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

Average Error: 7.9 → 6.5

Time: 28.4s

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 r8203740 = x0;
        double r8203741 = 1.0;
        double r8203742 = x1;
        double r8203743 = r8203741 - r8203742;
        double r8203744 = r8203740 / r8203743;
        double r8203745 = r8203744 - r8203740;
        return r8203745;
}

↓

double f(double x0, double x1) {
        double r8203746 = x0;
        double r8203747 = 1.0;
        double r8203748 = x1;
        double r8203749 = r8203747 - r8203748;
        double r8203750 = r8203746 / r8203749;
        double r8203751 = r8203750 * r8203750;
        double r8203752 = r8203751 / r8203749;
        double r8203753 = r8203746 * r8203746;
        double r8203754 = r8203752 - r8203753;
        double r8203755 = r8203746 * r8203754;
        double r8203756 = r8203753 * r8203753;
        double r8203757 = r8203756 * r8203756;
        double r8203758 = r8203750 * r8203746;
        double r8203759 = r8203758 * r8203758;
        double r8203760 = r8203759 * r8203759;
        double r8203761 = r8203757 - r8203760;
        double r8203762 = r8203759 + r8203756;
        double r8203763 = r8203761 / r8203762;
        double r8203764 = r8203749 * r8203763;
        double r8203765 = r8203753 - r8203758;
        double r8203766 = r8203758 * r8203765;
        double r8203767 = r8203764 + r8203766;
        double r8203768 = r8203749 * r8203765;
        double r8203769 = r8203767 / r8203768;
        double r8203770 = r8203755 / r8203769;
        return r8203770;
}

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 2019141 
(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))