-lft-neg-out 90.4%

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

      3. distribute-rgt-neg-in 90.4%

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

      4. associate-/r* 91.0%

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

      5. associate-/l/ 91.0%

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

      6. associate-/l/ 91.3%

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

      7. distribute-lft-neg-out 91.3%

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

      8. distribute-rgt-neg-in 91.3%

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

      9. distribute-lft-neg-in 91.3%

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

      10. remove-double-neg 91.3%

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

      11. sqr-neg 91.3%

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

      12. +-commutative 91.3%

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

      13. sqr-neg 91.3%

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

      14. fma-define 91.3%

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

      15. *-commutative 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in z around inf 59.3%

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

      1. *-un-lft-identity 59.3%

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

      2. pow2 59.3%

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

      3. associate-/r* 59.3%

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

      4. pow2 59.3%

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

      5. pow-flip 59.4%

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

      6. metadata-eval 59.4%

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

   7. Applied egg-rr 59.4%

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

      1. *-lft-identity 59.4%

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

   9. Simplified 59.4%

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

       1. sqr-pow 59.2%

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

       2. *-commutative 59.2%

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

       3. times-frac 63.0%

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

       4. metadata-eval 63.0%

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

       5. unpow-1 63.0%

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

       6. metadata-eval 63.0%

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

       7. unpow-1 63.0%

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

   11. Applied egg-rr 63.0%

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

Alternative 9: 94.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.01:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+248}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{z}}{y\_m \cdot x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* z z) 0.01)
    (/ (/ 1.0 y_m) x)
    (if (<= (* z z) 5e+248)
      (/ 1.0 (* y_m (* x (* z z))))
      (* (/ 1.0 z) (/ (/ 1.0 z) (* y_m x)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.01) {
		tmp = (1.0 / y_m) / x;
	} else if ((z * z) <= 5e+248) {
		tmp = 1.0 / (y_m * (x * (z * z)));
	} else {
		tmp = (1.0 / z) * ((1.0 / z) / (y_m * x));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 0.01d0) then
        tmp = (1.0d0 / y_m) / x
    else if ((z * z) <= 5d+248) then
        tmp = 1.0d0 / (y_m * (x * (z * z)))
    else
        tmp = (1.0d0 / z) * ((1.0d0 / z) / (y_m * x))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.01) {
		tmp = (1.0 / y_m) / x;
	} else if ((z * z) <= 5e+248) {
		tmp = 1.0 / (y_m * (x * (z * z)));
	} else {
		tmp = (1.0 / z) * ((1.0 / z) / (y_m * x));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z * z) <= 0.01:
		tmp = (1.0 / y_m) / x
	elif (z * z) <= 5e+248:
		tmp = 1.0 / (y_m * (x * (z * z)))
	else:
		tmp = (1.0 / z) * ((1.0 / z) / (y_m * x))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.01)
		tmp = Float64(Float64(1.0 / y_m) / x);
	elseif (Float64(z * z) <= 5e+248)
		tmp = Float64(1.0 / Float64(y_m * Float64(x * Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(1.0 / z) / Float64(y_m * x)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 0.01)
		tmp = (1.0 / y_m) / x;
	elseif ((z * z) <= 5e+248)
		tmp = 1.0 / (y_m * (x * (z * z)));
	else
		tmp = (1.0 / z) * ((1.0 / z) / (y_m * x));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 0.01], N[(N[(1.0 / y$95$m), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+248], N[(1.0 / N[(y$95$m * N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] / N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 0.0100000000000000002

   1. Initial program 99.7%

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

      1. associate-/l/ 99.7%

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

      2. remove-double-neg 99.7%

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

      3. distribute-rgt-neg-out 99.7%

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

      4. distribute-rgt-neg-out 99.7%

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

      5. remove-double-neg 99.7%

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

      6. associate-*l* 99.7%

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

      7. *-commutative 99.7%

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

      8. sqr-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. sqr-neg 99.7%

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

      11. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 99.0%

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

      1. add-sqr-sqrt 59.4%

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

      2. pow2 59.4%

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

      3. inv-pow 59.4%

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

      4. sqrt-pow1 59.4%

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

      5. *-commutative 59.4%

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

      6. metadata-eval 59.4%

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

   7. Applied egg-rr 59.4%

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

      1. pow-pow 99.0%

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

      2. metadata-eval 99.0%

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

      3. inv-pow 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-/r* 98.9%

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

   9. Applied egg-rr 98.9%

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

   if 0.0100000000000000002 < (*.f64 z z) < 4.9999999999999996e248

   1. Initial program 88.0%

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

      1. associate-/l/ 88.0%

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

      2. remove-double-neg 88.0%

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

      3. distribute-rgt-neg-out 88.0%

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

      4. distribute-rgt-neg-out 88.0%

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

      5. remove-double-neg 88.0%

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

      6. associate-*l* 86.2%

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

      7. *-commutative 86.2%

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

      8. sqr-neg 86.2%

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

      9. +-commutative 86.2%

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

      10. sqr-neg 86.2%

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

      11. fma-define 86.2%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in z around inf 86.2%

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

      1. pow2 86.2%

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

   7. Applied egg-rr 86.2%

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

   if 4.9999999999999996e248 < (*.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. remove-double-neg 79.0%

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

      2. distribute-lft-neg-out 79.0%

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

      3. distribute-rgt-neg-in 79.0%

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

      4. associate-/r* 78.6%

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

      5. associate-/l/ 78.6%

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

      6. associate-/l/ 78.7%

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

      7. distribute-lft-neg-out 78.7%

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

      8. distribute-rgt-neg-in 78.7%

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

      9. distribute-lft-neg-in 78.7%

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

      10. remove-double-neg 78.7%

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

      11. sqr-neg 78.7%

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

      12. +-commutative 78.7%

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

      13. sqr-neg 78.7%

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

      14. fma-define 78.7%

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

      15. *-commutative 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in z around inf 78.7%

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

      1. *-un-lft-identity 78.7%

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

      2. pow2 78.7%

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

      3. associate-/r* 78.7%

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

      4. pow2 78.7%

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

      5. pow-flip 78.8%

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

      6. metadata-eval 78.8%

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

   7. Applied egg-rr 78.8%

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

      1. *-lft-identity 78.8%

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

   9. Simplified 78.8%

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

       1. sqr-pow 78.8%

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

       2. associate-/l* 93.3%

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

       3. metadata-eval 93.3%

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

       4. unpow-1 93.3%

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

       5. metadata-eval 93.3%

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

       6. unpow-1 93.3%

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

   11. Applied egg-rr 93.3%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.01:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+248}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{z}}{y \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1.0], N[(N[(1.0 / y$95$m), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(y$95$m * N[(x * N[(z * z), $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. Step-by-step derivation

      1. associate-/l/ 99.7%

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

      2. remove-double-neg 99.7%

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

      3. distribute-rgt-neg-out 99.7%

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

      4. distribute-rgt-neg-out 99.7%

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

      5. remove-double-neg 99.7%

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

      6. associate-*l* 99.7%

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

      7. *-commutative 99.7%

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

      8. sqr-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. sqr-neg 99.7%

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

      11. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 99.0%

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

      1. add-sqr-sqrt 59.4%

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

      2. pow2 59.4%

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

      3. inv-pow 59.4%

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

      4. sqrt-pow1 59.4%

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

      5. *-commutative 59.4%

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

      6. metadata-eval 59.4%

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

   7. Applied egg-rr 59.4%

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

      1. pow-pow 99.0%

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

      2. metadata-eval 99.0%

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

      3. inv-pow 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-/r* 98.9%

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

   9. Applied egg-rr 98.9%

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

   if 1 < (*.f64 z z)

   1. Initial program 82.6%

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

      1. associate-/l/ 82.6%

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

      2. remove-double-neg 82.6%

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

      3. distribute-rgt-neg-out 82.6%

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

      4. distribute-rgt-neg-out 82.6%

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

      5. remove-double-neg 82.6%

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

      6. associate-*l* 80.6%

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

      7. *-commutative 80.6%

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

      8. sqr-neg 80.6%

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

      9. +-commutative 80.6%

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

      10. sqr-neg 80.6%

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

      11. fma-define 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in z around inf 80.6%

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

      1. pow2 80.6%

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

   7. Applied egg-rr 80.6%

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

Alternative 11: 58.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(1.0 / N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.4%

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

   1. associate-/l/ 90.4%

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

   2. remove-double-neg 90.4%

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

   3. distribute-rgt-neg-out 90.4%

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

   4. distribute-rgt-neg-out 90.4%

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

   5. remove-double-neg 90.4%

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

   6. associate-*l* 89.2%

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

   7. *-commutative 89.2%

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

   8. sqr-neg 89.2%

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

   9. +-commutative 89.2%

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

   10. sqr-neg 89.2%

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

   11. fma-define 89.2%

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

3. Simplified 89.2%

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

5. Taylor expanded in z around 0 52.9%

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

Developer Target 1: 92.3% 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 2024119 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* y (+ 1 (* z z))) -inf.0) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 868074325056725200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x)))))

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