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

Average Error: 7.9 → 6.5

Time: 26.6s

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

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

↓

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

C

double f(double x0, double x1) {
        double r6789530 = x0;
        double r6789531 = 1.0;
        double r6789532 = x1;
        double r6789533 = r6789531 - r6789532;
        double r6789534 = r6789530 / r6789533;
        double r6789535 = r6789534 - r6789530;
        return r6789535;
}

↓

double f(double x0, double x1) {
        double r6789536 = x0;
        double r6789537 = 1.0;
        double r6789538 = x1;
        double r6789539 = r6789537 - r6789538;
        double r6789540 = r6789536 / r6789539;
        double r6789541 = r6789540 * r6789540;
        double r6789542 = r6789541 / r6789539;
        double r6789543 = r6789536 * r6789536;
        double r6789544 = r6789542 - r6789543;
        double r6789545 = r6789536 * r6789544;
        double r6789546 = r6789543 * r6789543;
        double r6789547 = r6789546 * r6789546;
        double r6789548 = r6789540 * r6789536;
        double r6789549 = r6789548 * r6789548;
        double r6789550 = r6789549 * r6789549;
        double r6789551 = r6789547 - r6789550;
        double r6789552 = r6789549 + r6789546;
        double r6789553 = r6789551 / r6789552;
        double r6789554 = r6789539 * r6789553;
        double r6789555 = r6789543 - r6789548;
        double r6789556 = r6789548 * r6789555;
        double r6789557 = r6789554 + r6789556;
        double r6789558 = r6789539 * r6789555;
        double r6789559 = r6789557 / r6789558;
        double r6789560 = r6789545 / r6789559;
        return r6789560;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.9
Target	0.2
Herbie	6.5

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

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 7.4

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

6. Applied flip-+ 7.1

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

7. Applied associate-*r/ 7.1

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

8. Applied frac-add 7.1

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

10. Applied flip-- 6.5

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

11. Final simplification 6.5

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

Reproduce

herbie shell --seed 2019152 
(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))