Average Error: 6.4 → 6.3
Time: 9.2s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{1}{y}}{\mathsf{fma}\left(z, z, 1\right)}}{x}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{1}{y}}{\mathsf{fma}\left(z, z, 1\right)}}{x}
double f(double x, double y, double z) {
        double r323477 = 1.0;
        double r323478 = x;
        double r323479 = r323477 / r323478;
        double r323480 = y;
        double r323481 = z;
        double r323482 = r323481 * r323481;
        double r323483 = r323477 + r323482;
        double r323484 = r323480 * r323483;
        double r323485 = r323479 / r323484;
        return r323485;
}
↓
double f(double x, double y, double z) {
        double r323486 = 1.0;
        double r323487 = y;
        double r323488 = r323486 / r323487;
        double r323489 = z;
        double r323490 = fma(r323489, r323489, r323486);
        double r323491 = r323488 / r323490;
        double r323492 = x;
        double r323493 = r323491 / r323492;
        return r323493;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.4
Target5.8
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

  3. Applied clear-num6.7

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

  4. Simplified6.7

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

  6. Applied associate-/r/6.7

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

  7. Applied associate-/r*6.4

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

  8. Simplified6.3

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

  9. Final simplification6.3

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

Reproduce

herbie shell --seed 2020043 +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)))))