3frac (problem 3.3.3)

Average Error: 10.2 → 0.4

Time: 19.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le -288.66198405205364:\\ \;\;\;\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right)\\ \mathbf{elif}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le 2.0679515313825692 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right)\\ \end{array}\]

C

double f(double x) {
        double r3906267 = 1.0;
        double r3906268 = x;
        double r3906269 = r3906268 + r3906267;
        double r3906270 = r3906267 / r3906269;
        double r3906271 = 2.0;
        double r3906272 = r3906271 / r3906268;
        double r3906273 = r3906270 - r3906272;
        double r3906274 = r3906268 - r3906267;
        double r3906275 = r3906267 / r3906274;
        double r3906276 = r3906273 + r3906275;
        return r3906276;
}

↓

double f(double x) {
        double r3906277 = 1.0;
        double r3906278 = x;
        double r3906279 = r3906278 - r3906277;
        double r3906280 = r3906277 / r3906279;
        double r3906281 = r3906278 + r3906277;
        double r3906282 = r3906277 / r3906281;
        double r3906283 = 2.0;
        double r3906284 = r3906283 / r3906278;
        double r3906285 = r3906282 - r3906284;
        double r3906286 = r3906280 + r3906285;
        double r3906287 = -288.66198405205364;
        bool r3906288 = r3906286 <= r3906287;
        double r3906289 = 2.0679515313825692e-25;
        bool r3906290 = r3906286 <= r3906289;
        double r3906291 = 7.0;
        double r3906292 = pow(r3906278, r3906291);
        double r3906293 = r3906283 / r3906292;
        double r3906294 = r3906278 * r3906278;
        double r3906295 = r3906284 / r3906294;
        double r3906296 = 5.0;
        double r3906297 = pow(r3906278, r3906296);
        double r3906298 = r3906283 / r3906297;
        double r3906299 = r3906295 + r3906298;
        double r3906300 = r3906293 + r3906299;
        double r3906301 = r3906290 ? r3906300 : r3906286;
        double r3906302 = r3906288 ? r3906286 : r3906301;
        return r3906302;
}

Error

Bits error versus x

Target

Original	10.2
Target	0.3
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))) < -288.66198405205364 or 2.0679515313825692e-25 < (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))

   1. Initial program 0.4

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

   if -288.66198405205364 < (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))) < 2.0679515313825692e-25

   1. Initial program 20.2

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

   2. Taylor expanded around inf 0.7

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

   3. Simplified 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le -288.66198405205364:\\ \;\;\;\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right)\\ \mathbf{elif}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le 2.0679515313825692 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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))))