Average Error: 7.9 → 5.7
Time: 3.3s
Precision: 64
\[x0 = 1.854999999999999982236431605997495353222 \land x1 = 2.090000000000000115064208161541614572343 \cdot 10^{-4} \lor x0 = 2.984999999999999875655021241982467472553 \land x1 = 0.01859999999999999847899445626353553961962\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\frac{\frac{\frac{x0}{\sqrt{1 - x1}}}{\sqrt{1 - x1}} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{\left(\sqrt[3]{\frac{x0}{1 - x1} + x0} \cdot \sqrt[3]{\frac{x0}{1 - x1} + x0}\right) \cdot \sqrt[3]{\frac{x0}{1 - x1} + x0}}\]
\frac{x0}{1 - x1} - x0
↓
\frac{\frac{\frac{x0}{\sqrt{1 - x1}}}{\sqrt{1 - x1}} \cdot \frac{x0}{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 r164336 = x0;
        double r164337 = 1.0;
        double r164338 = x1;
        double r164339 = r164337 - r164338;
        double r164340 = r164336 / r164339;
        double r164341 = r164340 - r164336;
        return r164341;
}
↓
double f(double x0, double x1) {
        double r164342 = x0;
        double r164343 = 1.0;
        double r164344 = x1;
        double r164345 = r164343 - r164344;
        double r164346 = sqrt(r164345);
        double r164347 = r164342 / r164346;
        double r164348 = r164347 / r164346;
        double r164349 = r164342 / r164345;
        double r164350 = r164348 * r164349;
        double r164351 = r164342 * r164342;
        double r164352 = r164350 - r164351;
        double r164353 = r164349 + r164342;
        double r164354 = cbrt(r164353);
        double r164355 = r164354 * r164354;
        double r164356 = r164355 * r164354;
        double r164357 = r164352 / r164356;
        return r164357;
}

Error

Bits error versus x0

Bits error versus x1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.9
Target0.2
Herbie5.7
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 7.9

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

  3. Applied flip--7.3

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

  5. Applied add-sqr-sqrt5.6

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

  6. Applied associate-/r*5.6

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

  8. Applied add-cube-cbrt5.7

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

  9. Final simplification5.7

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

Reproduce

herbie shell --seed 2019362 
(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))