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

Average Error: 6.6 → 0.7

Time: 11.8s

Precision: binary64

Cost: 1736

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

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{\frac{1}{z \cdot \left(z \cdot y\right)}}{x}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z \cdot x}}{z}\\ \end{array} \]

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 (* z (* z y))) x)
     (if (<= t_0 2e+306) (/ (/ 1.0 x) t_0) (/ (/ (/ 1.0 y) (* z x)) z)))))

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 / (z * (z * y))) / x;
	} else if (t_0 <= 2e+306) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = ((1.0 / y) / (z * x)) / z;
	}
	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 / (z * (z * y))) / x;
	} else if (t_0 <= 2e+306) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = ((1.0 / y) / (z * x)) / z;
	}
	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 / (z * (z * y))) / x
	elif t_0 <= 2e+306:
		tmp = (1.0 / x) / t_0
	else:
		tmp = ((1.0 / y) / (z * x)) / z
	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 / Float64(z * Float64(z * y))) / x);
	elseif (t_0 <= 2e+306)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / y) / Float64(z * x)) / z);
	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 / (z * (z * y))) / x;
	elseif (t_0 <= 2e+306)
		tmp = (1.0 / x) / t_0;
	else
		tmp = ((1.0 / y) / (z * x)) / z;
	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 / N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 2e+306], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]]

Target

Original	6.6
Target	5.3
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 17.2

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

   2. Taylor expanded in z around inf 17.2

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

   3. Simplified 2.6

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

      [Start] 17.2	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      associate-*r* [=>] 17.2	\[ \frac{1}{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}} \]
      *-commutative [=>] 17.2	\[ \frac{1}{\color{blue}{\left({z}^{2} \cdot y\right)} \cdot x} \]
      unpow2 [=>] 17.2	\[ \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot x} \]
      associate-*r* [<=] 2.9	\[ \frac{1}{\color{blue}{\left(z \cdot \left(z \cdot y\right)\right)} \cdot x} \]
      associate-/r* [=>] 2.6	\[ \color{blue}{\frac{\frac{1}{z \cdot \left(z \cdot y\right)}}{x}} \]

   if -inf.0 < (*.f64 y (+.f64 1 (*.f64 z z))) < 2.00000000000000003e306

   1. Initial program 0.3

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

   if 2.00000000000000003e306 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 19.0

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

   2. Simplified 14.5

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

      [Start] 19.0	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [=>] 14.5	\[ \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}} \]

   3. Taylor expanded in z around inf 14.8

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

   4. Simplified 14.9

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

      [Start] 14.8	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      *-commutative [=>] 14.8	\[ \frac{1}{\color{blue}{\left({z}^{2} \cdot x\right) \cdot y}} \]
      associate-*l* [=>] 14.9	\[ \frac{1}{\color{blue}{{z}^{2} \cdot \left(x \cdot y\right)}} \]
      unpow2 [=>] 14.9	\[ \frac{1}{\color{blue}{\left(z \cdot z\right)} \cdot \left(x \cdot y\right)} \]
      *-commutative [<=] 14.9	\[ \frac{1}{\left(z \cdot z\right) \cdot \color{blue}{\left(y \cdot x\right)}} \]

   5. Applied egg-rr 6.7

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

   6. Taylor expanded in z around 0 14.8

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

   7. Simplified 1.3

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

      [Start] 14.8	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      associate-/r* [=>] 14.7	\[ \color{blue}{\frac{\frac{1}{y}}{{z}^{2} \cdot x}} \]
      unpow2 [=>] 14.7	\[ \frac{\frac{1}{y}}{\color{blue}{\left(z \cdot z\right)} \cdot x} \]
      associate-*l* [=>] 5.7	\[ \frac{\frac{1}{y}}{\color{blue}{z \cdot \left(z \cdot x\right)}} \]
      associate-/l/ [<=] 1.3	\[ \color{blue}{\frac{\frac{\frac{1}{y}}{z \cdot x}}{z}} \]
      associate-/r* [<=] 1.3	\[ \frac{\color{blue}{\frac{1}{y \cdot \left(z \cdot x\right)}}}{z} \]

   8. Applied egg-rr 14.1

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

   9. Simplified 1.3

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

      [Start] 14.1	\[ \frac{e^{\mathsf{log1p}\left(\frac{\frac{\frac{1}{y}}{z}}{x}\right)} - 1}{z} \]
      expm1-def [=>] 8.5	\[ \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\frac{1}{y}}{z}}{x}\right)\right)}}{z} \]
      expm1-log1p [=>] 3.2	\[ \frac{\color{blue}{\frac{\frac{\frac{1}{y}}{z}}{x}}}{z} \]
      associate-/r* [<=] 1.3	\[ \frac{\color{blue}{\frac{\frac{1}{y}}{z \cdot x}}}{z} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq -\infty:\\ \;\;\;\;\frac{\frac{1}{z \cdot \left(z \cdot y\right)}}{x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z \cdot x}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.9
Cost	13504

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

Alternative 2
Error	2.5
Cost	972

\[\begin{array}{l} t_0 := \frac{1}{\left(z \cdot y\right) \cdot \left(z \cdot x\right)}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+166}:\\ \;\;\;\;\frac{1}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	2.4
Cost	972

\[\begin{array}{l} t_0 := \frac{1}{\left(z \cdot y\right) \cdot \left(z \cdot x\right)}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{1}{z}}{z \cdot \left(y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.7
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+27} \lor \neg \left(z \leq 155000000\right):\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z \cdot x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x + x \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]

Alternative 5
Error	4.5
Cost	841

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

Alternative 6
Error	4.3
Cost	841

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

Alternative 7
Error	2.2
Cost	840

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

Alternative 8
Error	2.2
Cost	840

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

Alternative 9
Error	28.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq 4.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{y \cdot y}\\ \end{array} \]

Alternative 10
Error	29.4
Cost	320

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

Reproduce

herbie shell --seed 2023010 
(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)))))