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

Average Error: 7.9 → 4.7

Time: 4.0s

Precision: 64

• could not determine a ground truth for program pre

  1. x0 = 2.985
  2. x1 = 0.000209

\[x0 = 1.854999999999999982236431605997495353222 \land x1 = 2.090000000000000115064208161541614572343 \cdot 10^{-4} \lor x0 = 2.984999999999999875655021241982467472553 \land x1 = 0.01859999999999999847899445626353553961962\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x1 \le 2.12089080810546861321705391922876060562 \cdot 10^{-4}:\\ \;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \left(\frac{\sqrt{x0}}{\sqrt{1 - x1}} \cdot \frac{\sqrt{x0}}{\sqrt{1 - x1}}\right) - x0 \cdot x0}\right)}{\sqrt[3]{{\left(\frac{x0}{1 - x1}\right)}^{3}} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

C

double f(double x0, double x1) {
        double r144655 = x0;
        double r144656 = 1.0;
        double r144657 = x1;
        double r144658 = r144656 - r144657;
        double r144659 = r144655 / r144658;
        double r144660 = r144659 - r144655;
        return r144660;
}

↓

double f(double x0, double x1) {
        double r144661 = x1;
        double r144662 = 0.00021208908081054686;
        bool r144663 = r144661 <= r144662;
        double r144664 = x0;
        double r144665 = 1.0;
        double r144666 = r144665 - r144661;
        double r144667 = r144664 / r144666;
        double r144668 = sqrt(r144664);
        double r144669 = sqrt(r144666);
        double r144670 = r144668 / r144669;
        double r144671 = r144670 * r144670;
        double r144672 = r144667 * r144671;
        double r144673 = r144664 * r144664;
        double r144674 = r144672 - r144673;
        double r144675 = exp(r144674);
        double r144676 = log(r144675);
        double r144677 = 3.0;
        double r144678 = pow(r144667, r144677);
        double r144679 = cbrt(r144678);
        double r144680 = r144679 + r144664;
        double r144681 = r144676 / r144680;
        double r144682 = r144667 * r144667;
        double r144683 = r144682 - r144673;
        double r144684 = exp(r144683);
        double r144685 = log(r144684);
        double r144686 = r144667 + r144664;
        double r144687 = r144685 / r144686;
        double r144688 = r144663 ? r144681 : r144687;
        return r144688;
}

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 add-sqr-sqrt 8.1

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

   6. Applied add-sqr-sqrt 8.1

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

   7. Applied times-frac 8.1

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

   9. Applied add-log-exp 8.1

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

   10. Applied add-log-exp 8.1

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

   11. Applied diff-log 7.4

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

   12. Simplified 7.4

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

   14. Applied add-cbrt-cube 7.4

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

   15. Applied add-cbrt-cube 7.4

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

   16. Applied cbrt-undiv 7.4

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

   17. Simplified 7.4

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \le 2.12089080810546861321705391922876060562 \cdot 10^{-4}:\\ \;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \left(\frac{\sqrt{x0}}{\sqrt{1 - x1}} \cdot \frac{\sqrt{x0}}{\sqrt{1 - x1}}\right) - x0 \cdot x0}\right)}{\sqrt[3]{{\left(\frac{x0}{1 - x1}\right)}^{3}} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (x0 x1)
  :name "(- (/ x0 (- 1 x1)) x0)"
  :precision binary64
  :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))