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

Average Error: 7.9 → 6.1

Time: 11.6s

Precision: 64

• could not determine a ground truth for program pre

  1. x0 = 1.855
  2. x1 = 0.0186

\[x0 = 1.855 \land x1 = 0.000209 \lor x0 = 2.985 \land x1 = 0.0186\]

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 r7376379 = x0;
        double r7376380 = 1.0;
        double r7376381 = x1;
        double r7376382 = r7376380 - r7376381;
        double r7376383 = r7376379 / r7376382;
        double r7376384 = r7376383 - r7376379;
        return r7376384;
}

↓

double f(double x0, double x1) {
        double r7376385 = x0;
        double r7376386 = 1.0;
        double r7376387 = x1;
        double r7376388 = r7376386 - r7376387;
        double r7376389 = r7376385 / r7376388;
        double r7376390 = r7376389 * r7376389;
        double r7376391 = r7376390 * r7376385;
        double r7376392 = r7376391 / r7376388;
        double r7376393 = r7376385 * r7376385;
        double r7376394 = r7376385 * r7376393;
        double r7376395 = r7376392 - r7376394;
        double r7376396 = r7376389 * r7376390;
        double r7376397 = cbrt(r7376396);
        double r7376398 = r7376397 * r7376385;
        double r7376399 = r7376398 + r7376393;
        double r7376400 = r7376390 + r7376399;
        double r7376401 = r7376395 / r7376400;
        return r7376401;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.9
Target	0.2
Herbie	6.1

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

Derivation

1. Initial program 7.9

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

3. Applied flip3-- 7.7

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

4. Simplified 7.3

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

6. Applied associate-*l/ 6.1

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

8. Applied add-cbrt-cube 6.1

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

9. Applied add-cbrt-cube 6.1

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

10. Applied cbrt-undiv 6.1

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

11. Simplified 6.1

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

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 2019162 
(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))