3frac (problem 3.3.3)

Average Error: 9.6 → 0.2

Time: 21.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le -22.466348304101803:\\ \;\;\;\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right)\\ \mathbf{elif}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le 2.4319249413741473 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right)\\ \end{array}\]

C

double f(double x) {
        double r3788897 = 1.0;
        double r3788898 = x;
        double r3788899 = r3788898 + r3788897;
        double r3788900 = r3788897 / r3788899;
        double r3788901 = 2.0;
        double r3788902 = r3788901 / r3788898;
        double r3788903 = r3788900 - r3788902;
        double r3788904 = r3788898 - r3788897;
        double r3788905 = r3788897 / r3788904;
        double r3788906 = r3788903 + r3788905;
        return r3788906;
}

↓

double f(double x) {
        double r3788907 = 1.0;
        double r3788908 = x;
        double r3788909 = r3788908 - r3788907;
        double r3788910 = r3788907 / r3788909;
        double r3788911 = r3788908 + r3788907;
        double r3788912 = r3788907 / r3788911;
        double r3788913 = 2.0;
        double r3788914 = r3788913 / r3788908;
        double r3788915 = r3788912 - r3788914;
        double r3788916 = r3788910 + r3788915;
        double r3788917 = -22.466348304101803;
        bool r3788918 = r3788916 <= r3788917;
        double r3788919 = 2.4319249413741473e-12;
        bool r3788920 = r3788916 <= r3788919;
        double r3788921 = 7.0;
        double r3788922 = pow(r3788908, r3788921);
        double r3788923 = r3788913 / r3788922;
        double r3788924 = r3788908 * r3788908;
        double r3788925 = r3788914 / r3788924;
        double r3788926 = 5.0;
        double r3788927 = pow(r3788908, r3788926);
        double r3788928 = r3788913 / r3788927;
        double r3788929 = r3788925 + r3788928;
        double r3788930 = r3788923 + r3788929;
        double r3788931 = r3788920 ? r3788930 : r3788916;
        double r3788932 = r3788918 ? r3788916 : r3788931;
        return r3788932;
}

Error

Bits error versus x

Target

Original	9.6
Target	0.3
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))) < -22.466348304101803 or 2.4319249413741473e-12 < (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))

   1. Initial program 0.1

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

   if -22.466348304101803 < (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))) < 2.4319249413741473e-12

   1. Initial program 19.1

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

   2. Taylor expanded around inf 0.8

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

   3. Simplified 0.4

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

   4. Taylor expanded around -inf 0.8

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

   5. Simplified 0.4

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le -22.466348304101803:\\ \;\;\;\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right)\\ \mathbf{elif}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le 2.4319249413741473 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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