Average Error: 6.3 → 4.3
Time: 6.7s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq -\infty:\\ \;\;\;\;\frac{1}{y} \cdot \frac{1}{x + z \cdot \left(z \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(1 + z \cdot z\right)}}{x}\\ \end{array}\]
\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 -\infty:\\
\;\;\;\;\frac{1}{y} \cdot \frac{1}{x + z \cdot \left(z \cdot x\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{y \cdot \left(1 + z \cdot z\right)}}{x}\\

\end{array}
(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))) (- INFINITY))
   (* (/ 1.0 y) (/ 1.0 (+ x (* z (* z x)))))
   (/ (/ 1.0 (* y (+ 1.0 (* z z)))) x)))
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))) <= -((double) INFINITY)) {
		tmp = (1.0 / y) * (1.0 / (x + (z * (z * x))));
	} else {
		tmp = (1.0 / (y * (1.0 + (z * z)))) / x;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.3
Target5.7
Herbie4.3
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

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

    1. Initial program 18.2

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity_binary64_758018.2

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

    4. Applied add-sqr-sqrt_binary64_760218.2

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

    5. Applied times-frac_binary64_758618.2

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

    6. Applied times-frac_binary64_758614.4

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

    7. Simplified14.4

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

    8. Simplified14.4

      \[\leadsto \frac{1}{y} \cdot \color{blue}{\frac{1}{x + \left(z \cdot z\right) \cdot x}}\]
    9. Using strategy rm

    10. Applied associate-*l*_binary64_75215.8

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

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

    1. Initial program 4.0

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

    2. Taylor expanded around 0 4.3

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

    3. Simplified4.3

      \[\leadsto \color{blue}{\frac{1}{\left(y \cdot \left(z \cdot z + 1\right)\right) \cdot x}}\]
    4. Using strategy rm

    5. Applied associate-/r*_binary64_75244.0

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

  4. Final simplification4.3

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

Reproduce

herbie shell --seed 2021024 
(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)))))