qlog (example 3.10)

Average Error: 61.0 → 0.4

Time: 15.1s

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 r1641236 = 1.0;
        double r1641237 = x;
        double r1641238 = r1641236 - r1641237;
        double r1641239 = log(r1641238);
        double r1641240 = r1641236 + r1641237;
        double r1641241 = log(r1641240);
        double r1641242 = r1641239 / r1641241;
        return r1641242;
}

↓

double f(double x) {
        double r1641243 = x;
        double r1641244 = r1641243 * r1641243;
        double r1641245 = -0.5;
        double r1641246 = r1641244 * r1641245;
        double r1641247 = r1641246 - r1641243;
        double r1641248 = -1.0;
        double r1641249 = r1641247 + r1641248;
        return r1641249;
}

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

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

5. Applied associate--l+ 0.4

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

6. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019132 
(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))))