Average Error: 7.9 → 6.1
Time: 14.9s
Precision: 64
\[x0 = 1.854999999999999982236431605997495353222 \land x1 = 2.090000000000000115064208161541614572343 \cdot 10^{-4} \lor x0 = 2.984999999999999875655021241982467472553 \land x1 = 0.01859999999999999847899445626353553961962\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\begin{array}{l} \mathbf{if}\;x1 \le 0.01820459765624999823319107861152588156983:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)\\ \end{array}\]
\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)\\

\end{array}
double f(double x0, double x1) {
        double r9305081 = x0;
        double r9305082 = 1.0;
        double r9305083 = x1;
        double r9305084 = r9305082 - r9305083;
        double r9305085 = r9305081 / r9305084;
        double r9305086 = r9305085 - r9305081;
        return r9305086;
}
↓
double f(double x0, double x1) {
        double r9305087 = x1;
        double r9305088 = 0.018204597656249998;
        bool r9305089 = r9305087 <= r9305088;
        double r9305090 = x0;
        double r9305091 = cbrt(r9305090);
        double r9305092 = r9305091 * r9305091;
        double r9305093 = 1.0;
        double r9305094 = r9305093 - r9305087;
        double r9305095 = r9305091 / r9305094;
        double r9305096 = -r9305090;
        double r9305097 = fma(r9305092, r9305095, r9305096);
        double r9305098 = sqrt(r9305090);
        double r9305099 = sqrt(r9305093);
        double r9305100 = sqrt(r9305087);
        double r9305101 = r9305099 + r9305100;
        double r9305102 = r9305098 / r9305101;
        double r9305103 = r9305099 - r9305100;
        double r9305104 = r9305098 / r9305103;
        double r9305105 = fma(r9305102, r9305104, r9305096);
        double r9305106 = r9305089 ? r9305097 : r9305105;
        return r9305106;
}

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 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 4.6

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

    3. Applied add-sqr-sqrt4.6

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

    4. Applied add-sqr-sqrt4.6

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

    5. Applied difference-of-squares4.6

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

    6. Applied add-sqr-sqrt4.6

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

    7. Applied times-frac5.2

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

    8. Applied fma-neg3.3

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

  4. Final simplification6.1

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

Reproduce

herbie shell --seed 2019172 +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))