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

Average Error: 6.1 → 0.7

Time: 10.8s

Precision: binary64

Cost: 1736

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

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right)}\\ \mathbf{elif}\;t_0 \leq 10^{+308}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + y \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (if (<= t_0 (- INFINITY))
     (/ (/ 1.0 x) (* z (* y z)))
     (if (<= t_0 1e+308)
       (/ (/ 1.0 x) (+ y (* y (* z z))))
       (/ 1.0 (* z (* y (* z x))))))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 / x) / (z * (y * z));
	} else if (t_0 <= 1e+308) {
		tmp = (1.0 / x) / (y + (y * (z * z)));
	} else {
		tmp = 1.0 / (z * (y * (z * x)));
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (1.0 / x) / (z * (y * z));
	} else if (t_0 <= 1e+308) {
		tmp = (1.0 / x) / (y + (y * (z * z)));
	} else {
		tmp = 1.0 / (z * (y * (z * x)));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (1.0 / x) / (z * (y * z))
	elif t_0 <= 1e+308:
		tmp = (1.0 / x) / (y + (y * (z * z)))
	else:
		tmp = 1.0 / (z * (y * (z * x)))
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 / x) / Float64(z * Float64(y * z)));
	elseif (t_0 <= 1e+308)
		tmp = Float64(Float64(1.0 / x) / Float64(y + Float64(y * Float64(z * z))));
	else
		tmp = Float64(1.0 / Float64(z * Float64(y * Float64(z * x))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (1.0 / x) / (z * (y * z));
	elseif (t_0 <= 1e+308)
		tmp = (1.0 / x) / (y + (y * (z * z)));
	else
		tmp = 1.0 / (z * (y * (z * x)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 / x), $MachinePrecision] / N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+308], N[(N[(1.0 / x), $MachinePrecision] / N[(y + N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z * N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.1
Target	4.9
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 16.4

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

   2. Taylor expanded in z around inf 16.4

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

   3. Simplified 2.3

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{z \cdot \left(z \cdot y\right)}} \]
      Proof
      (*.f64 z (*.f64 z y)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z z) y)): 39 points increase in error, 26 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)) y): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 y (pow.f64 z 2))): 0 points increase in error, 0 points decrease in error

   if -inf.0 < (*.f64 y (+.f64 1 (*.f64 z z))) < 1e308

   1. Initial program 0.3

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

   2. Applied egg-rr 0.3

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

   if 1e308 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 18.3

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

   2. Simplified 18.3

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}} \]
      Proof
      (/.f64 1 (*.f64 x (*.f64 y (fma.f64 z z 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (*.f64 x (*.f64 y (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z z) 1))))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (*.f64 x (*.f64 y (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 z z)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 1 x) (*.f64 y (+.f64 1 (*.f64 z z))))): 25 points increase in error, 33 points decrease in error

   3. Taylor expanded in z around inf 13.5

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

   4. Simplified 1.4

      \[\leadsto \frac{1}{\color{blue}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}} \]
      Proof
      (*.f64 z (*.f64 y (*.f64 z x))): 0 points increase in error, 0 points decrease in error
      (*.f64 z (*.f64 y (Rewrite<= *-commutative_binary64 (*.f64 x z)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 y (*.f64 x z)) z)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 y (*.f64 (*.f64 x z) z))): 46 points increase in error, 23 points decrease in error
      (*.f64 y (Rewrite<= associate-*r*_binary64 (*.f64 x (*.f64 z z)))): 26 points increase in error, 18 points decrease in error
      (*.f64 y (*.f64 x (Rewrite<= unpow2_binary64 (pow.f64 z 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 y (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 z 2) x))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq -\infty:\\ \;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right)}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+308}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + y \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4440334674286477 \cdot 10^{+86}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \mathbf{elif}\;z \leq 1.1399150575872566 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z}}{y}}{z \cdot x}\\ \end{array} \]

Alternative 2
Error	1.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4440334674286477 \cdot 10^{+86}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \mathbf{elif}\;z \leq 15878.813093286084:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{x}}{y \cdot z}\\ \end{array} \]

Alternative 3
Error	4.2
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{\frac{\frac{1}{x}}{z}}{z}}{y}\\ \mathbf{if}\;z \leq -11509702650.103334:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.0020175744803694294:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	2.4
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{\frac{\frac{1}{x}}{z}}{y}}{z}\\ \mathbf{if}\;z \leq -11509702650.103334:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.0003839516014306086:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	2.0
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z}}{y}}{z \cdot x}\\ \end{array} \]

Alternative 6
Error	28.4
Cost	320

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

Reproduce

herbie shell --seed 2022316 
(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)))))