3frac (problem 3.3.3)

Average Error: 10.2 → 0.4

Time: 19.1s

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 r2298197 = 1.0;
        double r2298198 = x;
        double r2298199 = r2298198 + r2298197;
        double r2298200 = r2298197 / r2298199;
        double r2298201 = 2.0;
        double r2298202 = r2298201 / r2298198;
        double r2298203 = r2298200 - r2298202;
        double r2298204 = r2298198 - r2298197;
        double r2298205 = r2298197 / r2298204;
        double r2298206 = r2298203 + r2298205;
        return r2298206;
}

↓

double f(double x) {
        double r2298207 = 1.0;
        double r2298208 = x;
        double r2298209 = r2298208 - r2298207;
        double r2298210 = r2298207 / r2298209;
        double r2298211 = r2298208 + r2298207;
        double r2298212 = r2298207 / r2298211;
        double r2298213 = 2.0;
        double r2298214 = r2298213 / r2298208;
        double r2298215 = r2298212 - r2298214;
        double r2298216 = r2298210 + r2298215;
        double r2298217 = -288.66198405205364;
        bool r2298218 = r2298216 <= r2298217;
        double r2298219 = 2.0679515313825692e-25;
        bool r2298220 = r2298216 <= r2298219;
        double r2298221 = 7.0;
        double r2298222 = pow(r2298208, r2298221);
        double r2298223 = r2298213 / r2298222;
        double r2298224 = r2298208 * r2298208;
        double r2298225 = r2298214 / r2298224;
        double r2298226 = 5.0;
        double r2298227 = pow(r2298208, r2298226);
        double r2298228 = r2298213 / r2298227;
        double r2298229 = r2298225 + r2298228;
        double r2298230 = r2298223 + r2298229;
        double r2298231 = r2298220 ? r2298230 : r2298216;
        double r2298232 = r2298218 ? r2298216 : r2298231;
        return r2298232;
}

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))))