Average Error: 7.9 → 6.2
Time: 25.1s
Precision: 64
\[\frac{x0}{1 - x1} - x0\]
↓
\[\begin{array}{l} \mathbf{if}\;x0 \le 2.9451562499999997:\\ \;\;\;\;(\left(\frac{\sqrt{x0}}{\sqrt{x1} + 1}\right) \cdot \left(\frac{\sqrt{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{x1} + 1}\right) \cdot \left(\frac{\sqrt[3]{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*\\ \end{array}\]
double f(double x0, double x1) {
        double r6689542 = x0;
        double r6689543 = 1.0;
        double r6689544 = x1;
        double r6689545 = r6689543 - r6689544;
        double r6689546 = r6689542 / r6689545;
        double r6689547 = r6689546 - r6689542;
        return r6689547;
}
↓
double f(double x0, double x1) {
        double r6689548 = x0;
        double r6689549 = 2.9451562499999997;
        bool r6689550 = r6689548 <= r6689549;
        double r6689551 = sqrt(r6689548);
        double r6689552 = x1;
        double r6689553 = sqrt(r6689552);
        double r6689554 = 1.0;
        double r6689555 = r6689553 + r6689554;
        double r6689556 = r6689551 / r6689555;
        double r6689557 = r6689554 - r6689553;
        double r6689558 = r6689551 / r6689557;
        double r6689559 = -r6689548;
        double r6689560 = fma(r6689556, r6689558, r6689559);
        double r6689561 = cbrt(r6689548);
        double r6689562 = r6689561 * r6689561;
        double r6689563 = r6689562 / r6689555;
        double r6689564 = r6689561 / r6689557;
        double r6689565 = fma(r6689563, r6689564, r6689559);
        double r6689566 = r6689550 ? r6689560 : r6689565;
        return r6689566;
}


\frac{x0}{1 - x1} - x0
↓
\begin{array}{l}
\mathbf{if}\;x0 \le 2.9451562499999997:\\
\;\;\;\;(\left(\frac{\sqrt{x0}}{\sqrt{x1} + 1}\right) \cdot \left(\frac{\sqrt{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*\\

\mathbf{else}:\\
\;\;\;\;(\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{x1} + 1}\right) \cdot \left(\frac{\sqrt[3]{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*\\

\end{array}

Error

Bits error versus x0

Bits error versus x1

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x0 < 2.9451562499999997

    1. Initial program 7.4

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

    3. Applied add-sqr-sqrt7.4

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

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

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

    5. Applied difference-of-squares7.4

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

    6. Applied add-sqr-sqrt7.4

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

    7. Applied times-frac7.4

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

    8. Applied fma-neg5.3

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

    if 2.9451562499999997 < x0

    1. Initial program 8.4

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

    3. Applied add-sqr-sqrt8.4

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

    4. Applied *-un-lft-identity8.4

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

    5. Applied difference-of-squares8.4

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

    6. Applied add-cube-cbrt8.4

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

    7. Applied times-frac8.2

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

    8. Applied fma-neg7.0

      \[\leadsto \color{blue}{(\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{1 + \sqrt{x1}}\right) \cdot \left(\frac{\sqrt[3]{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x0 \le 2.9451562499999997:\\ \;\;\;\;(\left(\frac{\sqrt{x0}}{\sqrt{x1} + 1}\right) \cdot \left(\frac{\sqrt{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{x1} + 1}\right) \cdot \left(\frac{\sqrt[3]{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019101 +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))