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

Average Error: 7.8 → 3.3

Time: 6.7s

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

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

↓

\[\frac{x0 \cdot 2}{\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} \cdot \left(x0 + \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}\right) + x0 \cdot x0} \cdot \frac{\log \left(\sqrt{e^{{\left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}\right)}^{3} - {x0}^{3}}}\right)}{\frac{x0}{1 - x1} + x0}\]

C

double f(double x0, double x1) {
        double r169306 = x0;
        double r169307 = 1.0;
        double r169308 = x1;
        double r169309 = r169307 - r169308;
        double r169310 = r169306 / r169309;
        double r169311 = r169310 - r169306;
        return r169311;
}

↓

double f(double x0, double x1) {
        double r169312 = x0;
        double r169313 = 2.0;
        double r169314 = r169312 * r169313;
        double r169315 = 1.0;
        double r169316 = x1;
        double r169317 = r169315 - r169316;
        double r169318 = r169317 * r169317;
        double r169319 = r169312 / r169318;
        double r169320 = r169312 + r169319;
        double r169321 = r169319 * r169320;
        double r169322 = r169312 * r169312;
        double r169323 = r169321 + r169322;
        double r169324 = r169314 / r169323;
        double r169325 = 3.0;
        double r169326 = pow(r169319, r169325);
        double r169327 = pow(r169312, r169325);
        double r169328 = r169326 - r169327;
        double r169329 = exp(r169328);
        double r169330 = sqrt(r169329);
        double r169331 = log(r169330);
        double r169332 = r169312 / r169317;
        double r169333 = r169332 + r169312;
        double r169334 = r169331 / r169333;
        double r169335 = r169324 * r169334;
        return r169335;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.8
Target	0.3
Herbie	3.3

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

Derivation

1. Initial program 7.8

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

3. Applied flip-- 7.2

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

4. Simplified 6.3

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

6. Applied flip3-- 5.0

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

7. Simplified 5.0

   \[\leadsto \frac{x0 \cdot \frac{{\left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}\right)}^{3} - {x0}^{3}}{\color{blue}{x0 \cdot \left(x0 + \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}\right) + \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}}{\frac{x0}{1 - x1} + x0}\]
8. Using strategy rm

9. Applied add-log-exp 5.0

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

10. Applied add-log-exp 5.0

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

11. Applied diff-log 4.6

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

12. Simplified 4.6

    \[\leadsto \frac{x0 \cdot \frac{\log \color{blue}{\left(e^{{\left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}\right)}^{3} - {x0}^{3}}\right)}}{x0 \cdot \left(x0 + \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}\right) + \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]
13. Using strategy rm

14. Applied add-sqr-sqrt 3.8

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

15. Applied log-prod 3.3

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

16. Final simplification 3.3

    \[\leadsto \frac{x0 \cdot 2}{\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} \cdot \left(x0 + \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}\right) + x0 \cdot x0} \cdot \frac{\log \left(\sqrt{e^{{\left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}\right)}^{3} - {x0}^{3}}}\right)}{\frac{x0}{1 - x1} + x0}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x0 x1)
  :name "(- (/ x0 (- 1 x1)) x0)"
  :precision binary64
  :pre (or (and (== x0 1.855) (== x1 2.09000000000000012e-4)) (and (== x0 2.98499999999999988) (== x1 0.018599999999999998)))

  :herbie-target
  (/ (* x0 x1) (- 1 x1))

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