3frac (problem 3.3.3)

Average Error: 9.6 → 0.4

Time: 16.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -648.3628771614285142277367413043975830078:\\ \;\;\;\;\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x} + \frac{1}{x - 1}\\ \mathbf{elif}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 1.063446409383349278154895500847487710416 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{\frac{2}{x}}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \end{array}\]

C

double f(double x) {
        double r103454 = 1.0;
        double r103455 = x;
        double r103456 = r103455 + r103454;
        double r103457 = r103454 / r103456;
        double r103458 = 2.0;
        double r103459 = r103458 / r103455;
        double r103460 = r103457 - r103459;
        double r103461 = r103455 - r103454;
        double r103462 = r103454 / r103461;
        double r103463 = r103460 + r103462;
        return r103463;
}

↓

double f(double x) {
        double r103464 = 1.0;
        double r103465 = x;
        double r103466 = r103465 + r103464;
        double r103467 = r103464 / r103466;
        double r103468 = 2.0;
        double r103469 = r103468 / r103465;
        double r103470 = r103467 - r103469;
        double r103471 = r103465 - r103464;
        double r103472 = r103464 / r103471;
        double r103473 = r103470 + r103472;
        double r103474 = -648.3628771614285;
        bool r103475 = r103473 <= r103474;
        double r103476 = r103464 * r103465;
        double r103477 = r103466 * r103468;
        double r103478 = r103476 - r103477;
        double r103479 = r103466 * r103465;
        double r103480 = r103478 / r103479;
        double r103481 = r103480 + r103472;
        double r103482 = 1.0634464093833493e-11;
        bool r103483 = r103473 <= r103482;
        double r103484 = 7.0;
        double r103485 = pow(r103465, r103484);
        double r103486 = r103468 / r103485;
        double r103487 = 5.0;
        double r103488 = pow(r103465, r103487);
        double r103489 = r103468 / r103488;
        double r103490 = r103469 / r103465;
        double r103491 = r103490 / r103465;
        double r103492 = r103489 + r103491;
        double r103493 = r103486 + r103492;
        double r103494 = r103483 ? r103493 : r103473;
        double r103495 = r103475 ? r103481 : r103494;
        return r103495;
}

Error

Bits error versus x

Target

Original	9.6
Target	0.2
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))) < -648.3628771614285

   1. Initial program 0.0

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

   3. Applied frac-sub 0.0

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

   if -648.3628771614285 < (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))) < 1.0634464093833493e-11

   1. Initial program 19.3

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

   2. Taylor expanded around inf 1.0

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

   3. Simplified 1.0

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

   5. Applied unpow3 1.0

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

   6. Applied associate-/r* 0.6

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

   7. Simplified 0.6

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

   if 1.0634464093833493e-11 < (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))

   1. Initial program 0.2

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

   3. Applied frac-sub 0.2

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

   5. Applied *-un-lft-identity 0.2

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

   6. Applied *-un-lft-identity 0.2

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

   7. Applied distribute-lft-out 0.2

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

   8. Simplified 0.2

      \[\leadsto 1 \cdot \color{blue}{\left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -648.3628771614285142277367413043975830078:\\ \;\;\;\;\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x} + \frac{1}{x - 1}\\ \mathbf{elif}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 1.063446409383349278154895500847487710416 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{\frac{2}{x}}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(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))))