Average Error: 6.4 → 2.7
Time: 4.7s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{x} \leq -4.8362878226354626 \cdot 10^{+114} \lor \neg \left(\frac{1}{x} \leq 1.460151242421802 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{1}{y \cdot \left(x + z \cdot \left(x \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{x} \leq -4.8362878226354626 \cdot 10^{+114} \lor \neg \left(\frac{1}{x} \leq 1.460151242421802 \cdot 10^{-37}\right):\\
\;\;\;\;\frac{1}{y \cdot \left(x + z \cdot \left(x \cdot z\right)\right)}\\

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

\end{array}
(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z)
 :precision binary64
 (if (or (<= (/ 1.0 x) -4.8362878226354626e+114)
         (not (<= (/ 1.0 x) 1.460151242421802e-37)))
   (/ 1.0 (* y (+ x (* z (* x z)))))
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))))
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 (((1.0 / x) <= -4.8362878226354626e+114) || !((1.0 / x) <= 1.460151242421802e-37)) {
		tmp = 1.0 / (y * (x + (z * (x * z))));
	} else {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	}
	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

Original6.4
Target5.8
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 1 x) < -4.8362878226354626e114 or 1.46015124242180199e-37 < (/.f64 1 x)

    1. Initial program 12.5

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

    3. Applied *-un-lft-identity_binary6412.5

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

    4. Applied add-cube-cbrt_binary6412.5

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot x}}{y \cdot \left(1 + z \cdot z\right)}\]

    5. Applied times-frac_binary6412.5

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{x}}}{y \cdot \left(1 + z \cdot z\right)}\]

    6. Applied times-frac_binary649.6

      \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}{y} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{1 + z \cdot z}}\]

    7. Simplified9.6

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{1 + z \cdot z}\]

    8. Simplified9.6

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

    9. Taylor expanded around 0 9.6

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

    10. Simplified9.6

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

    12. Applied associate-*r*_binary643.3

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

    if -4.8362878226354626e114 < (/.f64 1 x) < 1.46015124242180199e-37

    1. Initial program 2.3

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

    3. Applied *-commutative_binary642.3

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x} \leq -4.8362878226354626 \cdot 10^{+114} \lor \neg \left(\frac{1}{x} \leq 1.460151242421802 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{1}{y \cdot \left(x + z \cdot \left(x \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021174 
(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)))))