logq (problem 3.4.3)

Average Error: 58.6 → 0.2

Time: 15.3s

Precision: 64

Math

\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]

↓

\[-\mathsf{fma}\left({\varepsilon}^{5}, \frac{2}{5}, 2 \cdot \varepsilon + \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{2}{3}\right)\right)\]

C

double f(double eps) {
        double r1380532 = 1.0;
        double r1380533 = eps;
        double r1380534 = r1380532 - r1380533;
        double r1380535 = r1380532 + r1380533;
        double r1380536 = r1380534 / r1380535;
        double r1380537 = log(r1380536);
        return r1380537;
}

↓

double f(double eps) {
        double r1380538 = eps;
        double r1380539 = 5.0;
        double r1380540 = pow(r1380538, r1380539);
        double r1380541 = 0.4;
        double r1380542 = 2.0;
        double r1380543 = r1380542 * r1380538;
        double r1380544 = r1380538 * r1380538;
        double r1380545 = 0.6666666666666666;
        double r1380546 = r1380544 * r1380545;
        double r1380547 = r1380538 * r1380546;
        double r1380548 = r1380543 + r1380547;
        double r1380549 = fma(r1380540, r1380541, r1380548);
        double r1380550 = -r1380549;
        return r1380550;
}

Error

Bits error versus eps

Target

Original	58.6
Target	0.2
Herbie	0.2

\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

1. Initial program 58.6

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]

2. Taylor expanded around 0 0.2

   \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot {\varepsilon}^{3} + \left(\frac{2}{5} \cdot {\varepsilon}^{5} + 2 \cdot \varepsilon\right)\right)}\]

3. Simplified 0.2

   \[\leadsto \color{blue}{-\mathsf{fma}\left({\varepsilon}^{5}, \frac{2}{5}, \varepsilon \cdot \left(2 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}\]
4. Using strategy rm

5. Applied distribute-rgt-in 0.2

   \[\leadsto -\mathsf{fma}\left({\varepsilon}^{5}, \frac{2}{5}, \color{blue}{2 \cdot \varepsilon + \left(\frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon}\right)\]

6. Final simplification 0.2

   \[\leadsto -\mathsf{fma}\left({\varepsilon}^{5}, \frac{2}{5}, 2 \cdot \varepsilon + \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{2}{3}\right)\right)\]

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))