Average Error: 7.9 → 6.1
Time: 21.6s
Precision: 64
\[x0 = 1.855 \land x1 = 0.000209 \lor x0 = 2.985 \land x1 = 0.0186\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\begin{array}{l} \mathbf{if}\;x0 \le 1.8749218749999998:\\ \;\;\;\;(\left(\frac{\sqrt{x0}}{\sqrt{x1} + 1}\right) \cdot \left(\frac{\sqrt{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{x1} + 1}\right) \cdot \left(\frac{\sqrt[3]{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*\\ \end{array}\]
\frac{x0}{1 - x1} - x0
↓
\begin{array}{l}
\mathbf{if}\;x0 \le 1.8749218749999998:\\
\;\;\;\;(\left(\frac{\sqrt{x0}}{\sqrt{x1} + 1}\right) \cdot \left(\frac{\sqrt{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*\\

\mathbf{else}:\\
\;\;\;\;(\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{x1} + 1}\right) \cdot \left(\frac{\sqrt[3]{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*\\

\end{array}
double f(double x0, double x1) {
        double r26626483 = x0;
        double r26626484 = 1.0;
        double r26626485 = x1;
        double r26626486 = r26626484 - r26626485;
        double r26626487 = r26626483 / r26626486;
        double r26626488 = r26626487 - r26626483;
        return r26626488;
}
↓
double f(double x0, double x1) {
        double r26626489 = x0;
        double r26626490 = 1.8749218749999998;
        bool r26626491 = r26626489 <= r26626490;
        double r26626492 = sqrt(r26626489);
        double r26626493 = x1;
        double r26626494 = sqrt(r26626493);
        double r26626495 = 1.0;
        double r26626496 = r26626494 + r26626495;
        double r26626497 = r26626492 / r26626496;
        double r26626498 = r26626495 - r26626494;
        double r26626499 = r26626492 / r26626498;
        double r26626500 = -r26626489;
        double r26626501 = fma(r26626497, r26626499, r26626500);
        double r26626502 = cbrt(r26626489);
        double r26626503 = r26626502 * r26626502;
        double r26626504 = r26626503 / r26626496;
        double r26626505 = r26626502 / r26626498;
        double r26626506 = fma(r26626504, r26626505, r26626500);
        double r26626507 = r26626491 ? r26626501 : r26626506;
        return r26626507;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original7.9
Target0.3
Herbie6.1
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Split input into 2 regimes

  2. if x0 < 1.8749218749999998

    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 *-un-lft-identity7.4

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

    5. Applied difference-of-squares7.4

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

    6. Applied add-sqr-sqrt7.4

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

    7. Applied times-frac7.4

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

    8. Applied fma-neg5.3

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

    if 1.8749218749999998 < x0

    1. Initial program 8.3

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

    3. Applied add-sqr-sqrt8.3

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

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

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

    5. Applied difference-of-squares8.3

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

    6. Applied add-cube-cbrt8.3

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

    7. Applied times-frac8.1

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

    8. Applied fma-neg6.9

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x0 \le 1.8749218749999998:\\ \;\;\;\;(\left(\frac{\sqrt{x0}}{\sqrt{x1} + 1}\right) \cdot \left(\frac{\sqrt{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{x1} + 1}\right) \cdot \left(\frac{\sqrt[3]{x0}}{1 - \sqrt{x1}}\right) + \left(-x0\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +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))