qlog (example 3.10)

Average Error: 61.4 → 0.4

Time: 20.8s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r3843435 = 1.0;
        double r3843436 = x;
        double r3843437 = r3843435 - r3843436;
        double r3843438 = log(r3843437);
        double r3843439 = r3843435 + r3843436;
        double r3843440 = log(r3843439);
        double r3843441 = r3843438 / r3843440;
        return r3843441;
}

↓

double f(double x) {
        double r3843442 = 1.0;
        double r3843443 = x;
        double r3843444 = 1.0;
        double r3843445 = r3843443 * r3843444;
        double r3843446 = r3843443 / r3843444;
        double r3843447 = r3843446 * r3843446;
        double r3843448 = 2.0;
        double r3843449 = r3843447 / r3843448;
        double r3843450 = r3843445 - r3843449;
        double r3843451 = log(r3843444);
        double r3843452 = r3843450 + r3843451;
        double r3843453 = r3843442 / r3843452;
        double r3843454 = r3843451 - r3843445;
        double r3843455 = r3843454 - r3843449;
        double r3843456 = r3843442 / r3843455;
        double r3843457 = r3843453 / r3843456;
        return r3843457;
}

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

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

4. Taylor expanded around 0 0.4

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

5. Simplified 0.4

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

7. Applied clear-num 0.4

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

8. Simplified 0.4

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

10. Applied div-inv 0.6

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

11. Applied associate-/r* 0.4

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

12. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019171 
(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))))