qlog (example 3.10)

Average Error: 60.8 → 0.5

Time: 19.8s

Precision: 64

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

Math

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

↓

\[\mathsf{fma}\left(\left(\mathsf{fma}\left(x, \frac{-1}{2}, -1\right)\right), x, -1\right)\]

C

double f(double x) {
        double r4324443 = 1.0;
        double r4324444 = x;
        double r4324445 = r4324443 - r4324444;
        double r4324446 = log(r4324445);
        double r4324447 = r4324443 + r4324444;
        double r4324448 = log(r4324447);
        double r4324449 = r4324446 / r4324448;
        return r4324449;
}

↓

double f(double x) {
        double r4324450 = x;
        double r4324451 = -0.5;
        double r4324452 = -1.0;
        double r4324453 = fma(r4324450, r4324451, r4324452);
        double r4324454 = fma(r4324453, r4324450, r4324452);
        return r4324454;
}

Error

Bits error versus x

Target

Original	60.8
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 60.8

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

2. Simplified 59.9

   \[\leadsto \color{blue}{\frac{\log \left(1 - x\right)}{\mathsf{log1p}\left(x\right)}}\]

3. Taylor expanded around 0 0.5

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

4. Simplified 0.5

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

5. Final simplification 0.5

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(x, \frac{-1}{2}, -1\right)\right), x, -1\right)\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(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))))