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

Percentage Accurate: 89.7% → 99.4%

Time: 11.5s

Alternatives: 9

Speedup: 0.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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 (* x_m t_0)) 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 / (x_m * t_0)) / 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 / (x_m * t_0)) / 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 / (x_m * t_0)) / 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 / Float64(x_m * t_0)) / 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 / (x_m * t_0)) / 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 / N[(x$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 92.8%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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. associate-*l* 90.4%

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

   3. *-commutative 90.4%

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

   4. sqr-neg 90.4%

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

   5. +-commutative 90.4%

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

   6. sqr-neg 90.4%

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

   7. fma-define 90.4%

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

3. Simplified 90.4%

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

   1. associate-*r* 91.8%

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

   2. *-commutative 91.8%

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

   3. associate-/r* 92.2%

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

   4. *-commutative 92.2%

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

   5. associate-/l/ 92.7%

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

   6. fma-undefine 92.7%

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

   7. +-commutative 92.7%

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

   8. associate-/r* 92.8%

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

   9. *-un-lft-identity 92.8%

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

   10. add-sqr-sqrt 47.7%

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

   11. times-frac 47.6%

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

   12. +-commutative 47.6%

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

   13. fma-undefine 47.6%

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

   14. *-commutative 47.6%

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

   15. sqrt-prod 47.7%

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

   16. fma-undefine 47.7%

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

   17. +-commutative 47.7%

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

   18. hypot-1-def 47.7%

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

   19. +-commutative 47.7%

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

6. Applied egg-rr 51.7%

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

   1. associate-*l/ 51.7%

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

   2. *-lft-identity 51.7%

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

   3. associate-/l/ 51.6%

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

   4. *-commutative 51.6%

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

8. Simplified 51.6%

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

9. Final simplification 51.6%

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

Alternative 2: 48.7% accurate, 0.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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 (* (hypot 1.0 z) (sqrt x_m)) -2.0) 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 * (pow((hypot(1.0, z) * sqrt(x_m)), -2.0) / 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 * (Math.pow((Math.hypot(1.0, z) * Math.sqrt(x_m)), -2.0) / 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 * (math.pow((math.hypot(1.0, z) * math.sqrt(x_m)), -2.0) / 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(hypot(1.0, z) * sqrt(x_m)) ^ -2.0) / 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 * (((hypot(1.0, z) * sqrt(x_m)) ^ -2.0) / 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[Power[N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.8%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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. associate-*l* 90.4%

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

   3. *-commutative 90.4%

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

   4. sqr-neg 90.4%

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

   5. +-commutative 90.4%

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

   6. sqr-neg 90.4%

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

   7. fma-define 90.4%

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

3. Simplified 90.4%

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

   1. associate-*r* 91.8%

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

   2. *-commutative 91.8%

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

   3. associate-/r* 92.2%

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

   4. *-commutative 92.2%

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

   5. associate-/l/ 92.7%

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

   6. fma-undefine 92.7%

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

   7. +-commutative 92.7%

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

   8. associate-/r* 92.8%

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

   9. *-un-lft-identity 92.8%

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

   10. add-sqr-sqrt 47.7%

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

   11. times-frac 47.6%

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

   12. +-commutative 47.6%

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

   13. fma-undefine 47.6%

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

   14. *-commutative 47.6%

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

   15. sqrt-prod 47.7%

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

   16. fma-undefine 47.7%

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

   17. +-commutative 47.7%

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

   18. hypot-1-def 47.7%

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

   19. +-commutative 47.7%

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

6. Applied egg-rr 51.7%

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

   1. associate-*l/ 51.7%

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

   2. *-lft-identity 51.7%

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

   3. associate-/l/ 51.6%

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

   4. *-commutative 51.6%

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

8. Simplified 51.6%

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

9. Taylor expanded in x around 0 91.9%

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

    1. +-commutative 91.9%

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

    2. unpow2 91.9%

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

    3. fma-undefine 91.9%

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

    4. associate-/l/ 92.8%

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

    5. associate-/r* 92.8%

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

    6. associate-/l/ 91.1%

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

    7. *-commutative 91.1%

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

    8. rem-square-sqrt 91.1%

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

    9. fma-undefine 91.1%

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

    10. unpow2 91.1%

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

    11. +-commutative 91.1%

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

    12. metadata-eval 91.1%

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

    13. unpow2 91.1%

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

    14. hypot-undefine 91.1%

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

    15. fma-undefine 91.1%

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

    16. unpow2 91.1%

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

    17. +-commutative 91.1%

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

    18. metadata-eval 91.1%

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

    19. unpow2 91.1%

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

    20. hypot-undefine 91.1%

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

    21. rem-square-sqrt 51.4%

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

11. Simplified 53.3%

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

12. Final simplification 53.3%

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

Alternative 3: 96.0% accurate, 0.1× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
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 (* (hypot 1.0 z) (* x_m (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 * (hypot(1.0, z) * (x_m * 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 * (Math.hypot(1.0, z) * (x_m * 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 * (math.hypot(1.0, z) * (x_m * 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(1.0 / Float64(y * Float64(hypot(1.0, z) * Float64(x_m * 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 * (hypot(1.0, z) * (x_m * hypot(1.0, z)))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[(x$95$m * N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.8%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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. associate-*l* 90.4%

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

   3. *-commutative 90.4%

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

   4. sqr-neg 90.4%

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

   5. +-commutative 90.4%

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

   6. sqr-neg 90.4%

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

   7. fma-define 90.4%

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

3. Simplified 90.4%

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

   1. add-sqr-sqrt 51.2%

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

   2. pow2 51.2%

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

   3. *-commutative 51.2%

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

   4. sqrt-prod 51.2%

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

   5. fma-undefine 51.2%

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

   6. +-commutative 51.2%

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

   7. hypot-1-def 53.1%

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

6. Applied egg-rr 53.1%

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

   1. unpow2 53.1%

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

   2. associate-*l* 53.1%

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

   3. *-commutative 53.1%

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

   4. associate-*l* 53.1%

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

   5. add-sqr-sqrt 93.7%

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

8. Applied egg-rr 93.7%

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

9. Final simplification 93.7%

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

Alternative 4: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(z \cdot y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{y} \cdot \frac{-1}{x\_m \cdot z}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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 (+ 1.0 (* z z))) 5e+306)
    (/ (/ 1.0 x_m) (fma (* z y) z y))
    (/ (* (/ -1.0 y) (/ -1.0 (* x_m 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) {
	double tmp;
	if ((y * (1.0 + (z * z))) <= 5e+306) {
		tmp = (1.0 / x_m) / fma((z * y), z, y);
	} else {
		tmp = ((-1.0 / y) * (-1.0 / (x_m * z))) / z;
	}
	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(y * Float64(1.0 + Float64(z * z))) <= 5e+306)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(z * y), z, y));
	else
		tmp = Float64(Float64(Float64(-1.0 / y) * Float64(-1.0 / Float64(x_m * z))) / z);
	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[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+306], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(z * y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / y), $MachinePrecision] * N[(-1.0 / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 4.99999999999999993e306

   1. Initial program 96.2%

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

      1. +-commutative 96.2%

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

      2. distribute-lft-in 96.2%

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

      3. associate-*r* 97.6%

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

      4. *-rgt-identity 97.6%

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

      5. fma-define 97.6%

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

   4. Applied egg-rr 97.6%

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

   if 4.99999999999999993e306 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 74.1%

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

      1. associate-/l/ 74.1%

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

      2. associate-*l* 78.7%

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

      3. *-commutative 78.7%

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

      4. sqr-neg 78.7%

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

      5. +-commutative 78.7%

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

      6. sqr-neg 78.7%

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

      7. fma-define 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in z around inf 74.1%

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

      1. associate-*r* 78.5%

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

      2. associate-/r* 78.6%

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

   7. Simplified 78.6%

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

      1. associate-/r* 78.5%

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

      2. div-inv 78.5%

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

      3. unpow2 78.5%

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

      4. times-frac 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. frac-2neg 99.7%

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

       2. associate-*r/ 97.4%

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

       3. associate-/l/ 95.1%

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

       4. *-commutative 95.1%

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

       5. distribute-neg-frac 95.1%

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

       6. metadata-eval 95.1%

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

   11. Applied egg-rr 95.1%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(z \cdot y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{y} \cdot \frac{-1}{x \cdot z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+80}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z \cdot \left(x\_m \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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) 1e+80)
    (/ (/ 1.0 x_m) (* y (+ 1.0 (* z z))))
    (/ (/ 1.0 y) (* z (* x_m 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) {
	double tmp;
	if ((z * z) <= 1e+80) {
		tmp = (1.0 / x_m) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / y) / (z * (x_m * z));
	}
	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) :: tmp
    if ((z * z) <= 1d+80) then
        tmp = (1.0d0 / x_m) / (y * (1.0d0 + (z * z)))
    else
        tmp = (1.0d0 / y) / (z * (x_m * z))
    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 tmp;
	if ((z * z) <= 1e+80) {
		tmp = (1.0 / x_m) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / y) / (z * (x_m * z));
	}
	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):
	tmp = 0
	if (z * z) <= 1e+80:
		tmp = (1.0 / x_m) / (y * (1.0 + (z * z)))
	else:
		tmp = (1.0 / y) / (z * (x_m * z))
	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) <= 1e+80)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / y) / Float64(z * Float64(x_m * z)));
	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)
	tmp = 0.0;
	if ((z * z) <= 1e+80)
		tmp = (1.0 / x_m) / (y * (1.0 + (z * z)));
	else
		tmp = (1.0 / y) / (z * (x_m * z));
	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_] := N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1e+80], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] / N[(z * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e80

   1. Initial program 99.7%

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

   if 1e80 < (*.f64 z z)

   1. Initial program 82.0%

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

      1. associate-/l/ 81.1%

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

      2. associate-*l* 77.2%

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

      3. *-commutative 77.2%

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

      4. sqr-neg 77.2%

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

      5. +-commutative 77.2%

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

      6. sqr-neg 77.2%

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

      7. fma-define 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in z around inf 81.1%

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

      1. associate-*r* 80.9%

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

      2. associate-/r* 81.8%

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

   7. Simplified 81.8%

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

      1. associate-/r* 81.7%

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

      2. div-inv 81.7%

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

      3. unpow2 81.7%

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

      4. times-frac 96.8%

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

   9. Applied egg-rr 96.8%

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

       1. clear-num 95.5%

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

       2. frac-times 86.2%

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

       3. *-un-lft-identity 86.2%

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

       4. div-inv 86.2%

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

       5. clear-num 86.3%

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

       6. /-rgt-identity 86.3%

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

       7. *-commutative 86.3%

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

   11. Applied egg-rr 86.3%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+80}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z \cdot \left(x \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 0.8× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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 1.0) (/ (/ 1.0 x_m) y) (/ 1.0 (* (* z y) (* x_m 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) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x_m) / y;
	} else {
		tmp = 1.0 / ((z * y) * (x_m * z));
	}
	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) :: tmp
    if (z <= 1.0d0) then
        tmp = (1.0d0 / x_m) / y
    else
        tmp = 1.0d0 / ((z * y) * (x_m * z))
    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 tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x_m) / y;
	} else {
		tmp = 1.0 / ((z * y) * (x_m * z));
	}
	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):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / x_m) / y
	else:
		tmp = 1.0 / ((z * y) * (x_m * z))
	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 (z <= 1.0)
		tmp = Float64(Float64(1.0 / x_m) / y);
	else
		tmp = Float64(1.0 / Float64(Float64(z * y) * Float64(x_m * z)));
	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)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (1.0 / x_m) / y;
	else
		tmp = 1.0 / ((z * y) * (x_m * z));
	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_] := N[(x$95$s * If[LessEqual[z, 1.0], N[(N[(1.0 / x$95$m), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(N[(z * y), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 95.6%

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

   3. Taylor expanded in z around 0 74.9%

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

   if 1 < z

   1. Initial program 82.5%

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

      1. associate-/l/ 82.5%

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

      2. associate-*l* 79.0%

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

      3. *-commutative 79.0%

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

      4. sqr-neg 79.0%

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

      5. +-commutative 79.0%

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

      6. sqr-neg 79.0%

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

      7. fma-define 79.0%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in z around inf 82.2%

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

      1. associate-*r* 81.9%

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

      2. associate-/r* 82.0%

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

   7. Simplified 82.0%

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

      1. associate-/r* 81.8%

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

      2. div-inv 81.9%

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

      3. unpow2 81.9%

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

      4. times-frac 98.9%

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

   9. Applied egg-rr 98.9%

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

       1. *-commutative 98.9%

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

       2. associate-/r* 97.4%

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

       3. clear-num 95.7%

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

       4. frac-times 95.7%

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

       5. metadata-eval 95.7%

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

       6. div-inv 95.7%

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

       7. clear-num 95.8%

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

       8. /-rgt-identity 95.8%

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

       9. *-commutative 95.8%

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

   11. Applied egg-rr 95.8%

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

4. Final simplification 79.3%

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

Alternative 7: 77.4% accurate, 0.8× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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 1.0) (/ (/ 1.0 x_m) y) (/ (/ 1.0 y) (* z (* x_m 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) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x_m) / y;
	} else {
		tmp = (1.0 / y) / (z * (x_m * z));
	}
	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) :: tmp
    if (z <= 1.0d0) then
        tmp = (1.0d0 / x_m) / y
    else
        tmp = (1.0d0 / y) / (z * (x_m * z))
    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 tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x_m) / y;
	} else {
		tmp = (1.0 / y) / (z * (x_m * z));
	}
	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):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / x_m) / y
	else:
		tmp = (1.0 / y) / (z * (x_m * z))
	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 (z <= 1.0)
		tmp = Float64(Float64(1.0 / x_m) / y);
	else
		tmp = Float64(Float64(1.0 / y) / Float64(z * Float64(x_m * z)));
	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)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (1.0 / x_m) / y;
	else
		tmp = (1.0 / y) / (z * (x_m * z));
	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_] := N[(x$95$s * If[LessEqual[z, 1.0], N[(N[(1.0 / x$95$m), $MachinePrecision] / y), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] / N[(z * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 95.6%

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

   3. Taylor expanded in z around 0 74.9%

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

   if 1 < z

   1. Initial program 82.5%

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

      1. associate-/l/ 82.5%

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

      2. associate-*l* 79.0%

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

      3. *-commutative 79.0%

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

      4. sqr-neg 79.0%

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

      5. +-commutative 79.0%

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

      6. sqr-neg 79.0%

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

      7. fma-define 79.0%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in z around inf 82.2%

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

      1. associate-*r* 81.9%

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

      2. associate-/r* 82.0%

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

   7. Simplified 82.0%

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

      1. associate-/r* 81.8%

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

      2. div-inv 81.9%

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

      3. unpow2 81.9%

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

      4. times-frac 98.9%

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

   9. Applied egg-rr 98.9%

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

       1. clear-num 97.3%

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

       2. frac-times 87.3%

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

       3. *-un-lft-identity 87.3%

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

       4. div-inv 87.3%

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

       5. clear-num 87.3%

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

       6. /-rgt-identity 87.3%

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

       7. *-commutative 87.3%

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

   11. Applied egg-rr 87.3%

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

4. Final simplification 77.5%

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

Alternative 8: 58.6% accurate, 2.2× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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(1.0 / Float64(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[(1.0 / N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.8%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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. associate-*l* 90.4%

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

   3. *-commutative 90.4%

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

   4. sqr-neg 90.4%

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

   5. +-commutative 90.4%

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

   6. sqr-neg 90.4%

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

   7. fma-define 90.4%

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

3. Simplified 90.4%

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

5. Taylor expanded in z around 0 63.1%

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

6. Final simplification 63.1%

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

Alternative 9: 58.6% 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 #s(literal 1 binary64) 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 92.8%

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

3. Taylor expanded in z around 0 63.2%

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

4. Final simplification 63.2%

   \[\leadsto \frac{\frac{1}{x}}{y} \]
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 2024066 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

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

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