Average Error: 7.8 → 6.0
Time: 8.7s
Precision: 64
\[x0 = 1.854999999999999982236431605997495353222 \land x1 = 2.090000000000000115064208161541614572343 \cdot 10^{-4} \lor x0 = 2.984999999999999875655021241982467472553 \land x1 = 0.01859999999999999847899445626353553961962\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\begin{array}{l} \mathbf{if}\;x1 \le 0.01820459765624999823319107861152588156983:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right) + \mathsf{fma}\left(x0, -1, x0\right)\\ \end{array}\]
\frac{x0}{1 - x1} - x0
↓
\begin{array}{l}
\mathbf{if}\;x1 \le 0.01820459765624999823319107861152588156983:\\
\;\;\;\;\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}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right) + \mathsf{fma}\left(x0, -1, x0\right)\\

\end{array}
double f(double x0, double x1) {
        double r182011 = x0;
        double r182012 = 1.0;
        double r182013 = x1;
        double r182014 = r182012 - r182013;
        double r182015 = r182011 / r182014;
        double r182016 = r182015 - r182011;
        return r182016;
}
↓
double f(double x0, double x1) {
        double r182017 = x1;
        double r182018 = 0.018204597656249998;
        bool r182019 = r182017 <= r182018;
        double r182020 = x0;
        double r182021 = cbrt(r182020);
        double r182022 = r182021 * r182021;
        double r182023 = 1.0;
        double r182024 = r182023 - r182017;
        double r182025 = r182021 / r182024;
        double r182026 = -r182020;
        double r182027 = fma(r182022, r182025, r182026);
        double r182028 = sqrt(r182020);
        double r182029 = sqrt(r182023);
        double r182030 = sqrt(r182017);
        double r182031 = r182029 + r182030;
        double r182032 = r182028 / r182031;
        double r182033 = r182029 - r182030;
        double r182034 = r182028 / r182033;
        double r182035 = fma(r182032, r182034, r182026);
        double r182036 = -1.0;
        double r182037 = fma(r182020, r182036, r182020);
        double r182038 = r182035 + r182037;
        double r182039 = r182019 ? r182027 : r182038;
        return r182039;
}

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

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

    4. Applied add-sqr-sqrt4.5

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

    5. Applied add-sqr-sqrt4.5

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

    6. Applied difference-of-squares4.5

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

    7. Applied add-sqr-sqrt4.5

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

    8. Applied times-frac5.2

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

    9. Applied prod-diff4.8

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

    10. Simplified4.8

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

    11. Simplified3.2

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \le 0.01820459765624999823319107861152588156983:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right) + \mathsf{fma}\left(x0, -1, x0\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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) (- 1.0 x1))

  (- (/ x0 (- 1.0 x1)) x0))