logq (problem 3.4.3)

Average Error: 58.6 → 0.2

Time: 15.9s

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 r1426089 = 1.0;
        double r1426090 = eps;
        double r1426091 = r1426089 - r1426090;
        double r1426092 = r1426089 + r1426090;
        double r1426093 = r1426091 / r1426092;
        double r1426094 = log(r1426093);
        return r1426094;
}

↓

double f(double eps) {
        double r1426095 = eps;
        double r1426096 = 5.0;
        double r1426097 = pow(r1426095, r1426096);
        double r1426098 = 0.4;
        double r1426099 = 2.0;
        double r1426100 = r1426099 * r1426095;
        double r1426101 = r1426095 * r1426095;
        double r1426102 = 0.6666666666666666;
        double r1426103 = r1426101 * r1426102;
        double r1426104 = r1426095 * r1426103;
        double r1426105 = r1426100 + r1426104;
        double r1426106 = fma(r1426097, r1426098, r1426105);
        double r1426107 = -r1426106;
        return r1426107;
}

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-lft-in 0.2

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

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 2019153 +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))))