Average Error: 6.7 → 6.8
Time: 7.3s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{y}}{\mathsf{fma}\left(z, z, 1\right) \cdot x}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{y}}{\mathsf{fma}\left(z, z, 1\right) \cdot x}
double code(double x, double y, double z) {
	return ((double) (((double) (1.0 / x)) / ((double) (y * ((double) (1.0 + ((double) (z * z))))))));
}
↓
double code(double x, double y, double z) {
	return ((double) (((double) (1.0 / y)) / ((double) (((double) fma(z, z, 1.0)) * x))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

Enter valid numbers for all inputs

Target

Original6.7
Target5.9
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.7

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

  2. Simplified6.6

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}{y}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity6.6

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

  5. Applied div-inv6.6

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

  6. Applied times-frac6.6

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

  7. Applied associate-/l*6.9

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

  8. Simplified7.0

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

  10. Applied associate-/r*6.8

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

  11. Simplified6.8

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

  12. Final simplification6.8

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

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1 (* z z))) #f) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.680743250567252e+305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

  (/ (/ 1 x) (* y (+ 1 (* z z)))))