qlog (example 3.10)

Average Error: 61.0 → 0.4

Time: 27.3s

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 r2763961 = 1.0;
        double r2763962 = x;
        double r2763963 = r2763961 - r2763962;
        double r2763964 = log(r2763963);
        double r2763965 = r2763961 + r2763962;
        double r2763966 = log(r2763965);
        double r2763967 = r2763964 / r2763966;
        return r2763967;
}

↓

double f(double x) {
        double r2763968 = -1.0;
        double r2763969 = x;
        double r2763970 = r2763969 * r2763969;
        double r2763971 = -0.5;
        double r2763972 = r2763970 * r2763971;
        double r2763973 = r2763969 - r2763972;
        double r2763974 = r2763968 - r2763973;
        return r2763974;
}

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

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

5. Applied associate-+l- 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 2019112 
(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))))