Average Error: 7.9 → 6.1
Time: 9.0s
Precision: 64
\[x0 = 1.855 \land x1 = 0.000209 \lor x0 = 2.985 \land x1 = 0.0186\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\begin{array}{l} \mathbf{if}\;x1 \le 0.018204597656249998:\\ \;\;\;\;\mathsf{fma}\left(\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}\right), \left(\frac{\sqrt[3]{x0}}{1 - x1}\right), \left(-x0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\sqrt{x0}}{1 + \sqrt{x1}}\right), \left(\frac{\sqrt{x0}}{1 - \sqrt{x1}}\right), \left(-x0\right)\right)\\ \end{array}\]
\frac{x0}{1 - x1} - x0
↓
\begin{array}{l}
\mathbf{if}\;x1 \le 0.018204597656249998:\\
\;\;\;\;\mathsf{fma}\left(\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}\right), \left(\frac{\sqrt[3]{x0}}{1 - x1}\right), \left(-x0\right)\right)\\

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

\end{array}
double f(double x0, double x1) {
        double r5584363 = x0;
        double r5584364 = 1.0;
        double r5584365 = x1;
        double r5584366 = r5584364 - r5584365;
        double r5584367 = r5584363 / r5584366;
        double r5584368 = r5584367 - r5584363;
        return r5584368;
}
↓
double f(double x0, double x1) {
        double r5584369 = x1;
        double r5584370 = 0.018204597656249998;
        bool r5584371 = r5584369 <= r5584370;
        double r5584372 = x0;
        double r5584373 = cbrt(r5584372);
        double r5584374 = r5584373 * r5584373;
        double r5584375 = 1.0;
        double r5584376 = r5584375 - r5584369;
        double r5584377 = r5584373 / r5584376;
        double r5584378 = -r5584372;
        double r5584379 = fma(r5584374, r5584377, r5584378);
        double r5584380 = sqrt(r5584372);
        double r5584381 = sqrt(r5584369);
        double r5584382 = r5584375 + r5584381;
        double r5584383 = r5584380 / r5584382;
        double r5584384 = r5584375 - r5584381;
        double r5584385 = r5584380 / r5584384;
        double r5584386 = fma(r5584383, r5584385, r5584378);
        double r5584387 = r5584371 ? r5584379 : r5584386;
        return r5584387;
}

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 x1 < 0.018204597656249998

    1. Initial program 11.2

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

    3. Applied *-un-lft-identity11.2

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

    4. Applied *-un-lft-identity11.2

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

    5. Applied distribute-lft-out--11.2

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

    6. Applied add-cube-cbrt11.2

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

    7. Applied times-frac10.9

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

    8. Applied fma-neg8.9

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

    if 0.018204597656249998 < x1

    1. Initial program 4.5

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

    3. Applied add-sqr-sqrt4.5

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

    4. Applied *-un-lft-identity4.5

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

    5. Applied difference-of-squares4.5

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

    6. Applied add-sqr-sqrt4.5

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

    7. Applied times-frac5.2

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

    8. Applied fma-neg3.2

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \le 0.018204597656249998:\\ \;\;\;\;\mathsf{fma}\left(\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}\right), \left(\frac{\sqrt[3]{x0}}{1 - x1}\right), \left(-x0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\sqrt{x0}}{1 + \sqrt{x1}}\right), \left(\frac{\sqrt{x0}}{1 - \sqrt{x1}}\right), \left(-x0\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +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))