qlog (example 3.10)

Average Error: 61.4 → 0.4

Time: 21.4s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r4158527 = 1.0;
        double r4158528 = x;
        double r4158529 = r4158527 - r4158528;
        double r4158530 = log(r4158529);
        double r4158531 = r4158527 + r4158528;
        double r4158532 = log(r4158531);
        double r4158533 = r4158530 / r4158532;
        return r4158533;
}

↓

double f(double x) {
        double r4158534 = 1.0;
        double r4158535 = -0.5;
        double r4158536 = x;
        double r4158537 = 1.0;
        double r4158538 = r4158536 / r4158537;
        double r4158539 = r4158538 * r4158538;
        double r4158540 = log(r4158537);
        double r4158541 = fma(r4158537, r4158536, r4158540);
        double r4158542 = fma(r4158535, r4158539, r4158541);
        double r4158543 = r4158534 / r4158542;
        double r4158544 = 0.5;
        double r4158545 = r4158536 * r4158537;
        double r4158546 = fma(r4158539, r4158544, r4158545);
        double r4158547 = r4158540 - r4158546;
        double r4158548 = r4158534 / r4158547;
        double r4158549 = r4158543 / r4158548;
        return r4158549;
}

Error

Bits error versus x

Target

Original	61.4
Target	0.3
Herbie	0.4

\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666666851703837437526090070605 \cdot {x}^{3}\right)\]

Derivation

1. Initial program 61.4

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

2. Taylor expanded around 0 60.5

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

3. Simplified 60.5

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

4. Taylor expanded around 0 0.4

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

5. Simplified 0.4

   \[\leadsto \frac{\color{blue}{\log 1 - \mathsf{fma}\left(\frac{x}{1} \cdot \frac{x}{1}, \frac{1}{2}, 1 \cdot x\right)}}{\mathsf{fma}\left(\frac{-1}{2}, \frac{x}{1} \cdot \frac{x}{1}, \mathsf{fma}\left(1, x, \log 1\right)\right)}\]
6. Using strategy rm

7. Applied clear-num 0.4

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{x}{1} \cdot \frac{x}{1}, \mathsf{fma}\left(1, x, \log 1\right)\right)}{\log 1 - \mathsf{fma}\left(\frac{x}{1} \cdot \frac{x}{1}, \frac{1}{2}, 1 \cdot x\right)}}}\]
8. Using strategy rm

9. Applied div-inv 0.6

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

10. Applied associate-/r* 0.4

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

11. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "qlog (example 3.10)"
  :pre (and (< -1.0 x) (< x 1.0))

  :herbie-target
  (- (+ (+ (+ 1.0 x) (/ (* x x) 2.0)) (* 0.4166666666666667 (pow x 3.0))))

  (/ (log (- 1.0 x)) (log (+ 1.0 x))))