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

Average Error: 7.8 → 4.4

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

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

↓

\[\begin{array}{l} \mathbf{if}\;x0 \le 1.874921874999999849009668650978710502386:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right)}^{3} + {\left(\mathsf{fma}\left(\frac{\sqrt[3]{x0}}{1 - x1}, {x0}^{\frac{2}{3}}, \log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right)\right)}^{3}}{\mathsf{fma}\left(\frac{\sqrt[3]{x0}}{1 - x1}, {x0}^{\frac{2}{3}}, \log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right) \cdot \left(\frac{\sqrt[3]{x0}}{1 - x1} \cdot {x0}^{\frac{2}{3}}\right) + \log \left(\sqrt{e^{x0}}\right) \cdot \log \left(\sqrt{e^{x0}}\right)}\\ \end{array}\]

C

double f(double x0, double x1) {
        double r171441 = x0;
        double r171442 = 1.0;
        double r171443 = x1;
        double r171444 = r171442 - r171443;
        double r171445 = r171441 / r171444;
        double r171446 = r171445 - r171441;
        return r171446;
}

↓

double f(double x0, double x1) {
        double r171447 = x0;
        double r171448 = 1.8749218749999998;
        bool r171449 = r171447 <= r171448;
        double r171450 = sqrt(r171447);
        double r171451 = 1.0;
        double r171452 = sqrt(r171451);
        double r171453 = x1;
        double r171454 = sqrt(r171453);
        double r171455 = r171452 + r171454;
        double r171456 = r171450 / r171455;
        double r171457 = r171452 - r171454;
        double r171458 = r171450 / r171457;
        double r171459 = -r171447;
        double r171460 = fma(r171456, r171458, r171459);
        double r171461 = 1.0;
        double r171462 = exp(r171447);
        double r171463 = sqrt(r171462);
        double r171464 = r171461 / r171463;
        double r171465 = log(r171464);
        double r171466 = 3.0;
        double r171467 = pow(r171465, r171466);
        double r171468 = cbrt(r171447);
        double r171469 = r171451 - r171453;
        double r171470 = r171468 / r171469;
        double r171471 = 0.6666666666666666;
        double r171472 = pow(r171447, r171471);
        double r171473 = fma(r171470, r171472, r171465);
        double r171474 = pow(r171473, r171466);
        double r171475 = r171467 + r171474;
        double r171476 = r171470 * r171472;
        double r171477 = r171473 * r171476;
        double r171478 = log(r171463);
        double r171479 = r171478 * r171478;
        double r171480 = r171477 + r171479;
        double r171481 = r171475 / r171480;
        double r171482 = r171449 ? r171460 : r171481;
        return r171482;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.8
Target	0.3
Herbie	4.4

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

Derivation

1. Split input into 2 regimes

2. if x0 < 1.8749218749999998

   1. Initial program 7.4

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

   3. Applied add-sqr-sqrt 7.4

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

   4. Applied add-sqr-sqrt 7.4

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

   5. Applied difference-of-squares 7.4

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

   6. Applied add-sqr-sqrt 7.4

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

   7. Applied times-frac 7.4

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

   8. Applied fma-neg 5.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)}\]

   if 1.8749218749999998 < x0

   1. Initial program 8.2

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

   3. Applied *-un-lft-identity 8.2

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

   4. Applied add-cube-cbrt 8.2

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

   5. Applied times-frac 8.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{1} \cdot \frac{\sqrt[3]{x0}}{1 - x1}} - x0\]

   6. Applied fma-neg 7.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{1}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)}\]
   7. Using strategy rm

   8. Applied add-log-exp 7.5

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

   9. Simplified 5.7

      \[\leadsto \log \color{blue}{\left(\frac{{\left(e^{{x0}^{\frac{2}{3}}}\right)}^{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}}{e^{x0}}\right)}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt 6.4

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

   12. Applied *-un-lft-identity 6.4

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

   13. Applied unpow-prod-down 6.4

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

   14. Applied times-frac 5.7

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

   15. Applied log-prod 5.7

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

   16. Simplified 5.7

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

   17. Simplified 5.7

       \[\leadsto \log \left(\frac{1}{\sqrt{e^{x0}}}\right) + \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{x0}}{1 - x1}, {x0}^{\frac{2}{3}}, \log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right)}\]
   18. Using strategy rm

   19. Applied flip3-+ 3.5

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

   20. Simplified 3.5

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

4. Final simplification 4.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x0 \le 1.874921874999999849009668650978710502386:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right)}^{3} + {\left(\mathsf{fma}\left(\frac{\sqrt[3]{x0}}{1 - x1}, {x0}^{\frac{2}{3}}, \log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right)\right)}^{3}}{\mathsf{fma}\left(\frac{\sqrt[3]{x0}}{1 - x1}, {x0}^{\frac{2}{3}}, \log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right) \cdot \left(\frac{\sqrt[3]{x0}}{1 - x1} \cdot {x0}^{\frac{2}{3}}\right) + \log \left(\sqrt{e^{x0}}\right) \cdot \log \left(\sqrt{e^{x0}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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))