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

Average Error: 7.9 → 4.4

Time: 3.9s

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

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

↓

\[\begin{array}{l} \mathbf{if}\;x0 \le 1.87492187499999985:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right)}^{3} + {\left(\mathsf{fma}\left(\frac{\sqrt[3]{x0}}{1 - x1}, {x0}^{\frac{2}{3}}, \log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right)\right)}^{3}}{\mathsf{fma}\left(\frac{\sqrt[3]{x0}}{1 - x1}, {x0}^{\frac{2}{3}}, \log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right) \cdot \left(\frac{\sqrt[3]{x0}}{1 - x1} \cdot {x0}^{\frac{2}{3}}\right) + \log \left(\sqrt{e^{x0}}\right) \cdot \log \left(\sqrt{e^{x0}}\right)}\\ \end{array}\]

C

double f(double x0, double x1) {
        double r155484 = x0;
        double r155485 = 1.0;
        double r155486 = x1;
        double r155487 = r155485 - r155486;
        double r155488 = r155484 / r155487;
        double r155489 = r155488 - r155484;
        return r155489;
}

↓

double f(double x0, double x1) {
        double r155490 = x0;
        double r155491 = 1.8749218749999998;
        bool r155492 = r155490 <= r155491;
        double r155493 = sqrt(r155490);
        double r155494 = 1.0;
        double r155495 = sqrt(r155494);
        double r155496 = x1;
        double r155497 = sqrt(r155496);
        double r155498 = r155495 + r155497;
        double r155499 = r155493 / r155498;
        double r155500 = r155495 - r155497;
        double r155501 = r155493 / r155500;
        double r155502 = -r155490;
        double r155503 = fma(r155499, r155501, r155502);
        double r155504 = 1.0;
        double r155505 = exp(r155490);
        double r155506 = sqrt(r155505);
        double r155507 = r155504 / r155506;
        double r155508 = log(r155507);
        double r155509 = 3.0;
        double r155510 = pow(r155508, r155509);
        double r155511 = cbrt(r155490);
        double r155512 = r155494 - r155496;
        double r155513 = r155511 / r155512;
        double r155514 = 0.6666666666666666;
        double r155515 = pow(r155490, r155514);
        double r155516 = fma(r155513, r155515, r155508);
        double r155517 = pow(r155516, r155509);
        double r155518 = r155510 + r155517;
        double r155519 = r155513 * r155515;
        double r155520 = r155516 * r155519;
        double r155521 = log(r155506);
        double r155522 = r155521 * r155521;
        double r155523 = r155520 + r155522;
        double r155524 = r155518 / r155523;
        double r155525 = r155492 ? r155503 : r155524;
        return r155525;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.9
Target	0.2
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 5.3

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

   if 1.8749218749999998 < x0

   1. Initial program 8.4

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

   3. Applied *-un-lft-identity 8.4

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

   4. Applied add-cube-cbrt 8.4

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

   5. Applied times-frac 8.4

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

   6. Applied fma-neg 7.2

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

   8. Applied add-log-exp 7.6

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

   9. Simplified 5.9

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

   11. Applied add-sqr-sqrt 6.7

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

   12. Applied *-un-lft-identity 6.7

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

   13. Applied unpow-prod-down 6.7

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

   14. Applied times-frac 5.9

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

   15. Applied log-prod 5.8

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

   16. Simplified 5.8

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

   17. Simplified 5.8

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

   19. Applied flip3-+ 3.6

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x0 \le 1.87492187499999985:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right)}^{3} + {\left(\mathsf{fma}\left(\frac{\sqrt[3]{x0}}{1 - x1}, {x0}^{\frac{2}{3}}, \log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right)\right)}^{3}}{\mathsf{fma}\left(\frac{\sqrt[3]{x0}}{1 - x1}, {x0}^{\frac{2}{3}}, \log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right) \cdot \left(\frac{\sqrt[3]{x0}}{1 - x1} \cdot {x0}^{\frac{2}{3}}\right) + \log \left(\sqrt{e^{x0}}\right) \cdot \log \left(\sqrt{e^{x0}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 +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))