qlog (example 3.10)

Average Error: 61.0 → 0.4

Time: 18.6s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r2031067 = 1.0;
        double r2031068 = x;
        double r2031069 = r2031067 - r2031068;
        double r2031070 = log(r2031069);
        double r2031071 = r2031067 + r2031068;
        double r2031072 = log(r2031071);
        double r2031073 = r2031070 / r2031072;
        return r2031073;
}

↓

double f(double x) {
        double r2031074 = -1.0;
        double r2031075 = x;
        double r2031076 = 0.5;
        double r2031077 = r2031076 * r2031075;
        double r2031078 = r2031077 * r2031075;
        double r2031079 = r2031075 + r2031078;
        double r2031080 = r2031074 - r2031079;
        return r2031080;
}

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 0.5

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

3. Simplified 0.5

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

5. Applied associate--l- 0.4

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

6. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019151 
(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))))