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

Percentage Accurate: 90.4% → 99.6%

Time: 8.1s

Alternatives: 10

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: 90.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \sqrt{y} \cdot \mathsf{hypot}\left(1, z\right)\\ t_1 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{\frac{-1}{x}}}{\sqrt{-y}}}{z}\right)}^{2}\\ \mathbf{elif}\;t_1 \leq 10^{+292}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + y \cdot {z}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0} \cdot \frac{\frac{1}{x}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sqrt y) (hypot 1.0 z))) (t_1 (* y (+ 1.0 (* z z)))))
   (if (<= t_1 (- INFINITY))
     (pow (/ (/ (sqrt (/ -1.0 x)) (sqrt (- y))) z) 2.0)
     (if (<= t_1 1e+292)
       (/ (/ 1.0 x) (+ y (* y (pow z 2.0))))
       (* (/ 1.0 t_0) (/ (/ 1.0 x) t_0))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = sqrt(y) * hypot(1.0, z);
	double t_1 = y * (1.0 + (z * z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = pow(((sqrt((-1.0 / x)) / sqrt(-y)) / z), 2.0);
	} else if (t_1 <= 1e+292) {
		tmp = (1.0 / x) / (y + (y * pow(z, 2.0)));
	} else {
		tmp = (1.0 / t_0) * ((1.0 / x) / t_0);
	}
	return tmp;
}

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = Math.sqrt(y) * Math.hypot(1.0, z);
	double t_1 = y * (1.0 + (z * z));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.pow(((Math.sqrt((-1.0 / x)) / Math.sqrt(-y)) / z), 2.0);
	} else if (t_1 <= 1e+292) {
		tmp = (1.0 / x) / (y + (y * Math.pow(z, 2.0)));
	} else {
		tmp = (1.0 / t_0) * ((1.0 / x) / t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = math.sqrt(y) * math.hypot(1.0, z)
	t_1 = y * (1.0 + (z * z))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.pow(((math.sqrt((-1.0 / x)) / math.sqrt(-y)) / z), 2.0)
	elif t_1 <= 1e+292:
		tmp = (1.0 / x) / (y + (y * math.pow(z, 2.0)))
	else:
		tmp = (1.0 / t_0) * ((1.0 / x) / t_0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(sqrt(y) * hypot(1.0, z))
	t_1 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(-1.0 / x)) / sqrt(Float64(-y))) / z) ^ 2.0;
	elseif (t_1 <= 1e+292)
		tmp = Float64(Float64(1.0 / x) / Float64(y + Float64(y * (z ^ 2.0))));
	else
		tmp = Float64(Float64(1.0 / t_0) * Float64(Float64(1.0 / x) / t_0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = sqrt(y) * hypot(1.0, z);
	t_1 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = ((sqrt((-1.0 / x)) / sqrt(-y)) / z) ^ 2.0;
	elseif (t_1 <= 1e+292)
		tmp = (1.0 / x) / (y + (y * (z ^ 2.0)));
	else
		tmp = (1.0 / t_0) * ((1.0 / x) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sqrt[y], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[Power[N[(N[(N[Sqrt[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-y)], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$1, 1e+292], N[(N[(1.0 / x), $MachinePrecision] / N[(y + N[(y * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < -inf.0

   1. Initial program 69.0%

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

      1. associate-/r* 69.0%

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

      2. associate-/l/ 69.0%

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

      3. associate-/r* 69.0%

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

      4. associate-/l/ 69.0%

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

      5. sqr-neg 69.0%

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

      6. +-commutative 69.0%

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

      7. sqr-neg 69.0%

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

      8. fma-def 69.0%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in z around inf 69.0%

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

      1. add-sqr-sqrt 69.0%

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

      2. pow2 69.0%

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

      3. associate-/r* 68.2%

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

      4. sqrt-div 44.4%

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

      5. associate-/l/ 44.4%

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

      6. inv-pow 44.4%

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

      7. sqrt-pow1 44.5%

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

      8. metadata-eval 44.5%

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

      9. unpow2 44.5%

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

      10. sqrt-prod 23.0%

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

      11. add-sqr-sqrt 51.1%

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

   6. Applied egg-rr 51.1%

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

      1. metadata-eval 51.1%

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

      2. sqrt-pow1 49.0%

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

      3. inv-pow 49.0%

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

      4. associate-/r* 49.0%

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

      5. frac-2neg 49.0%

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

      6. sqrt-div 46.5%

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

      7. distribute-neg-frac 46.5%

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

      8. metadata-eval 46.5%

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

   8. Applied egg-rr 46.5%

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

   if -inf.0 < (*.f64 y (+.f64 1 (*.f64 z z))) < 1e292

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. pow2 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Applied egg-rr 99.7%

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

   if 1e292 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 69.9%

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

      1. *-un-lft-identity 69.9%

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

      2. add-sqr-sqrt 69.9%

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

      3. times-frac 69.9%

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

      4. sqrt-prod 69.9%

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

      5. hypot-1-def 69.9%

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

      6. sqrt-prod 79.8%

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

      7. hypot-1-def 99.7%

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

   3. Applied egg-rr 99.7%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq -\infty:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{\frac{-1}{x}}}{\sqrt{-y}}}{z}\right)}^{2}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+292}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + y \cdot {z}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y} \cdot \mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{x}}{\sqrt{y} \cdot \mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Alternative 2: 78.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + y \cdot {z}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\frac{1}{x}}{y}}{z}}{\sqrt{z}}}{\sqrt{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+145)
   (/ (/ 1.0 x) (+ y (* y (pow z 2.0))))
   (/ (/ (/ (/ (/ 1.0 x) y) z) (sqrt z)) (sqrt z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+145) {
		tmp = (1.0 / x) / (y + (y * pow(z, 2.0)));
	} else {
		tmp = ((((1.0 / x) / y) / z) / sqrt(z)) / sqrt(z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+145) then
        tmp = (1.0d0 / x) / (y + (y * (z ** 2.0d0)))
    else
        tmp = ((((1.0d0 / x) / y) / z) / sqrt(z)) / sqrt(z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+145) {
		tmp = (1.0 / x) / (y + (y * Math.pow(z, 2.0)));
	} else {
		tmp = ((((1.0 / x) / y) / z) / Math.sqrt(z)) / Math.sqrt(z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+145:
		tmp = (1.0 / x) / (y + (y * math.pow(z, 2.0)))
	else:
		tmp = ((((1.0 / x) / y) / z) / math.sqrt(z)) / math.sqrt(z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+145)
		tmp = Float64(Float64(1.0 / x) / Float64(y + Float64(y * (z ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / x) / y) / z) / sqrt(z)) / sqrt(z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+145)
		tmp = (1.0 / x) / (y + (y * (z ^ 2.0)));
	else
		tmp = ((((1.0 / x) / y) / z) / sqrt(z)) / sqrt(z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+145], N[(N[(1.0 / x), $MachinePrecision] / N[(y + N[(y * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e145

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

      2. distribute-lft-in 99.1%

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

      3. pow2 99.1%

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

      4. *-rgt-identity 99.1%

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

   3. Applied egg-rr 99.1%

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

   if 2e145 < (*.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-/r* 72.6%

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

      2. associate-/l/ 72.6%

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

      3. associate-/r* 72.6%

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

      4. associate-/l/ 72.6%

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

      5. sqr-neg 72.6%

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

      6. +-commutative 72.6%

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

      7. sqr-neg 72.6%

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

      8. fma-def 72.6%

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

   3. Simplified 72.6%

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

   4. Taylor expanded in z around inf 72.6%

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

      1. add-sqr-sqrt 67.4%

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

      2. pow2 67.4%

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

      3. associate-/r* 66.8%

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

      4. sqrt-div 45.6%

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

      5. associate-/l/ 45.6%

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

      6. inv-pow 45.6%

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

      7. sqrt-pow1 45.7%

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

      8. metadata-eval 45.7%

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

      9. unpow2 45.7%

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

      10. sqrt-prod 26.6%

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

      11. add-sqr-sqrt 52.4%

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

   6. Applied egg-rr 52.4%

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

      1. unpow2 52.4%

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

      2. add-sqr-sqrt 34.9%

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

      3. associate-*l* 34.9%

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

      4. sqrt-div 26.6%

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

      5. sqrt-pow1 26.6%

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

      6. metadata-eval 26.6%

         \[\leadsto \frac{{\left(x \cdot y\right)}^{\color{blue}{-0.25}}}{\sqrt{z}} \cdot \left(\sqrt{\frac{{\left(x \cdot y\right)}^{-0.5}}{z}} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z}\right) \]

      7. sqrt-div 26.6%

         \[\leadsto \frac{{\left(x \cdot y\right)}^{-0.25}}{\sqrt{z}} \cdot \left(\color{blue}{\frac{\sqrt{{\left(x \cdot y\right)}^{-0.5}}}{\sqrt{z}}} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z}\right) \]

      8. sqrt-pow1 26.6%

         \[\leadsto \frac{{\left(x \cdot y\right)}^{-0.25}}{\sqrt{z}} \cdot \left(\frac{\color{blue}{{\left(x \cdot y\right)}^{\left(\frac{-0.5}{2}\right)}}}{\sqrt{z}} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z}\right) \]

      9. metadata-eval 26.6%

         \[\leadsto \frac{{\left(x \cdot y\right)}^{-0.25}}{\sqrt{z}} \cdot \left(\frac{{\left(x \cdot y\right)}^{\color{blue}{-0.25}}}{\sqrt{z}} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z}\right) \]

   8. Applied egg-rr 26.6%

      \[\leadsto \color{blue}{\frac{{\left(x \cdot y\right)}^{-0.25}}{\sqrt{z}} \cdot \left(\frac{{\left(x \cdot y\right)}^{-0.25}}{\sqrt{z}} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z}\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 26.6%

         \[\leadsto \color{blue}{\frac{{\left(x \cdot y\right)}^{-0.25} \cdot \left(\frac{{\left(x \cdot y\right)}^{-0.25}}{\sqrt{z}} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z}\right)}{\sqrt{z}}} \]

      2. associate-*l/ 26.6%

         \[\leadsto \frac{{\left(x \cdot y\right)}^{-0.25} \cdot \color{blue}{\frac{{\left(x \cdot y\right)}^{-0.25} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z}}{\sqrt{z}}}}{\sqrt{z}} \]

      3. associate-*r/ 26.5%

         \[\leadsto \frac{\color{blue}{\frac{{\left(x \cdot y\right)}^{-0.25} \cdot \left({\left(x \cdot y\right)}^{-0.25} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z}\right)}{\sqrt{z}}}}{\sqrt{z}} \]

      4. associate-*l* 26.5%

         \[\leadsto \frac{\frac{\color{blue}{\left({\left(x \cdot y\right)}^{-0.25} \cdot {\left(x \cdot y\right)}^{-0.25}\right) \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z}}}{\sqrt{z}}}{\sqrt{z}} \]

      5. pow-sqr 26.6%

         \[\leadsto \frac{\frac{\color{blue}{{\left(x \cdot y\right)}^{\left(2 \cdot -0.25\right)}} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z}}{\sqrt{z}}}{\sqrt{z}} \]

      6. metadata-eval 26.6%

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

      7. associate-*r/ 25.7%

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

      8. pow-sqr 43.2%

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

      9. metadata-eval 43.2%

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

      10. unpow-1 43.2%

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

      11. associate-/r* 43.2%

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

   10. Simplified 43.2%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + y \cdot {z}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\frac{1}{x}}{y}}{z}}{\sqrt{z}}}{\sqrt{z}}\\ \end{array} \]

Alternative 3: 94.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-297}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{elif}\;y \leq 10^{-41}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + {\left(z \cdot \sqrt{y}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 6e-297)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (if (<= y 1e-41)
     (/ (/ 1.0 x) (+ y (pow (* z (sqrt y)) 2.0)))
     (/ (/ 1.0 y) (* x (fma z z 1.0))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e-297) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else if (y <= 1e-41) {
		tmp = (1.0 / x) / (y + pow((z * sqrt(y)), 2.0));
	} else {
		tmp = (1.0 / y) / (x * fma(z, z, 1.0));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 6e-297)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))));
	elseif (y <= 1e-41)
		tmp = Float64(Float64(1.0 / x) / Float64(y + (Float64(z * sqrt(y)) ^ 2.0)));
	else
		tmp = Float64(Float64(1.0 / y) / Float64(x * fma(z, z, 1.0)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 6e-297], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e-41], N[(N[(1.0 / x), $MachinePrecision] / N[(y + N[Power[N[(z * N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] / N[(x * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.9999999999999999e-297

   1. Initial program 89.7%

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

   if 5.9999999999999999e-297 < y < 1.00000000000000001e-41

   1. Initial program 87.9%

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

      1. +-commutative 87.9%

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

      2. distribute-lft-in 87.9%

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

      3. pow2 87.9%

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

      4. *-rgt-identity 87.9%

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

   3. Applied egg-rr 87.9%

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

      1. add-sqr-sqrt 87.8%

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

      2. pow2 87.8%

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

      3. *-commutative 87.8%

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

      4. sqrt-prod 87.9%

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

      5. unpow2 87.9%

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

      6. sqrt-prod 45.0%

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

      7. add-sqr-sqrt 92.6%

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

   5. Applied egg-rr 92.6%

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

   if 1.00000000000000001e-41 < y

   1. Initial program 88.6%

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

      1. associate-/r* 86.6%

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

      2. associate-/l/ 88.6%

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

      3. associate-/r* 88.6%

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

      4. associate-/l/ 94.2%

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

      5. sqr-neg 94.2%

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

      6. +-commutative 94.2%

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

      7. sqr-neg 94.2%

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

      8. fma-def 94.2%

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

   3. Simplified 94.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-297}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{elif}\;y \leq 10^{-41}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + {\left(z \cdot \sqrt{y}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \end{array} \]

Alternative 4: 90.4% accurate, 0.1× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (/ (/ 1.0 y) (* x (fma z z 1.0))))

C

assert(x < y);
double code(double x, double y, double z) {
	return (1.0 / y) / (x * fma(z, z, 1.0));
}

Julia

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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(1.0 / y), $MachinePrecision] / N[(x * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.1%

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

   1. associate-/r* 88.1%

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

   2. associate-/l/ 89.1%

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

   3. associate-/r* 89.0%

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

   4. associate-/l/ 88.6%

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

   5. sqr-neg 88.6%

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

   6. +-commutative 88.6%

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

   7. sqr-neg 88.6%

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

   8. fma-def 88.6%

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

3. Simplified 88.6%

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

4. Final simplification 88.6%

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

Alternative 5: 92.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot x} \cdot \left(\frac{1}{z} \cdot \frac{1}{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (if (<= t_0 2e+305)
     (/ (/ 1.0 x) t_0)
     (* (/ 1.0 (* y x)) (* (/ 1.0 z) (/ 1.0 z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+305) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (1.0 / (y * x)) * ((1.0 / z) * (1.0 / z));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    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+305) then
        tmp = (1.0d0 / x) / t_0
    else
        tmp = (1.0d0 / (y * x)) * ((1.0d0 / z) * (1.0d0 / z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+305) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (1.0 / (y * x)) * ((1.0 / z) * (1.0 / z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 2e+305:
		tmp = (1.0 / x) / t_0
	else:
		tmp = (1.0 / (y * x)) * ((1.0 / z) * (1.0 / z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 2e+305)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(Float64(1.0 / Float64(y * x)) * Float64(Float64(1.0 / z) * Float64(1.0 / z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 2e+305)
		tmp = (1.0 / x) / t_0;
	else
		tmp = (1.0 / (y * x)) * ((1.0 / z) * (1.0 / z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+305], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / N[(y * x), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1.9999999999999999e305

   1. Initial program 93.5%

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

   if 1.9999999999999999e305 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 67.1%

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

      1. associate-/r* 67.1%

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

      2. associate-/l/ 67.1%

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

      3. associate-/r* 67.1%

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

      4. associate-/l/ 78.0%

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

      5. sqr-neg 78.0%

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

      6. +-commutative 78.0%

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

      7. sqr-neg 78.0%

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

      8. fma-def 78.0%

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

   3. Simplified 78.0%

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

   4. Taylor expanded in z around inf 78.0%

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

      1. add-sqr-sqrt 73.3%

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

      2. pow2 73.3%

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

      3. associate-/r* 73.2%

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

      4. sqrt-div 49.3%

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

      5. associate-/l/ 49.3%

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

      6. inv-pow 49.3%

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

      7. sqrt-pow1 49.3%

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

      8. metadata-eval 49.3%

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

      9. unpow2 49.3%

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

      10. sqrt-prod 32.4%

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

      11. add-sqr-sqrt 58.0%

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

   6. Applied egg-rr 58.0%

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

      1. unpow2 58.0%

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

      2. div-inv 58.0%

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

      3. div-inv 58.0%

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

      4. swap-sqr 49.4%

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

      5. pow-prod-up 80.0%

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

      6. metadata-eval 80.0%

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

      7. inv-pow 80.0%

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

   8. Applied egg-rr 80.0%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot x} \cdot \left(\frac{1}{z} \cdot \frac{1}{z}\right)\\ \end{array} \]

Alternative 6: 92.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{y \cdot x}}{z \cdot \left(-z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (if (<= t_0 2e+305) (/ (/ 1.0 x) t_0) (/ (/ -1.0 (* y x)) (* z (- z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+305) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (-1.0 / (y * x)) / (z * -z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    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+305) then
        tmp = (1.0d0 / x) / t_0
    else
        tmp = ((-1.0d0) / (y * x)) / (z * -z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+305) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (-1.0 / (y * x)) / (z * -z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 2e+305:
		tmp = (1.0 / x) / t_0
	else:
		tmp = (-1.0 / (y * x)) / (z * -z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 2e+305)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(Float64(-1.0 / Float64(y * x)) / Float64(z * Float64(-z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 2e+305)
		tmp = (1.0 / x) / t_0;
	else
		tmp = (-1.0 / (y * x)) / (z * -z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+305], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(-1.0 / N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(z * (-z)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1.9999999999999999e305

   1. Initial program 93.5%

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

   if 1.9999999999999999e305 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 67.1%

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

      1. associate-/r* 67.1%

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

      2. associate-/l/ 67.1%

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

      3. associate-/r* 67.1%

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

      4. associate-/l/ 78.0%

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

      5. sqr-neg 78.0%

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

      6. +-commutative 78.0%

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

      7. sqr-neg 78.0%

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

      8. fma-def 78.0%

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

   3. Simplified 78.0%

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

   4. Taylor expanded in z around inf 78.0%

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

      1. add-sqr-sqrt 73.3%

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

      2. pow2 73.3%

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

      3. associate-/r* 73.2%

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

      4. sqrt-div 49.3%

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

      5. associate-/l/ 49.3%

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

      6. inv-pow 49.3%

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

      7. sqrt-pow1 49.3%

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

      8. metadata-eval 49.3%

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

      9. unpow2 49.3%

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

      10. sqrt-prod 32.4%

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

      11. add-sqr-sqrt 58.0%

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

   6. Applied egg-rr 58.0%

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

      1. unpow2 58.0%

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

      2. frac-2neg 58.0%

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

      3. frac-times 49.4%

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

      4. distribute-lft-neg-in 49.4%

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

      5. pow-prod-up 77.9%

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

      6. metadata-eval 77.9%

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

      7. inv-pow 77.9%

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

      8. clear-num 77.9%

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

      9. distribute-neg-frac 77.9%

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

      10. metadata-eval 77.9%

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

      11. /-rgt-identity 77.9%

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

   8. Applied egg-rr 77.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{y \cdot x}}{z \cdot \left(-z\right)}\\ \end{array} \]

Alternative 7: 74.4% accurate, 0.9× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 1.0) (/ (/ 1.0 y) x) (/ (/ -1.0 (* y x)) (* z (- z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y) / x;
	} else {
		tmp = (-1.0 / (y * x)) / (z * -z);
	}
	return tmp;
}

Fortran

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

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y) / x;
	} else {
		tmp = (-1.0 / (y * x)) / (z * -z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / y) / x
	else:
		tmp = (-1.0 / (y * x)) / (z * -z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= 1.0)
		tmp = Float64(Float64(1.0 / y) / x);
	else
		tmp = Float64(Float64(-1.0 / Float64(y * x)) / Float64(z * Float64(-z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (1.0 / y) / x;
	else
		tmp = (-1.0 / (y * x)) / (z * -z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 1.0], N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision], N[(N[(-1.0 / N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(z * (-z)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 92.1%

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

   2. Taylor expanded in z around 0 66.4%

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

      1. associate-/l/ 67.2%

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

   4. Simplified 67.2%

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

   if 1 < z

   1. Initial program 80.9%

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

      1. associate-/r* 80.5%

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

      2. associate-/l/ 81.0%

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

      3. associate-/r* 80.9%

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

      4. associate-/l/ 79.6%

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

      5. sqr-neg 79.6%

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

      6. +-commutative 79.6%

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

      7. sqr-neg 79.6%

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

      8. fma-def 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in z around inf 78.5%

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

      1. add-sqr-sqrt 60.0%

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

      2. pow2 60.0%

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

      3. associate-/r* 61.0%

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

      4. sqrt-div 43.5%

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

      5. associate-/l/ 43.5%

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

      6. inv-pow 43.5%

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

      7. sqrt-pow1 43.5%

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

      8. metadata-eval 43.5%

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

      9. unpow2 43.5%

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

      10. sqrt-prod 51.4%

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

      11. add-sqr-sqrt 51.4%

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

   6. Applied egg-rr 51.4%

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

      1. unpow2 51.4%

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

      2. frac-2neg 51.4%

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

      3. frac-times 43.6%

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

      4. distribute-lft-neg-in 43.6%

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

      5. pow-prod-up 82.5%

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

      6. metadata-eval 82.5%

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

      7. inv-pow 82.5%

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

      8. clear-num 82.5%

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

      9. distribute-neg-frac 82.5%

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

      10. metadata-eval 82.5%

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

      11. /-rgt-identity 82.5%

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

   8. Applied egg-rr 82.5%

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

4. Final simplification 71.4%

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

Alternative 8: 58.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{y \cdot x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (/ 1.0 (* y x)))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 / (y * x)
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return 1.0 / (y * x)

Julia

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

MATLAB

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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(1.0 / N[(y * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.1%

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

2. Taylor expanded in z around 0 52.6%

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

3. Final simplification 52.6%

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

Alternative 9: 58.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{1}{x}}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) y))

C

assert(x < y);
double code(double x, double y, double z) {
	return (1.0 / x) / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return (1.0 / x) / y

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(1.0 / x) / y)
end

MATLAB

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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision]

Derivation

1. Initial program 89.1%

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

2. Taylor expanded in z around 0 53.2%

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

3. Final simplification 53.2%

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

Alternative 10: 58.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{1}{y}}{x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (/ (/ 1.0 y) x))

C

assert(x < y);
double code(double x, double y, double z) {
	return (1.0 / y) / x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 / y) / x
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return (1.0 / y) / x

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(1.0 / y) / x)
end

MATLAB

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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 89.1%

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

2. Taylor expanded in z around 0 52.6%

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

   1. associate-/l/ 53.2%

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

4. Simplified 53.2%

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

5. Final simplification 53.2%

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

Developer target: 92.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023318 
(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)))))