Average Error: 7.8 → 6.1
Time: 32.0s
Precision: 64
\[x0 = 1.855 \land x1 = 0.000209 \lor x0 = 2.985 \land x1 = 0.0186\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\begin{array}{l} \mathbf{if}\;x0 \le 1.8749218749999998:\\ \;\;\;\;(\left(\frac{\sqrt{x0}}{\sqrt{x1} + 1}\right) \cdot \left(\frac{\sqrt{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{x1} + 1}\right) \cdot \left(\frac{\sqrt[3]{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*\\ \end{array}\]
\frac{x0}{1 - x1} - x0
↓
\begin{array}{l}
\mathbf{if}\;x0 \le 1.8749218749999998:\\
\;\;\;\;(\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 r33425554 = x0;
        double r33425555 = 1.0;
        double r33425556 = x1;
        double r33425557 = r33425555 - r33425556;
        double r33425558 = r33425554 / r33425557;
        double r33425559 = r33425558 - r33425554;
        return r33425559;
}
↓
double f(double x0, double x1) {
        double r33425560 = x0;
        double r33425561 = 1.8749218749999998;
        bool r33425562 = r33425560 <= r33425561;
        double r33425563 = sqrt(r33425560);
        double r33425564 = x1;
        double r33425565 = sqrt(r33425564);
        double r33425566 = 1.0;
        double r33425567 = r33425565 + r33425566;
        double r33425568 = r33425563 / r33425567;
        double r33425569 = r33425566 - r33425565;
        double r33425570 = r33425563 / r33425569;
        double r33425571 = -r33425560;
        double r33425572 = fma(r33425568, r33425570, r33425571);
        double r33425573 = cbrt(r33425560);
        double r33425574 = r33425573 * r33425573;
        double r33425575 = r33425574 / r33425567;
        double r33425576 = r33425573 / r33425569;
        double r33425577 = fma(r33425575, r33425576, r33425571);
        double r33425578 = r33425562 ? r33425572 : r33425577;
        return r33425578;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original7.8
Target0.2
Herbie6.1
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Split input into 2 regimes

  2. if x0 < 1.8749218749999998

    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 1.8749218749999998 < x0

    1. Initial program 8.3

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

    3. Applied add-sqr-sqrt8.3

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

    4. Applied *-un-lft-identity8.3

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

    5. Applied difference-of-squares8.3

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

    6. Applied add-cube-cbrt8.3

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

    7. Applied times-frac8.1

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

    8. Applied fma-neg6.9

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x0 \le 1.8749218749999998:\\ \;\;\;\;(\left(\frac{\sqrt{x0}}{\sqrt{x1} + 1}\right) \cdot \left(\frac{\sqrt{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{x1} + 1}\right) \cdot \left(\frac{\sqrt[3]{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 +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))