Average Error: 7.9 → 6.1
Time: 25.4s
Precision: 64
\[x0 = 1.855 \land x1 = 0.000209 \lor x0 = 2.985 \land x1 = 0.0186\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\begin{array}{l} \mathbf{if}\;x0 \le 1.8749218749999998:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\sqrt{x0}}{\sqrt{x1} + 1}\right), \left(\frac{\sqrt{x0}}{1 - \sqrt{x1}}\right), \left(-x0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{x1} + 1}\right), \left(\frac{\sqrt[3]{x0}}{1 - \sqrt{x1}}\right), \left(-x0\right)\right)\\ \end{array}\]
\frac{x0}{1 - x1} - x0
↓
\begin{array}{l}
\mathbf{if}\;x0 \le 1.8749218749999998:\\
\;\;\;\;\mathsf{fma}\left(\left(\frac{\sqrt{x0}}{\sqrt{x1} + 1}\right), \left(\frac{\sqrt{x0}}{1 - \sqrt{x1}}\right), \left(-x0\right)\right)\\

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

\end{array}
double f(double x0, double x1) {
        double r21097406 = x0;
        double r21097407 = 1.0;
        double r21097408 = x1;
        double r21097409 = r21097407 - r21097408;
        double r21097410 = r21097406 / r21097409;
        double r21097411 = r21097410 - r21097406;
        return r21097411;
}
↓
double f(double x0, double x1) {
        double r21097412 = x0;
        double r21097413 = 1.8749218749999998;
        bool r21097414 = r21097412 <= r21097413;
        double r21097415 = sqrt(r21097412);
        double r21097416 = x1;
        double r21097417 = sqrt(r21097416);
        double r21097418 = 1.0;
        double r21097419 = r21097417 + r21097418;
        double r21097420 = r21097415 / r21097419;
        double r21097421 = r21097418 - r21097417;
        double r21097422 = r21097415 / r21097421;
        double r21097423 = -r21097412;
        double r21097424 = fma(r21097420, r21097422, r21097423);
        double r21097425 = cbrt(r21097412);
        double r21097426 = r21097425 * r21097425;
        double r21097427 = r21097426 / r21097419;
        double r21097428 = r21097425 / r21097421;
        double r21097429 = fma(r21097427, r21097428, r21097423);
        double r21097430 = r21097414 ? r21097424 : r21097429;
        return r21097430;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original7.9
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

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

    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-frac8.2

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

    8. Applied fma-neg7.0

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

  4. Final simplification6.1

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

Reproduce

herbie shell --seed 2019121 +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))