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

Average Error: 6.4 → 0.8

Time: 10.3s

Precision: binary64

Cost: 7492

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* y (+ 1.0 (* z z))) 2e+306)
   (/ (/ 1.0 (fma z (* y z) y)) x)
   (/ (/ 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 tmp;
	if ((y * (1.0 + (z * z))) <= 2e+306) {
		tmp = (1.0 / fma(z, (y * z), y)) / x;
	} 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)
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 2e+306)
		tmp = Float64(Float64(1.0 / fma(z, Float64(y * z), y)) / x);
	else
		tmp = Float64(Float64(1.0 / Float64(y * Float64(z * x))) / z);
	end
	return 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_] := If[LessEqual[N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+306], N[(N[(1.0 / N[(z * N[(y * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Target

Original	6.4
Target	5.0
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 1.8

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

   2. Taylor expanded in x around 0 3.9

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

   3. Simplified 3.4

      \[\leadsto \color{blue}{\frac{1}{y \cdot \mathsf{fma}\left(z, z \cdot x, x\right)}} \]
      Proof
      (/.f64 1 (*.f64 y (fma.f64 z (*.f64 z x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (*.f64 y (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 z (*.f64 z x)) x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (*.f64 y (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z z) x)) x))): 11 points increase in error, 9 points decrease in error
      (/.f64 1 (*.f64 y (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)) x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (*.f64 y (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 (pow.f64 z 2) 1) x)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in y around 0 3.9

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

   5. Simplified 0.5

      \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(z, z \cdot y, y\right)}}{x}} \]
      Proof
      (/.f64 (/.f64 1 (fma.f64 z (*.f64 z y) y)) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 1 (fma.f64 z (Rewrite<= *-commutative_binary64 (*.f64 y z)) y)) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 1 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z (*.f64 y z)) y))) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 1 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 y z) z)) y)) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 1 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 y (*.f64 z z))) y)) x): 10 points increase in error, 1 points decrease in error
      (/.f64 (/.f64 1 (+.f64 (*.f64 y (Rewrite<= unpow2_binary64 (pow.f64 z 2))) y)) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 1 (+.f64 (*.f64 y (pow.f64 z 2)) (Rewrite<= *-rgt-identity_binary64 (*.f64 y 1)))) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 1 (Rewrite<= distribute-lft-in_binary64 (*.f64 y (+.f64 (pow.f64 z 2) 1)))) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 1 y) (+.f64 (pow.f64 z 2) 1))) x): 5 points increase in error, 6 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 1 y) (*.f64 (+.f64 (pow.f64 z 2) 1) x))): 14 points increase in error, 10 points decrease in error
      (/.f64 (/.f64 1 y) (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 (pow.f64 z 2) x) x))): 0 points increase in error, 1 points decrease in error
      (Rewrite=> associate-/l/_binary64 (/.f64 1 (*.f64 (+.f64 (*.f64 (pow.f64 z 2) x) x) y))): 34 points increase in error, 23 points decrease in error

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

   1. Initial program 18.5

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

   2. Taylor expanded in z around inf 14.1

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

   3. Simplified 14.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{z \cdot z}} \]
      Proof
      (/.f64 (/.f64 (/.f64 1 x) y) (*.f64 z z)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 1 x) y) (Rewrite<= unpow2_binary64 (pow.f64 z 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/l/_binary64 (/.f64 (/.f64 1 x) (*.f64 (pow.f64 z 2) y))): 29 points increase in error, 20 points decrease in error
      (/.f64 (/.f64 1 x) (Rewrite<= *-commutative_binary64 (*.f64 y (pow.f64 z 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/l/_binary64 (/.f64 1 (*.f64 (*.f64 y (pow.f64 z 2)) x))): 15 points increase in error, 9 points decrease in error
      (/.f64 1 (Rewrite<= associate-*r*_binary64 (*.f64 y (*.f64 (pow.f64 z 2) x)))): 20 points increase in error, 22 points decrease in error

   4. Applied egg-rr 3.0

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

   5. Applied egg-rr 1.5

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

   6. Taylor expanded in x around 0 1.5

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

4. Final simplification 0.8

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

Alternatives

Alternative 1
Error	1.7
Cost	968

\[\begin{array}{l} t_0 := \frac{\frac{1}{x}}{z} \cdot \frac{\frac{1}{y}}{z}\\ \mathbf{if}\;z \leq -7.74659560908662 \cdot 10^{+124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.057631571651298 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	2.3
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{z} \cdot \frac{\frac{1}{y}}{z}\\ \end{array} \]

Alternative 3
Error	3.5
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \mathbf{if}\;z \leq -428125795482.03235:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.405546041062985 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	3.4
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{1}{y \cdot z}}{z \cdot x}\\ \mathbf{if}\;z \leq -428125795482.03235:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.405546041062985 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	29.0
Cost	320

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

Reproduce

herbie shell --seed 2022308 
(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)))))