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

Percentage Accurate: 89.8% → 99.3%

Time: 12.8s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.8% 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: 99.3% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. associate-/l/ 91.0%

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

   2. metadata-eval 91.0%

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

   3. associate-*r/ 91.0%

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

   4. associate-/l/ 91.6%

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

   5. associate-*r/ 91.6%

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

   6. associate-/l* 90.9%

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

   7. associate-/r/ 91.0%

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

   8. /-rgt-identity 91.0%

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

   9. associate-*l* 92.5%

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

   10. *-commutative 92.5%

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

   11. sqr-neg 92.5%

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

   12. +-commutative 92.5%

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

   13. sqr-neg 92.5%

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

   14. fma-def 92.5%

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

3. Simplified 92.5%

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

   1. fma-udef 92.5%

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

   2. +-commutative 92.5%

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

   3. *-commutative 92.5%

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

   4. associate-*l* 91.0%

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

   5. associate-/l/ 91.6%

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

   6. add-sqr-sqrt 50.1%

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

   7. *-un-lft-identity 50.1%

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

   8. times-frac 50.1%

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

   9. *-commutative 50.1%

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

   10. sqrt-prod 50.1%

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

   11. hypot-1-def 50.1%

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

   12. *-commutative 50.1%

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

   13. sqrt-prod 51.5%

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

   14. hypot-1-def 55.5%

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

6. Applied egg-rr 55.5%

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

7. Final simplification 55.5%

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

Alternative 2: 49.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot {\left(\frac{{x\_m}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}\right)}^{2} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (pow (/ (pow x_m -0.5) (* (hypot 1.0 z) (sqrt y))) 2.0)))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	return x_s * pow((pow(x_m, -0.5) / (hypot(1.0, z) * sqrt(y))), 2.0);
}

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * Math.pow((Math.pow(x_m, -0.5) / (Math.hypot(1.0, z) * Math.sqrt(y))), 2.0);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	return x_s * math.pow((math.pow(x_m, -0.5) / (math.hypot(1.0, z) * math.sqrt(y))), 2.0)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	return Float64(x_s * (Float64((x_m ^ -0.5) / Float64(hypot(1.0, z) * sqrt(y))) ^ 2.0))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (((x_m ^ -0.5) / (hypot(1.0, z) * sqrt(y))) ^ 2.0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[Power[N[(N[Power[x$95$m, -0.5], $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

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

   1. associate-/l/ 91.0%

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

   2. metadata-eval 91.0%

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

   3. associate-*r/ 91.0%

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

   4. associate-/l/ 91.6%

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

   5. associate-*r/ 91.6%

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

   6. associate-/l* 90.9%

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

   7. associate-/r/ 91.0%

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

   8. /-rgt-identity 91.0%

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

   9. associate-*l* 92.5%

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

   10. *-commutative 92.5%

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

   11. sqr-neg 92.5%

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

   12. +-commutative 92.5%

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

   13. sqr-neg 92.5%

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

   14. fma-def 92.5%

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

3. Simplified 92.5%

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

   1. fma-udef 92.5%

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

   2. +-commutative 92.5%

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

   3. *-commutative 92.5%

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

   4. associate-*l* 91.0%

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

   5. associate-/l/ 91.6%

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

   6. add-sqr-sqrt 62.2%

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

   7. pow2 62.2%

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

   8. sqrt-div 24.0%

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

   9. inv-pow 24.0%

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

   10. sqrt-pow1 24.0%

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

   11. metadata-eval 24.0%

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

   12. *-commutative 24.0%

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

   13. sqrt-prod 25.0%

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

   14. hypot-1-def 27.1%

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

6. Applied egg-rr 27.1%

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

7. Final simplification 27.1%

   \[\leadsto {\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}\right)}^{2} \]
8. Add Preprocessing

Alternative 3: 97.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. associate-/l/ 91.0%

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

   2. metadata-eval 91.0%

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

   3. associate-*r/ 91.0%

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

   4. associate-/l/ 91.6%

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

   5. associate-*r/ 91.6%

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

   6. associate-/l* 90.9%

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

   7. associate-/r/ 91.0%

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

   8. /-rgt-identity 91.0%

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

   9. associate-*l* 92.5%

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

   10. *-commutative 92.5%

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

   11. sqr-neg 92.5%

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

   12. +-commutative 92.5%

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

   13. sqr-neg 92.5%

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

   14. fma-def 92.5%

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

3. Simplified 92.5%

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

   1. associate-/r* 93.0%

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

   2. div-inv 92.9%

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

6. Applied egg-rr 92.9%

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

   1. associate-/r* 93.1%

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

   2. *-un-lft-identity 93.1%

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

   3. add-sqr-sqrt 93.1%

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

   4. times-frac 93.0%

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

8. Applied egg-rr 93.0%

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

   1. associate-*r/ 93.1%

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

   2. fma-udef 93.1%

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

   3. unpow2 93.1%

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

   4. +-commutative 93.1%

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

   5. unpow2 93.1%

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

   6. hypot-1-def 93.1%

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

   7. associate-*r/ 93.0%

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

   8. fma-udef 93.0%

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

   9. unpow2 93.0%

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

   10. +-commutative 93.0%

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

   11. unpow2 93.0%

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

   12. hypot-1-def 96.8%

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

   13. associate-*l/ 96.8%

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

   14. *-lft-identity 96.8%

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

10. Simplified 96.8%

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

11. Final simplification 96.8%

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

Alternative 4: 97.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. associate-/l/ 91.0%

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

   2. metadata-eval 91.0%

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

   3. associate-*r/ 91.0%

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

   4. associate-/l/ 91.6%

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

   5. associate-*r/ 91.6%

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

   6. associate-/l* 90.9%

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

   7. associate-/r/ 91.0%

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

   8. /-rgt-identity 91.0%

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

   9. associate-*l* 92.5%

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

   10. *-commutative 92.5%

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

   11. sqr-neg 92.5%

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

   12. +-commutative 92.5%

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

   13. sqr-neg 92.5%

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

   14. fma-def 92.5%

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

3. Simplified 92.5%

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

   1. associate-/r* 93.0%

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

   2. div-inv 92.9%

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

6. Applied egg-rr 92.9%

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

   1. associate-/r* 93.1%

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

   2. *-un-lft-identity 93.1%

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

   3. add-sqr-sqrt 93.1%

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

   4. times-frac 93.0%

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

8. Applied egg-rr 93.0%

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

   1. associate-*r/ 93.1%

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

   2. fma-udef 93.1%

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

   3. unpow2 93.1%

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

   4. +-commutative 93.1%

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

   5. unpow2 93.1%

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

   6. hypot-1-def 93.1%

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

   7. associate-*r/ 93.0%

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

   8. fma-udef 93.0%

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

   9. unpow2 93.0%

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

   10. +-commutative 93.0%

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

   11. unpow2 93.0%

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

   12. hypot-1-def 96.8%

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

   13. associate-*l/ 96.8%

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

   14. *-lft-identity 96.8%

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

10. Simplified 96.8%

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

    1. frac-times 97.6%

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

    2. *-un-lft-identity 97.6%

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

    3. associate-/l/ 97.6%

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

    4. *-commutative 97.6%

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

    5. *-commutative 97.6%

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

12. Applied egg-rr 97.6%

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

13. Final simplification 97.6%

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

Alternative 5: 92.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{z}^{2} \cdot \left(y \cdot x\_m\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (*
    x_s
    (if (<= t_0 2e+302)
      (/ (/ 1.0 x_m) t_0)
      (/ 1.0 (* (pow z 2.0) (* y x_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+302) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = 1.0 / (pow(z, 2.0) * (y * x_m));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 + (z * z))
    if (t_0 <= 2d+302) then
        tmp = (1.0d0 / x_m) / t_0
    else
        tmp = 1.0d0 / ((z ** 2.0d0) * (y * x_m))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+302) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = 1.0 / (Math.pow(z, 2.0) * (y * x_m));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 2e+302:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = 1.0 / (math.pow(z, 2.0) * (y * x_m))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 2e+302)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(1.0 / Float64((z ^ 2.0) * Float64(y * x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 2e+302)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = 1.0 / ((z ^ 2.0) * (y * x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 2e+302], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(1.0 / N[(N[Power[z, 2.0], $MachinePrecision] * N[(y * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 2.0000000000000002e302

   1. Initial program 96.3%

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

   if 2.0000000000000002e302 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 72.6%

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

      1. associate-/l/ 72.6%

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

      2. metadata-eval 72.6%

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

      3. associate-*r/ 72.6%

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

      4. associate-/l/ 72.6%

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

      5. associate-*r/ 72.6%

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

      6. associate-/l* 72.6%

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

      7. associate-/r/ 72.6%

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

      8. /-rgt-identity 72.6%

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

      9. associate-*l* 79.9%

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

      10. *-commutative 79.9%

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

      11. sqr-neg 79.9%

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

      12. +-commutative 79.9%

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

      13. sqr-neg 79.9%

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

      14. fma-def 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in z around inf 72.6%

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

      1. associate-*r* 79.7%

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

      2. *-commutative 79.7%

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

   7. Simplified 79.7%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{z}^{2} \cdot \left(y \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{x\_m \cdot {z}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (*
    x_s
    (if (<= t_0 2e+302)
      (/ (/ 1.0 x_m) t_0)
      (/ (/ 1.0 y) (* x_m (pow z 2.0)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+302) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (1.0 / y) / (x_m * pow(z, 2.0));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 + (z * z))
    if (t_0 <= 2d+302) then
        tmp = (1.0d0 / x_m) / t_0
    else
        tmp = (1.0d0 / y) / (x_m * (z ** 2.0d0))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+302) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (1.0 / y) / (x_m * Math.pow(z, 2.0));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 2e+302:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = (1.0 / y) / (x_m * math.pow(z, 2.0))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 2e+302)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(Float64(1.0 / y) / Float64(x_m * (z ^ 2.0)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 2e+302)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = (1.0 / y) / (x_m * (z ^ 2.0));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 2e+302], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] / N[(x$95$m * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 2.0000000000000002e302

   1. Initial program 96.3%

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

   if 2.0000000000000002e302 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 72.6%

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

      1. associate-/l/ 72.6%

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

      2. metadata-eval 72.6%

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

      3. associate-*r/ 72.6%

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

      4. associate-/l/ 72.6%

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

      5. associate-*r/ 72.6%

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

      6. associate-/l* 72.6%

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

      7. associate-/r/ 72.6%

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

      8. /-rgt-identity 72.6%

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

      9. associate-*l* 79.9%

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

      10. *-commutative 79.9%

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

      11. sqr-neg 79.9%

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

      12. +-commutative 79.9%

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

      13. sqr-neg 79.9%

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

      14. fma-def 79.9%

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

   3. Simplified 79.9%

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

      1. fma-udef 79.9%

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

      2. +-commutative 79.9%

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

      3. *-commutative 79.9%

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

      4. associate-*l* 72.6%

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

      5. associate-/l/ 72.6%

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

      6. add-sqr-sqrt 72.6%

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

      7. *-un-lft-identity 72.6%

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

      8. times-frac 72.6%

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

      9. *-commutative 72.6%

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

      10. sqrt-prod 72.6%

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

      11. hypot-1-def 72.6%

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

      12. *-commutative 72.6%

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

      13. sqrt-prod 79.8%

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

      14. hypot-1-def 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 72.6%

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

      1. *-commutative 72.6%

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

      2. +-commutative 72.6%

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

      3. unpow2 72.6%

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

      4. fma-udef 72.6%

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

      5. associate-*l* 79.9%

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

      6. associate-/l/ 79.9%

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

   9. Simplified 79.9%

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

   10. Taylor expanded in z around inf 79.9%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot {z}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{{z}^{-2}}{y \cdot x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (*
    x_s
    (if (<= t_0 2e+302) (/ (/ 1.0 x_m) t_0) (/ (pow z -2.0) (* y x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+302) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = pow(z, -2.0) / (y * x_m);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 + (z * z))
    if (t_0 <= 2d+302) then
        tmp = (1.0d0 / x_m) / t_0
    else
        tmp = (z ** (-2.0d0)) / (y * x_m)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+302) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = Math.pow(z, -2.0) / (y * x_m);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 2e+302:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = math.pow(z, -2.0) / (y * x_m)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 2e+302)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64((z ^ -2.0) / Float64(y * x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 2e+302)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = (z ^ -2.0) / (y * x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 2e+302], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Power[z, -2.0], $MachinePrecision] / N[(y * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 2.0000000000000002e302

   1. Initial program 96.3%

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

   if 2.0000000000000002e302 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 72.6%

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

      1. associate-/l/ 72.6%

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

      2. metadata-eval 72.6%

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

      3. associate-*r/ 72.6%

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

      4. associate-/l/ 72.6%

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

      5. associate-*r/ 72.6%

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

      6. associate-/l* 72.6%

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

      7. associate-/r/ 72.6%

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

      8. /-rgt-identity 72.6%

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

      9. associate-*l* 79.9%

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

      10. *-commutative 79.9%

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

      11. sqr-neg 79.9%

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

      12. +-commutative 79.9%

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

      13. sqr-neg 79.9%

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

      14. fma-def 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in z around inf 72.6%

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

      1. associate-*r* 79.7%

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

      2. *-commutative 79.7%

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

   7. Simplified 79.7%

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

      1. expm1-log1p-u 79.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{z}^{2} \cdot \left(x \cdot y\right)}\right)\right)} \]

      2. expm1-udef 72.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{z}^{2} \cdot \left(x \cdot y\right)}\right)} - 1} \]

      3. associate-/r* 72.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{{z}^{2}}}{x \cdot y}}\right)} - 1 \]

      4. pow-flip 72.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{z}^{\left(-2\right)}}}{x \cdot y}\right)} - 1 \]

      5. metadata-eval 72.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{{z}^{\color{blue}{-2}}}{x \cdot y}\right)} - 1 \]

   9. Applied egg-rr 72.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{z}^{-2}}{x \cdot y}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 79.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{z}^{-2}}{x \cdot y}\right)\right)} \]

       2. expm1-log1p 79.7%

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

   11. Simplified 79.7%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{z}^{-2}}{y \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.6%

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

      1. associate-/l/ 91.0%

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

      2. metadata-eval 91.0%

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

      3. associate-*r/ 91.0%

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

      4. associate-/l/ 91.6%

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

      5. associate-*r/ 91.6%

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

      6. associate-/l* 90.9%

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

      7. associate-/r/ 91.0%

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

      8. /-rgt-identity 91.0%

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

      9. associate-*l* 92.5%

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

      10. *-commutative 92.5%

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

      11. sqr-neg 92.5%

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

      12. +-commutative 92.5%

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

      13. sqr-neg 92.5%

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

      14. fma-def 92.5%

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

   3. Simplified 92.5%

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

      1. fma-udef 92.5%

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

      2. +-commutative 92.5%

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

      3. *-commutative 92.5%

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

      4. associate-*l* 91.0%

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

      5. associate-/l/ 91.6%

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

      6. add-sqr-sqrt 50.1%

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

      7. *-un-lft-identity 50.1%

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

      8. times-frac 50.1%

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

      9. *-commutative 50.1%

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

      10. sqrt-prod 50.1%

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

      11. hypot-1-def 50.1%

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

      12. *-commutative 50.1%

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

      13. sqrt-prod 51.5%

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

      14. hypot-1-def 55.5%

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

   6. Applied egg-rr 55.5%

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

   7. Taylor expanded in y around 0 91.0%

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

      1. *-commutative 91.0%

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

      2. +-commutative 91.0%

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

      3. unpow2 91.0%

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

      4. fma-udef 91.0%

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

      5. associate-*l* 92.5%

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

      6. associate-/l/ 93.0%

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

   9. Simplified 93.0%

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

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

   1. Initial program 91.6%

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

      1. associate-/l/ 91.0%

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

      2. metadata-eval 91.0%

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

      3. associate-*r/ 91.0%

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

      4. associate-/l/ 91.6%

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

      5. associate-*r/ 91.6%

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

      6. associate-/l* 90.9%

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

      7. associate-/r/ 91.0%

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

      8. /-rgt-identity 91.0%

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

      9. associate-*l* 92.5%

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

      10. *-commutative 92.5%

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

      11. sqr-neg 92.5%

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

      12. +-commutative 92.5%

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

      13. sqr-neg 92.5%

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

      14. fma-def 92.5%

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

   3. Simplified 92.5%

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

      1. associate-/r* 93.0%

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

      2. div-inv 92.9%

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

   6. Applied egg-rr 92.9%

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

      1. associate-/r* 93.1%

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

      2. *-un-lft-identity 93.1%

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

      3. add-sqr-sqrt 93.1%

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

      4. times-frac 93.0%

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

   8. Applied egg-rr 93.0%

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

      1. associate-*r/ 93.1%

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

      2. fma-udef 93.1%

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

      3. unpow2 93.1%

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

      4. +-commutative 93.1%

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

      5. unpow2 93.1%

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

      6. hypot-1-def 93.1%

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

      7. associate-*r/ 93.0%

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

      8. fma-udef 93.0%

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

      9. unpow2 93.0%

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

      10. +-commutative 93.0%

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

      11. unpow2 93.0%

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

      12. hypot-1-def 96.8%

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

      13. associate-*l/ 96.8%

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

      14. *-lft-identity 96.8%

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

   10. Simplified 96.8%

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

       1. frac-times 97.6%

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

       2. *-un-lft-identity 97.6%

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

       3. associate-/l/ 97.6%

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

       4. *-commutative 97.6%

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

       5. *-commutative 97.6%

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

   12. Applied egg-rr 97.6%

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

   13. Taylor expanded in z around inf 41.4%

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

4. Final simplification 93.0%

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

Alternative 9: 95.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(z \cdot y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{x\_m \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 2.5e+110)
    (/ (/ 1.0 x_m) (fma (* z y) z y))
    (/ (/ 1.0 y) (* x_m (fma z z 1.0))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 2.5e+110) {
		tmp = (1.0 / x_m) / fma((z * y), z, y);
	} else {
		tmp = (1.0 / y) / (x_m * fma(z, z, 1.0));
	}
	return x_s * tmp;
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 2.5e+110)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(z * y), z, y));
	else
		tmp = Float64(Float64(1.0 / y) / Float64(x_m * fma(z, z, 1.0)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 2.5e+110], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(z * y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.49999999999999989e110

   1. Initial program 91.4%

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

      1. +-commutative 91.4%

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

      2. distribute-lft-in 91.5%

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

      3. associate-*r* 94.6%

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

      4. *-rgt-identity 94.6%

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

      5. fma-def 94.6%

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

   4. Applied egg-rr 94.6%

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

   if 2.49999999999999989e110 < y

   1. Initial program 91.9%

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

      1. associate-/l/ 91.9%

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

      2. metadata-eval 91.9%

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

      3. associate-*r/ 91.9%

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

      4. associate-/l/ 91.9%

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

      5. associate-*r/ 91.9%

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

      6. associate-/l* 91.9%

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

      7. associate-/r/ 91.9%

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

      8. /-rgt-identity 91.9%

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

      9. associate-*l* 98.2%

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

      10. *-commutative 98.2%

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

      11. sqr-neg 98.2%

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

      12. +-commutative 98.2%

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

      13. sqr-neg 98.2%

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

      14. fma-def 98.2%

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

   3. Simplified 98.2%

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

      1. fma-udef 98.2%

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

      2. +-commutative 98.2%

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

      3. *-commutative 98.2%

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

      4. associate-*l* 91.9%

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

      5. associate-/l/ 91.9%

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

      6. add-sqr-sqrt 91.7%

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

      7. *-un-lft-identity 91.7%

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

      8. times-frac 91.7%

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

      9. *-commutative 91.7%

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

      10. sqrt-prod 91.7%

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

      11. hypot-1-def 91.7%

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

      12. *-commutative 91.7%

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

      13. sqrt-prod 98.0%

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

      14. hypot-1-def 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around 0 91.9%

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

      1. *-commutative 91.9%

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

      2. +-commutative 91.9%

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

      3. unpow2 91.9%

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

      4. fma-udef 91.9%

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

      5. associate-*l* 98.2%

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

      6. associate-/l/ 98.3%

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

   9. Simplified 98.3%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(z \cdot y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. associate-/l/ 91.0%

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

   2. metadata-eval 91.0%

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

   3. associate-*r/ 91.0%

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

   4. associate-/l/ 91.6%

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

   5. associate-*r/ 91.6%

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

   6. associate-/l* 90.9%

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

   7. associate-/r/ 91.0%

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

   8. /-rgt-identity 91.0%

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

   9. associate-*l* 92.5%

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

   10. *-commutative 92.5%

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

   11. sqr-neg 92.5%

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

   12. +-commutative 92.5%

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

   13. sqr-neg 92.5%

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

   14. fma-def 92.5%

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

3. Simplified 92.5%

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

5. Final simplification 92.5%

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

Alternative 11: 89.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. associate-/l/ 91.0%

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

3. Simplified 91.0%

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

5. Final simplification 91.0%

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

Alternative 12: 89.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Final simplification 91.6%

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

Alternative 13: 58.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. associate-/l/ 91.0%

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

   2. metadata-eval 91.0%

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

   3. associate-*r/ 91.0%

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

   4. associate-/l/ 91.6%

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

   5. associate-*r/ 91.6%

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

   6. associate-/l* 90.9%

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

   7. associate-/r/ 91.0%

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

   8. /-rgt-identity 91.0%

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

   9. associate-*l* 92.5%

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

   10. *-commutative 92.5%

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

   11. sqr-neg 92.5%

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

   12. +-commutative 92.5%

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

   13. sqr-neg 92.5%

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

   14. fma-def 92.5%

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

3. Simplified 92.5%

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

5. Taylor expanded in z around 0 60.7%

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

6. Final simplification 60.7%

   \[\leadsto \frac{1}{y \cdot x} \]
7. Add Preprocessing

Alternative 14: 58.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in z around 0 61.0%

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

4. Final simplification 61.0%

   \[\leadsto \frac{\frac{1}{x}}{y} \]
5. Add Preprocessing

Developer target: 92.9% 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 2024031 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

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

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