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

Average Error: 7.8 → 6.4

Time: 27.5s

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 r6668978 = x0;
        double r6668979 = 1.0;
        double r6668980 = x1;
        double r6668981 = r6668979 - r6668980;
        double r6668982 = r6668978 / r6668981;
        double r6668983 = r6668982 - r6668978;
        return r6668983;
}

↓

double f(double x0, double x1) {
        double r6668984 = x0;
        double r6668985 = 1.0;
        double r6668986 = x1;
        double r6668987 = r6668985 - r6668986;
        double r6668988 = r6668984 / r6668987;
        double r6668989 = r6668988 * r6668988;
        double r6668990 = r6668989 / r6668987;
        double r6668991 = r6668984 * r6668984;
        double r6668992 = r6668990 - r6668991;
        double r6668993 = r6668984 * r6668992;
        double r6668994 = r6668991 * r6668991;
        double r6668995 = r6668994 * r6668994;
        double r6668996 = r6668988 * r6668984;
        double r6668997 = r6668996 * r6668996;
        double r6668998 = r6668997 * r6668997;
        double r6668999 = r6668995 - r6668998;
        double r6669000 = r6668997 + r6668994;
        double r6669001 = r6668999 / r6669000;
        double r6669002 = r6668987 * r6669001;
        double r6669003 = r6668991 - r6668996;
        double r6669004 = r6668996 * r6669003;
        double r6669005 = r6669002 + r6669004;
        double r6669006 = r6668987 * r6669003;
        double r6669007 = r6669005 / r6669006;
        double r6669008 = r6668993 / r6669007;
        return r6669008;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.8
Target	0.3
Herbie	6.4

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

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 7.3

   \[\leadsto \frac{\color{blue}{x0 \cdot \left(\frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}}{1 - x1} - x0 \cdot x0\right)}}{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} + \left(x0 \cdot x0 + \frac{x0}{1 - x1} \cdot x0\right)}\]
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-- 6.4

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

11. Final simplification 6.4

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

Reproduce

herbie shell --seed 2019138 
(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))