Average Error: 7.9 → 5.5
Time: 3.6s
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 r140281 = x0;
        double r140282 = 1.0;
        double r140283 = x1;
        double r140284 = r140282 - r140283;
        double r140285 = r140281 / r140284;
        double r140286 = r140285 - r140281;
        return r140286;
}
↓
double f(double x0, double x1) {
        double r140287 = x0;
        double r140288 = 2.9451562499999997;
        bool r140289 = r140287 <= r140288;
        double r140290 = sqrt(r140287);
        double r140291 = 1.0;
        double r140292 = sqrt(r140291);
        double r140293 = x1;
        double r140294 = sqrt(r140293);
        double r140295 = r140292 + r140294;
        double r140296 = r140290 / r140295;
        double r140297 = r140292 - r140294;
        double r140298 = r140290 / r140297;
        double r140299 = -r140287;
        double r140300 = fma(r140296, r140298, r140299);
        double r140301 = 0.6666666666666666;
        double r140302 = pow(r140287, r140301);
        double r140303 = exp(r140302);
        double r140304 = cbrt(r140287);
        double r140305 = r140291 - r140293;
        double r140306 = r140304 / r140305;
        double r140307 = pow(r140303, r140306);
        double r140308 = exp(r140287);
        double r140309 = r140307 / r140308;
        double r140310 = log(r140309);
        double r140311 = log(r140310);
        double r140312 = exp(r140311);
        double r140313 = r140289 ? r140300 : r140312;
        return r140313;
}

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 7.4

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

    3. Applied add-sqr-sqrt7.4

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

    4. Applied add-sqr-sqrt7.4

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

    5. Applied difference-of-squares7.4

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

    6. Applied add-sqr-sqrt7.4

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

    7. Applied times-frac7.4

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

    8. Applied fma-neg5.3

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

    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-neg7.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{1}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)}\]
    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 2020060 +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))