Average Error: 7.8 → 5.6
Time: 3.1s
Precision: 64
\[x0 = 1.855 \land x1 = 2.09000000000000012 \cdot 10^{-4} \lor x0 = 2.98499999999999988 \land x1 = 0.018599999999999998\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\frac{\frac{x0}{1 - x1} \cdot \frac{\frac{x0}{\sqrt{1 - x1}}}{\sqrt{1 - x1}} - x0 \cdot x0}{\left(\sqrt[3]{\frac{x0}{1 - x1} + x0} \cdot \sqrt[3]{\frac{x0}{1 - x1} + x0}\right) \cdot \sqrt[3]{\frac{x0}{1 - x1} + x0}}\]
\frac{x0}{1 - x1} - x0
↓
\frac{\frac{x0}{1 - x1} \cdot \frac{\frac{x0}{\sqrt{1 - x1}}}{\sqrt{1 - x1}} - x0 \cdot x0}{\left(\sqrt[3]{\frac{x0}{1 - x1} + x0} \cdot \sqrt[3]{\frac{x0}{1 - x1} + x0}\right) \cdot \sqrt[3]{\frac{x0}{1 - x1} + x0}}
double f(double x0, double x1) {
        double r176536 = x0;
        double r176537 = 1.0;
        double r176538 = x1;
        double r176539 = r176537 - r176538;
        double r176540 = r176536 / r176539;
        double r176541 = r176540 - r176536;
        return r176541;
}
↓
double f(double x0, double x1) {
        double r176542 = x0;
        double r176543 = 1.0;
        double r176544 = x1;
        double r176545 = r176543 - r176544;
        double r176546 = r176542 / r176545;
        double r176547 = sqrt(r176545);
        double r176548 = r176542 / r176547;
        double r176549 = r176548 / r176547;
        double r176550 = r176546 * r176549;
        double r176551 = r176542 * r176542;
        double r176552 = r176550 - r176551;
        double r176553 = r176546 + r176542;
        double r176554 = cbrt(r176553);
        double r176555 = r176554 * r176554;
        double r176556 = r176555 * r176554;
        double r176557 = r176552 / r176556;
        return r176557;
}

Error

Bits error versus x0

Bits error versus x1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.8
Target0.3
Herbie5.6
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 7.8

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

  3. Applied flip--7.2

    \[\leadsto \color{blue}{\frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{\frac{x0}{1 - x1} + x0}}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt5.6

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

  6. Applied associate-/r*5.6

    \[\leadsto \frac{\frac{x0}{1 - x1} \cdot \color{blue}{\frac{\frac{x0}{\sqrt{1 - x1}}}{\sqrt{1 - x1}}} - x0 \cdot x0}{\frac{x0}{1 - x1} + x0}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt5.6

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

  9. Final simplification5.6

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

Reproduce

herbie shell --seed 2020018 
(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))