Average Error: 7.8 → 6.9
Time: 38.4s
Precision: 64
\[x0 = 1.855 \land x1 = 0.000209 \lor x0 = 2.985 \land x1 = 0.0186\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\mathsf{fma}\left(\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}\right), \left(\frac{\sqrt[3]{x0}}{1 - x1}\right), \left(-x0\right)\right)\]
\frac{x0}{1 - x1} - x0
↓
\mathsf{fma}\left(\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}\right), \left(\frac{\sqrt[3]{x0}}{1 - x1}\right), \left(-x0\right)\right)
double f(double x0, double x1) {
        double r12808473 = x0;
        double r12808474 = 1.0;
        double r12808475 = x1;
        double r12808476 = r12808474 - r12808475;
        double r12808477 = r12808473 / r12808476;
        double r12808478 = r12808477 - r12808473;
        return r12808478;
}
↓
double f(double x0, double x1) {
        double r12808479 = x0;
        double r12808480 = cbrt(r12808479);
        double r12808481 = r12808480 * r12808480;
        double r12808482 = 1.0;
        double r12808483 = x1;
        double r12808484 = r12808482 - r12808483;
        double r12808485 = r12808480 / r12808484;
        double r12808486 = -r12808479;
        double r12808487 = fma(r12808481, r12808485, r12808486);
        return r12808487;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original7.8
Target0.3
Herbie6.9
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 7.8

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

  3. Applied *-un-lft-identity7.8

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

  4. Applied *-un-lft-identity7.8

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

  5. Applied distribute-lft-out--7.8

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

  6. Applied add-cube-cbrt7.8

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

  7. Applied times-frac8.1

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

  8. Applied fma-neg6.9

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

  9. Final simplification6.9

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

Reproduce

herbie shell --seed 2019128 +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))