Average Error: 7.9 → 7.0
Time: 38.9s
Precision: 64
\[x0 = 1.855 \land x1 = 0.000209 \lor x0 = 2.985 \land x1 = 0.0186\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\mathsf{fma}\left(\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}\right), \left(\frac{\sqrt[3]{x0}}{1 - x1}\right), \left(-x0\right)\right)\]
\frac{x0}{1 - x1} - x0
↓
\mathsf{fma}\left(\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}\right), \left(\frac{\sqrt[3]{x0}}{1 - x1}\right), \left(-x0\right)\right)
double f(double x0, double x1) {
        double r13249782 = x0;
        double r13249783 = 1.0;
        double r13249784 = x1;
        double r13249785 = r13249783 - r13249784;
        double r13249786 = r13249782 / r13249785;
        double r13249787 = r13249786 - r13249782;
        return r13249787;
}
↓
double f(double x0, double x1) {
        double r13249788 = x0;
        double r13249789 = cbrt(r13249788);
        double r13249790 = r13249789 * r13249789;
        double r13249791 = 1.0;
        double r13249792 = x1;
        double r13249793 = r13249791 - r13249792;
        double r13249794 = r13249789 / r13249793;
        double r13249795 = -r13249788;
        double r13249796 = fma(r13249790, r13249794, r13249795);
        return r13249796;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original7.9
Target0.3
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

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

  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-neg7.0

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

  9. Final simplification7.0

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

Reproduce

herbie shell --seed 2019125 +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))