qlog (example 3.10)

Average Error: 61.0 → 0.0

Time: 21.8s

Precision: 64

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

Math

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

↓

\[\frac{\mathsf{log1p}\left(-x\right)}{\mathsf{log1p}\left(x\right)}\]

C

double f(double x) {
        double r3906400 = 1.0;
        double r3906401 = x;
        double r3906402 = r3906400 - r3906401;
        double r3906403 = log(r3906402);
        double r3906404 = r3906400 + r3906401;
        double r3906405 = log(r3906404);
        double r3906406 = r3906403 / r3906405;
        return r3906406;
}

↓

double f(double x) {
        double r3906407 = x;
        double r3906408 = -r3906407;
        double r3906409 = log1p(r3906408);
        double r3906410 = log1p(r3906407);
        double r3906411 = r3906409 / r3906410;
        return r3906411;
}

Error

Bits error versus x

Target

Original	61.0
Target	0.3
Herbie	0.0

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

Derivation

1. Initial program 61.0

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

2. Simplified 60.1

   \[\leadsto \color{blue}{\frac{\log \left(1 - x\right)}{\mathsf{log1p}\left(x\right)}}\]
3. Using strategy rm

4. Applied log1p-expm1-u 60.1

   \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(1 - x\right)\right)\right)}}{\mathsf{log1p}\left(x\right)}\]

5. Simplified 0.0

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

6. Final simplification 0.0

   \[\leadsto \frac{\mathsf{log1p}\left(-x\right)}{\mathsf{log1p}\left(x\right)}\]

Reproduce

herbie shell --seed 2019163 +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))))