Average Error: 6.9 → 6.5
Time: 3.4s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{1}{y \cdot \sqrt{1 + z \cdot z}} \cdot \frac{1}{\sqrt{1 + z \cdot z} \cdot x}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{1}{y \cdot \sqrt{1 + z \cdot z}} \cdot \frac{1}{\sqrt{1 + z \cdot z} \cdot x}
double code(double x, double y, double z) {
	return ((1.0 / x) / ((double) (y * ((double) (1.0 + ((double) (z * z)))))));
}
↓
double code(double x, double y, double z) {
	return ((double) ((1.0 / ((double) (y * ((double) sqrt(((double) (1.0 + ((double) (z * z))))))))) * (1.0 / ((double) (((double) sqrt(((double) (1.0 + ((double) (z * z)))))) * x)))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original6.9
Target6.2
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.9

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

  3. Applied associate-/r*6.8

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

  4. Simplified7.0

    \[\leadsto \frac{\color{blue}{\frac{1}{y \cdot x}}}{1 + z \cdot z}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt7.0

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

  7. Applied *-un-lft-identity7.0

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

  8. Applied times-frac6.9

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

  9. Applied times-frac6.4

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

  10. Simplified6.5

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

  11. Simplified6.5

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

  12. Final simplification6.5

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

Reproduce

herbie shell --seed 2020196 
(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)))))