Average Error: 7.9 → 7.0
Time: 12.8s
Precision: 64
\[x0 = 1.854999999999999982236431605997495353222 \land x1 = 2.090000000000000115064208161541614572343 \cdot 10^{-4} \lor x0 = 2.984999999999999875655021241982467472553 \land x1 = 0.01859999999999999847899445626353553961962\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\mathsf{fma}\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)\]
\frac{x0}{1 - x1} - x0
↓
\mathsf{fma}\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)
double f(double x0, double x1) {
        double r128327 = x0;
        double r128328 = 1.0;
        double r128329 = x1;
        double r128330 = r128328 - r128329;
        double r128331 = r128327 / r128330;
        double r128332 = r128331 - r128327;
        return r128332;
}
↓
double f(double x0, double x1) {
        double r128333 = x0;
        double r128334 = cbrt(r128333);
        double r128335 = r128334 * r128334;
        double r128336 = 1.0;
        double r128337 = x1;
        double r128338 = r128336 - r128337;
        double r128339 = r128334 / r128338;
        double r128340 = -r128333;
        double r128341 = fma(r128335, r128339, r128340);
        return r128341;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original7.9
Target0.2
Herbie7.0
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 7.9

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

  3. Applied *-un-lft-identity7.9

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

  4. Applied add-cube-cbrt7.9

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

  5. Applied times-frac8.2

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

  6. Applied fma-neg7.0

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

  7. Final simplification7.0

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

Reproduce

herbie shell --seed 2019325 +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))