qlog (example 3.10)

Average Error: 60.9 → 0.5

Time: 22.1s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r3926419 = 1.0;
        double r3926420 = x;
        double r3926421 = r3926419 - r3926420;
        double r3926422 = log(r3926421);
        double r3926423 = r3926419 + r3926420;
        double r3926424 = log(r3926423);
        double r3926425 = r3926422 / r3926424;
        return r3926425;
}

↓

double f(double x) {
        double r3926426 = -0.5;
        double r3926427 = x;
        double r3926428 = r3926426 * r3926427;
        double r3926429 = r3926428 * r3926427;
        double r3926430 = -1.0;
        double r3926431 = r3926430 - r3926427;
        double r3926432 = r3926429 + r3926431;
        return r3926432;
}

Error

Bits error versus x

Target

Original	60.9
Target	0.3
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 60.9

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

2. Taylor expanded around 0 0.5

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

3. Simplified 0.5

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

4. Taylor expanded around 0 0.5

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

5. Simplified 0.5

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

6. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019158 
(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))))