(- (/ x0 (- 1 x1)) x0)

Average Error: 7.8 → 6.1

Time: 11.2s

Precision: 64

• could not determine a ground truth for program pre

  1. x0 = 1.855
  2. x1 = 0.0186

\[x0 = 1.854999999999999982236431605997495353222 \land x1 = 2.090000000000000115064208161541614572343 \cdot 10^{-4} \lor x0 = 2.984999999999999875655021241982467472553 \land x1 = 0.01859999999999999847899445626353553961962\]

Math

\[\frac{x0}{1 - x1} - x0\]

↓

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

C

double f(double x0, double x1) {
        double r6758342 = x0;
        double r6758343 = 1.0;
        double r6758344 = x1;
        double r6758345 = r6758343 - r6758344;
        double r6758346 = r6758342 / r6758345;
        double r6758347 = r6758346 - r6758342;
        return r6758347;
}

↓

double f(double x0, double x1) {
        double r6758348 = x0;
        double r6758349 = 1.0;
        double r6758350 = x1;
        double r6758351 = r6758349 - r6758350;
        double r6758352 = r6758348 / r6758351;
        double r6758353 = r6758352 * r6758352;
        double r6758354 = r6758353 * r6758348;
        double r6758355 = r6758354 / r6758351;
        double r6758356 = r6758348 * r6758348;
        double r6758357 = r6758348 * r6758356;
        double r6758358 = r6758355 - r6758357;
        double r6758359 = r6758352 * r6758353;
        double r6758360 = cbrt(r6758359);
        double r6758361 = r6758360 * r6758348;
        double r6758362 = r6758361 + r6758356;
        double r6758363 = r6758353 + r6758362;
        double r6758364 = r6758358 / r6758363;
        return r6758364;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.8
Target	0.3
Herbie	6.1

\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

1. Initial program 7.8

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

3. Applied flip3-- 7.6

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

4. Simplified 7.2

   \[\leadsto \frac{\color{blue}{\left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right) \cdot \frac{x0}{1 - x1} - \left(x0 \cdot x0\right) \cdot x0}}{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} + \left(x0 \cdot x0 + \frac{x0}{1 - x1} \cdot x0\right)}\]
5. Using strategy rm

6. Applied associate-*r/ 6.1

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

8. Applied add-cbrt-cube 6.1

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

9. Applied add-cbrt-cube 6.1

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

10. Applied cbrt-undiv 6.1

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

11. Simplified 6.1

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

12. Final simplification 6.1

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

Reproduce

herbie shell --seed 2019170 
(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) (- 1.0 x1))

  (- (/ x0 (- 1.0 x1)) x0))