qlog (example 3.10)

Average Error: 60.9 → 0.5

Time: 17.2s

Precision: 64

\[-1 \lt x \land x \lt 1\]

Math

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

↓

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

C

double f(double x) {
        double r2618987 = 1.0;
        double r2618988 = x;
        double r2618989 = r2618987 - r2618988;
        double r2618990 = log(r2618989);
        double r2618991 = r2618987 + r2618988;
        double r2618992 = log(r2618991);
        double r2618993 = r2618990 / r2618992;
        return r2618993;
}

↓

double f(double x) {
        double r2618994 = x;
        double r2618995 = r2618994 * r2618994;
        double r2618996 = -0.5;
        double r2618997 = r2618995 * r2618996;
        double r2618998 = r2618997 - r2618994;
        double r2618999 = -1.0;
        double r2619000 = r2618998 + r2618999;
        return r2619000;
}

Error

Bits error versus x

Target

Original	60.9
Target	0.4
Herbie	0.5

\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

1. Initial program 60.9

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

2. Taylor expanded around 0 0.5

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

3. Simplified 0.5

   \[\leadsto \color{blue}{\left(-1 + \left(x \cdot x\right) \cdot \frac{-1}{2}\right) - x}\]
4. Using strategy rm

5. Applied associate--l+ 0.5

   \[\leadsto \color{blue}{-1 + \left(\left(x \cdot x\right) \cdot \frac{-1}{2} - x\right)}\]

6. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019141 
(FPCore (x)
  :name "qlog (example 3.10)"
  :pre (and (< -1 x) (< x 1))

  :herbie-target
  (- (+ (+ (+ 1 x) (/ (* x x) 2)) (* 5/12 (pow x 3))))

  (/ (log (- 1 x)) (log (+ 1 x))))