Average Error: 7.9 → 6.9
Time: 20.8s
Precision: 64
\[x0 = 1.855 \land x1 = 0.000209 \lor x0 = 2.985 \land x1 = 0.0186\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\mathsf{fma}\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)\]
\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 r4771496 = x0;
        double r4771497 = 1.0;
        double r4771498 = x1;
        double r4771499 = r4771497 - r4771498;
        double r4771500 = r4771496 / r4771499;
        double r4771501 = r4771500 - r4771496;
        return r4771501;
}
↓
double f(double x0, double x1) {
        double r4771502 = x0;
        double r4771503 = cbrt(r4771502);
        double r4771504 = r4771503 * r4771503;
        double r4771505 = 1.0;
        double r4771506 = x1;
        double r4771507 = r4771505 - r4771506;
        double r4771508 = r4771503 / r4771507;
        double r4771509 = -r4771502;
        double r4771510 = fma(r4771504, r4771508, r4771509);
        return r4771510;
}

Error

Bits error versus x0

Bits error versus x1

Target

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

Derivation

  1. Initial program 7.9

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

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

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

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

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

  5. Applied distribute-lft-out--7.9

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

  6. Applied add-cube-cbrt7.9

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

  7. Applied times-frac8.2

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

  8. Applied fma-neg6.9

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

  9. Final simplification6.9

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

Reproduce

herbie shell --seed 2019151 +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))