3frac (problem 3.3.3)

Average Error: 9.7 → 0.1

Time: 20.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -137.4367895734916373839951120316982269287:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{7}}\right)\\ \mathbf{elif}\;x \le 108.745999215554448369402962271124124527:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{7}}\right)\\ \end{array}\]

C

double f(double x) {
        double r3745283 = 1.0;
        double r3745284 = x;
        double r3745285 = r3745284 + r3745283;
        double r3745286 = r3745283 / r3745285;
        double r3745287 = 2.0;
        double r3745288 = r3745287 / r3745284;
        double r3745289 = r3745286 - r3745288;
        double r3745290 = r3745284 - r3745283;
        double r3745291 = r3745283 / r3745290;
        double r3745292 = r3745289 + r3745291;
        return r3745292;
}

↓

double f(double x) {
        double r3745293 = x;
        double r3745294 = -137.43678957349164;
        bool r3745295 = r3745293 <= r3745294;
        double r3745296 = 2.0;
        double r3745297 = 5.0;
        double r3745298 = pow(r3745293, r3745297);
        double r3745299 = r3745296 / r3745298;
        double r3745300 = r3745296 / r3745293;
        double r3745301 = r3745293 * r3745293;
        double r3745302 = r3745300 / r3745301;
        double r3745303 = 7.0;
        double r3745304 = pow(r3745293, r3745303);
        double r3745305 = r3745296 / r3745304;
        double r3745306 = r3745302 + r3745305;
        double r3745307 = r3745299 + r3745306;
        double r3745308 = 108.74599921555445;
        bool r3745309 = r3745293 <= r3745308;
        double r3745310 = 1.0;
        double r3745311 = r3745310 + r3745293;
        double r3745312 = r3745310 / r3745311;
        double r3745313 = r3745312 - r3745300;
        double r3745314 = r3745293 - r3745310;
        double r3745315 = r3745310 / r3745314;
        double r3745316 = r3745313 + r3745315;
        double r3745317 = r3745309 ? r3745316 : r3745307;
        double r3745318 = r3745295 ? r3745307 : r3745317;
        return r3745318;
}

Error

Bits error versus x

Target

Original	9.7
Target	0.2
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if x < -137.43678957349164 or 108.74599921555445 < x

   1. Initial program 19.5

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

   2. Taylor expanded around inf 0.4

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

   3. Simplified 0.4

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

   5. Applied associate-/r* 0.1

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

   if -137.43678957349164 < x < 108.74599921555445

   1. Initial program 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -137.4367895734916373839951120316982269287:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{7}}\right)\\ \mathbf{elif}\;x \le 108.745999215554448369402962271124124527:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{7}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))