Average Error: 7.9 → 6.9
Time: 17.5s
Precision: 64
\[x0 = 1.855 \land x1 = 0.000209 \lor x0 = 2.985 \land x1 = 0.0186\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\mathsf{fma}\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)\]
\frac{x0}{1 - x1} - x0
↓
\mathsf{fma}\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)
double f(double x0, double x1) {
        double r5777446 = x0;
        double r5777447 = 1.0;
        double r5777448 = x1;
        double r5777449 = r5777447 - r5777448;
        double r5777450 = r5777446 / r5777449;
        double r5777451 = r5777450 - r5777446;
        return r5777451;
}
↓
double f(double x0, double x1) {
        double r5777452 = x0;
        double r5777453 = cbrt(r5777452);
        double r5777454 = r5777453 * r5777453;
        double r5777455 = 1.0;
        double r5777456 = x1;
        double r5777457 = r5777455 - r5777456;
        double r5777458 = r5777453 / r5777457;
        double r5777459 = -r5777452;
        double r5777460 = fma(r5777454, r5777458, r5777459);
        return r5777460;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original7.9
Target0.2
Herbie6.9
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 7.9

    \[\frac{x0}{1 - x1} - x0\]
  2. Using strategy rm

  3. Applied *-un-lft-identity7.9

    \[\leadsto \frac{x0}{\color{blue}{1 \cdot \left(1 - x1\right)}} - x0\]

  4. Applied add-cube-cbrt7.9

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}\right) \cdot \sqrt[3]{x0}}}{1 \cdot \left(1 - x1\right)} - x0\]

  5. Applied times-frac8.2

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{1} \cdot \frac{\sqrt[3]{x0}}{1 - x1}} - x0\]

  6. Applied fma-neg6.9

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{1}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)}\]

  7. Final simplification6.9

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x0 x1)
  :name "(- (/ x0 (- 1 x1)) x0)"
  :pre (or (and (== x0 1.855) (== x1 0.000209)) (and (== x0 2.985) (== x1 0.0186)))

  :herbie-target
  (/ (* x0 x1) (- 1 x1))

  (- (/ x0 (- 1 x1)) x0))