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

Average Error: 7.9 → 4.7

Time: 8.7s

Precision: 64

• could not determine a ground truth for program pre

  1. x0 = 2.985
  2. x1 = 0.000209

\[x0 = 1.855 \land x1 = 2.09000000000000012 \cdot 10^{-4} \lor x0 = 2.98499999999999988 \land x1 = 0.018599999999999998\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x1 \le 2.12089080810546861 \cdot 10^{-4}:\\ \;\;\;\;\frac{\log \left(e^{\left(\frac{x0}{1 \cdot 1 - x1 \cdot x1} \cdot \left(1 + x1\right)\right) \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{\sqrt[3]{{\left(\frac{x0}{1 - x1}\right)}^{3}} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

C

double f(double x0, double x1) {
        double r95627 = x0;
        double r95628 = 1.0;
        double r95629 = x1;
        double r95630 = r95628 - r95629;
        double r95631 = r95627 / r95630;
        double r95632 = r95631 - r95627;
        return r95632;
}

↓

double f(double x0, double x1) {
        double r95633 = x1;
        double r95634 = 0.00021208908081054686;
        bool r95635 = r95633 <= r95634;
        double r95636 = x0;
        double r95637 = 1.0;
        double r95638 = r95637 * r95637;
        double r95639 = r95633 * r95633;
        double r95640 = r95638 - r95639;
        double r95641 = r95636 / r95640;
        double r95642 = r95637 + r95633;
        double r95643 = r95641 * r95642;
        double r95644 = r95637 - r95633;
        double r95645 = r95636 / r95644;
        double r95646 = r95643 * r95645;
        double r95647 = r95636 * r95636;
        double r95648 = r95646 - r95647;
        double r95649 = exp(r95648);
        double r95650 = log(r95649);
        double r95651 = 3.0;
        double r95652 = pow(r95645, r95651);
        double r95653 = cbrt(r95652);
        double r95654 = r95653 + r95636;
        double r95655 = r95650 / r95654;
        double r95656 = r95645 * r95645;
        double r95657 = r95656 - r95647;
        double r95658 = exp(r95657);
        double r95659 = log(r95658);
        double r95660 = r95645 + r95636;
        double r95661 = r95659 / r95660;
        double r95662 = r95635 ? r95655 : r95661;
        return r95662;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.9
Target	0.3
Herbie	4.7

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

Derivation

1. Split input into 2 regimes

2. if x1 < 0.00021208908081054686

   1. Initial program 11.2

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

   3. Applied flip-- 11.4

      \[\leadsto \color{blue}{\frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{\frac{x0}{1 - x1} + x0}}\]
   4. Using strategy rm

   5. Applied flip-- 8.1

      \[\leadsto \frac{\frac{x0}{\color{blue}{\frac{1 \cdot 1 - x1 \cdot x1}{1 + x1}}} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{\frac{x0}{1 - x1} + x0}\]

   6. Applied associate-/r/ 8.1

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

   8. Applied add-log-exp 8.1

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

   9. Applied add-log-exp 8.1

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

   10. Applied diff-log 7.4

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

   11. Simplified 7.4

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

   13. Applied add-cbrt-cube 7.4

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

   14. Applied add-cbrt-cube 7.4

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

   15. Applied cbrt-undiv 7.4

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

   16. Simplified 7.4

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

   if 0.00021208908081054686 < x1

   1. Initial program 4.6

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

   3. Applied flip-- 3.2

      \[\leadsto \color{blue}{\frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{\frac{x0}{1 - x1} + x0}}\]
   4. Using strategy rm

   5. Applied add-log-exp 3.2

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

   6. Applied add-log-exp 3.2

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

   7. Applied diff-log 3.5

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

   8. Simplified 2.0

      \[\leadsto \frac{\log \color{blue}{\left(e^{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}}{\frac{x0}{1 - x1} + x0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \le 2.12089080810546861 \cdot 10^{-4}:\\ \;\;\;\;\frac{\log \left(e^{\left(\frac{x0}{1 \cdot 1 - x1 \cdot x1} \cdot \left(1 + x1\right)\right) \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{\sqrt[3]{{\left(\frac{x0}{1 - x1}\right)}^{3}} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(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))