3frac (problem 3.3.3)

Average Error: 9.4 → 0.2

Time: 9.5s

Precision: 64

Math

\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -111.0202622422814613400987582281231880188:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{\frac{2}{x}}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 109.8975046823589565292422776110470294952:\\ \;\;\;\;1 \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)\right)\\ \end{array}\]

C

double f(double x) {
        double r170421 = 1.0;
        double r170422 = x;
        double r170423 = r170422 + r170421;
        double r170424 = r170421 / r170423;
        double r170425 = 2.0;
        double r170426 = r170425 / r170422;
        double r170427 = r170424 - r170426;
        double r170428 = r170422 - r170421;
        double r170429 = r170421 / r170428;
        double r170430 = r170427 + r170429;
        return r170430;
}

↓

double f(double x) {
        double r170431 = x;
        double r170432 = -111.02026224228146;
        bool r170433 = r170431 <= r170432;
        double r170434 = 1.0;
        double r170435 = 2.0;
        double r170436 = 7.0;
        double r170437 = pow(r170431, r170436);
        double r170438 = r170434 / r170437;
        double r170439 = 5.0;
        double r170440 = pow(r170431, r170439);
        double r170441 = r170434 / r170440;
        double r170442 = r170435 / r170431;
        double r170443 = r170431 * r170431;
        double r170444 = r170442 / r170443;
        double r170445 = fma(r170435, r170441, r170444);
        double r170446 = fma(r170435, r170438, r170445);
        double r170447 = r170434 * r170446;
        double r170448 = 109.89750468235896;
        bool r170449 = r170431 <= r170448;
        double r170450 = 1.0;
        double r170451 = r170431 + r170450;
        double r170452 = r170450 / r170451;
        double r170453 = r170452 - r170442;
        double r170454 = r170431 - r170450;
        double r170455 = r170450 / r170454;
        double r170456 = r170453 + r170455;
        double r170457 = r170434 * r170456;
        double r170458 = 3.0;
        double r170459 = pow(r170431, r170458);
        double r170460 = r170435 / r170459;
        double r170461 = fma(r170435, r170441, r170460);
        double r170462 = fma(r170435, r170438, r170461);
        double r170463 = r170434 * r170462;
        double r170464 = r170449 ? r170457 : r170463;
        double r170465 = r170433 ? r170447 : r170464;
        return r170465;
}

Error

Bits error versus x

Target

Original	9.4
Target	0.3
Herbie	0.2

\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

1. Split input into 3 regimes

2. if x < -111.02026224228146

   1. Initial program 18.3

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
   2. Using strategy rm

   3. Applied add-exp-log 18.9

      \[\leadsto \color{blue}{e^{\log \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 18.9

      \[\leadsto e^{\log \color{blue}{\left(1 \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)\right)}}\]

   6. Applied log-prod 18.9

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

   7. Applied exp-sum 18.9

      \[\leadsto \color{blue}{e^{\log 1} \cdot e^{\log \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)}}\]

   8. Simplified 18.9

      \[\leadsto \color{blue}{1} \cdot e^{\log \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)}\]

   9. Simplified 18.3

      \[\leadsto 1 \cdot \color{blue}{\left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)}\]

   10. Taylor expanded around inf 0.4

       \[\leadsto 1 \cdot \color{blue}{\left(2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}\right)\right)}\]

   11. Simplified 0.4

       \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)\right)}\]
   12. Using strategy rm

   13. Applied cube-mult 0.5

       \[\leadsto 1 \cdot \mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right)\]

   14. Applied associate-/r* 0.1

       \[\leadsto 1 \cdot \mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \color{blue}{\frac{\frac{2}{x}}{x \cdot x}}\right)\right)\]

   if -111.02026224228146 < x < 109.89750468235896

   1. Initial program 0.0

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
   2. Using strategy rm

   3. Applied add-exp-log 35.6

      \[\leadsto \color{blue}{e^{\log \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 35.6

      \[\leadsto e^{\log \color{blue}{\left(1 \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)\right)}}\]

   6. Applied log-prod 35.6

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

   7. Applied exp-sum 35.6

      \[\leadsto \color{blue}{e^{\log 1} \cdot e^{\log \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)}}\]

   8. Simplified 35.6

      \[\leadsto \color{blue}{1} \cdot e^{\log \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)}\]

   9. Simplified 0.0

      \[\leadsto 1 \cdot \color{blue}{\left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)}\]

   if 109.89750468235896 < x

   1. Initial program 19.4

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
   2. Using strategy rm

   3. Applied add-exp-log 19.4

      \[\leadsto \color{blue}{e^{\log \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 19.4

      \[\leadsto e^{\log \color{blue}{\left(1 \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)\right)}}\]

   6. Applied log-prod 19.4

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

   7. Applied exp-sum 19.4

      \[\leadsto \color{blue}{e^{\log 1} \cdot e^{\log \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)}}\]

   8. Simplified 19.4

      \[\leadsto \color{blue}{1} \cdot e^{\log \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)}\]

   9. Simplified 19.4

      \[\leadsto 1 \cdot \color{blue}{\left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)}\]

   10. Taylor expanded around inf 0.6

       \[\leadsto 1 \cdot \color{blue}{\left(2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}\right)\right)}\]

   11. Simplified 0.6

       \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -111.0202622422814613400987582281231880188:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{\frac{2}{x}}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 109.8975046823589565292422776110470294952:\\ \;\;\;\;1 \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))