qlog (example 3.10)

Average Error: 61.0 → 0.4

Time: 19.0s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r3142544 = 1.0;
        double r3142545 = x;
        double r3142546 = r3142544 - r3142545;
        double r3142547 = log(r3142546);
        double r3142548 = r3142544 + r3142545;
        double r3142549 = log(r3142548);
        double r3142550 = r3142547 / r3142549;
        return r3142550;
}

↓

double f(double x) {
        double r3142551 = -1.0;
        double r3142552 = x;
        double r3142553 = r3142552 * r3142552;
        double r3142554 = -0.5;
        double r3142555 = r3142553 * r3142554;
        double r3142556 = r3142552 - r3142555;
        double r3142557 = r3142551 - r3142556;
        return r3142557;
}

Error

Bits error versus x

Target

Original	61.0
Target	0.3
Herbie	0.4

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

Derivation

1. Initial program 61.0

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

2. Taylor expanded around 0 0.4

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

3. Simplified 0.4

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

4. Taylor expanded around 0 0.4

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

5. Simplified 0.4

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

6. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019164 
(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))))