3frac (problem 3.3.3)

Average Error: 9.7 → 0.2

Time: 48.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -118.48907173078551:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 109.57651104194328:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\right)\\ \end{array}\]

C

double f(double x) {
        double r3341411 = 1.0;
        double r3341412 = x;
        double r3341413 = r3341412 + r3341411;
        double r3341414 = r3341411 / r3341413;
        double r3341415 = 2.0;
        double r3341416 = r3341415 / r3341412;
        double r3341417 = r3341414 - r3341416;
        double r3341418 = r3341412 - r3341411;
        double r3341419 = r3341411 / r3341418;
        double r3341420 = r3341417 + r3341419;
        return r3341420;
}

↓

double f(double x) {
        double r3341421 = x;
        double r3341422 = -118.48907173078551;
        bool r3341423 = r3341421 <= r3341422;
        double r3341424 = 2.0;
        double r3341425 = 7.0;
        double r3341426 = pow(r3341421, r3341425);
        double r3341427 = r3341424 / r3341426;
        double r3341428 = r3341424 / r3341421;
        double r3341429 = r3341421 * r3341421;
        double r3341430 = r3341428 / r3341429;
        double r3341431 = 5.0;
        double r3341432 = pow(r3341421, r3341431);
        double r3341433 = r3341424 / r3341432;
        double r3341434 = r3341430 + r3341433;
        double r3341435 = r3341427 + r3341434;
        double r3341436 = 109.57651104194328;
        bool r3341437 = r3341421 <= r3341436;
        double r3341438 = 1.0;
        double r3341439 = r3341438 + r3341421;
        double r3341440 = r3341438 / r3341439;
        double r3341441 = r3341440 - r3341428;
        double r3341442 = r3341421 - r3341438;
        double r3341443 = r3341438 / r3341442;
        double r3341444 = r3341441 + r3341443;
        double r3341445 = 3.0;
        double r3341446 = pow(r3341421, r3341445);
        double r3341447 = r3341424 / r3341446;
        double r3341448 = r3341433 + r3341447;
        double r3341449 = r3341427 + r3341448;
        double r3341450 = r3341437 ? r3341444 : r3341449;
        double r3341451 = r3341423 ? r3341435 : r3341450;
        return r3341451;
}

Error

Bits error versus x

Target

Original	9.7
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 < -118.48907173078551

   1. Initial program 19.4

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

   3. Applied *-un-lft-identity 19.4

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

   4. Applied *-un-lft-identity 19.4

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

   5. Applied distribute-lft-out-- 19.4

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

   6. Applied associate-/r* 19.4

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

   7. Simplified 19.4

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

   9. Applied flip-+ 19.4

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

   10. Taylor expanded around -inf 0.5

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

   11. Simplified 0.5

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

   12. Taylor expanded around -inf 0.5

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

   13. Simplified 0.1

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

   if -118.48907173078551 < x < 109.57651104194328

   1. Initial program 0.0

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

   3. Applied *-un-lft-identity 0.0

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

   4. Applied *-un-lft-identity 0.0

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

   5. Applied distribute-lft-out-- 0.0

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

   6. Applied associate-/r* 0.0

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

   7. Simplified 0.0

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

   if 109.57651104194328 < x

   1. Initial program 19.7

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

   3. Applied *-un-lft-identity 19.7

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

   4. Applied *-un-lft-identity 19.7

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

   5. Applied distribute-lft-out-- 19.7

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

   6. Applied associate-/r* 19.7

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

   7. Simplified 19.7

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

   9. Applied flip-+ 19.7

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

   10. Taylor expanded around -inf 0.5

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

   11. Simplified 0.6

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

   13. Applied pow2 0.6

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

   14. Applied pow-plus 0.5

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

   15. Simplified 0.5

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -118.48907173078551:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 109.57651104194328:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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