Average Error: 7.9 → 5.5
Time: 3.2s
Precision: 64
\[x0 = 1.855 \land x1 = 2.09000000000000012 \cdot 10^{-4} \lor x0 = 2.98499999999999988 \land x1 = 0.018599999999999998\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\begin{array}{l} \mathbf{if}\;x0 \le 2.9451562499999997:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\log \left(\frac{{\left(e^{{x0}^{\frac{2}{3}}}\right)}^{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}}{e^{x0}}\right)\right)}\\ \end{array}\]
\frac{x0}{1 - x1} - x0
↓
\begin{array}{l}
\mathbf{if}\;x0 \le 2.9451562499999997:\\
\;\;\;\;\mathsf{fma}\left(\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)\\

\mathbf{else}:\\
\;\;\;\;e^{\log \left(\log \left(\frac{{\left(e^{{x0}^{\frac{2}{3}}}\right)}^{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}}{e^{x0}}\right)\right)}\\

\end{array}
double f(double x0, double x1) {
        double r224843 = x0;
        double r224844 = 1.0;
        double r224845 = x1;
        double r224846 = r224844 - r224845;
        double r224847 = r224843 / r224846;
        double r224848 = r224847 - r224843;
        return r224848;
}
↓
double f(double x0, double x1) {
        double r224849 = x0;
        double r224850 = 2.9451562499999997;
        bool r224851 = r224849 <= r224850;
        double r224852 = sqrt(r224849);
        double r224853 = 1.0;
        double r224854 = sqrt(r224853);
        double r224855 = x1;
        double r224856 = sqrt(r224855);
        double r224857 = r224854 + r224856;
        double r224858 = r224852 / r224857;
        double r224859 = r224854 - r224856;
        double r224860 = r224852 / r224859;
        double r224861 = -r224849;
        double r224862 = fma(r224858, r224860, r224861);
        double r224863 = 0.6666666666666666;
        double r224864 = pow(r224849, r224863);
        double r224865 = exp(r224864);
        double r224866 = cbrt(r224849);
        double r224867 = r224853 - r224855;
        double r224868 = r224866 / r224867;
        double r224869 = pow(r224865, r224868);
        double r224870 = exp(r224849);
        double r224871 = r224869 / r224870;
        double r224872 = log(r224871);
        double r224873 = log(r224872);
        double r224874 = exp(r224873);
        double r224875 = r224851 ? r224862 : r224874;
        return r224875;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original7.9
Target0.2
Herbie5.5
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Split input into 2 regimes

  2. if x0 < 2.9451562499999997

    1. Initial program 7.5

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

    3. Applied add-sqr-sqrt7.5

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

    4. Applied add-sqr-sqrt7.5

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

    5. Applied difference-of-squares7.5

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

    6. Applied add-sqr-sqrt7.5

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

    7. Applied times-frac7.5

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

    8. Applied fma-neg5.4

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

    if 2.9451562499999997 < x0

    1. Initial program 8.4

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

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

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

    4. Applied add-cube-cbrt8.4

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

    5. Applied times-frac8.4

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

    6. Applied fma-neg7.1

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

    8. Applied add-exp-log7.2

      \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{1}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)\right)}}\]
    9. Using strategy rm

    10. Applied add-log-exp7.6

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

    11. Simplified5.6

      \[\leadsto e^{\log \left(\log \color{blue}{\left(\frac{{\left(e^{{x0}^{\frac{2}{3}}}\right)}^{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}}{e^{x0}}\right)}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x0 \le 2.9451562499999997:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\log \left(\frac{{\left(e^{{x0}^{\frac{2}{3}}}\right)}^{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}}{e^{x0}}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x0 x1)
  :name "(- (/ x0 (- 1 x1)) x0)"
  :precision binary64
  :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))