Average Error: 7.8 → 6.0
Time: 10.0s
Precision: 64
\[x0 = 1.855 \land x1 = 0.000209 \lor x0 = 2.985 \land x1 = 0.0186\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\begin{array}{l} \mathbf{if}\;x1 \le 0.018204597656249998:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{x0}}{1 + \sqrt{x1}}, \frac{\sqrt{x0}}{1 - \sqrt{x1}}, -x0\right)\\ \end{array}\]
\frac{x0}{1 - x1} - x0
↓
\begin{array}{l}
\mathbf{if}\;x1 \le 0.018204597656249998:\\
\;\;\;\;\mathsf{fma}\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)\\

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

\end{array}
double f(double x0, double x1) {
        double r6801612 = x0;
        double r6801613 = 1.0;
        double r6801614 = x1;
        double r6801615 = r6801613 - r6801614;
        double r6801616 = r6801612 / r6801615;
        double r6801617 = r6801616 - r6801612;
        return r6801617;
}
↓
double f(double x0, double x1) {
        double r6801618 = x1;
        double r6801619 = 0.018204597656249998;
        bool r6801620 = r6801618 <= r6801619;
        double r6801621 = x0;
        double r6801622 = cbrt(r6801621);
        double r6801623 = r6801622 * r6801622;
        double r6801624 = 1.0;
        double r6801625 = r6801624 - r6801618;
        double r6801626 = r6801622 / r6801625;
        double r6801627 = -r6801621;
        double r6801628 = fma(r6801623, r6801626, r6801627);
        double r6801629 = sqrt(r6801621);
        double r6801630 = sqrt(r6801618);
        double r6801631 = r6801624 + r6801630;
        double r6801632 = r6801629 / r6801631;
        double r6801633 = r6801624 - r6801630;
        double r6801634 = r6801629 / r6801633;
        double r6801635 = fma(r6801632, r6801634, r6801627);
        double r6801636 = r6801620 ? r6801628 : r6801635;
        return r6801636;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original7.8
Target0.2
Herbie6.0
\[\frac{x0 \cdot x1}{1 - x1}\]

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

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

    4. Applied add-cube-cbrt11.2

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

    5. Applied times-frac10.9

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

    6. Applied fma-neg8.9

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

    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-neg3.1

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

  4. Final simplification6.0

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

Reproduce

herbie shell --seed 2019168 +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))