qlog (example 3.10)

Average Error: 60.8 → 0.4

Time: 17.5s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r1830581 = 1.0;
        double r1830582 = x;
        double r1830583 = r1830581 - r1830582;
        double r1830584 = log(r1830583);
        double r1830585 = r1830581 + r1830582;
        double r1830586 = log(r1830585);
        double r1830587 = r1830584 / r1830586;
        return r1830587;
}

↓

double f(double x) {
        double r1830588 = x;
        double r1830589 = r1830588 * r1830588;
        double r1830590 = -0.5;
        double r1830591 = r1830589 * r1830590;
        double r1830592 = r1830591 - r1830588;
        double r1830593 = -1.0;
        double r1830594 = r1830592 + r1830593;
        return r1830594;
}

Error

Bits error versus x

Target

Original	60.8
Target	0.3
Herbie	0.4

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

Derivation

1. Initial program 60.8

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

2. Taylor expanded around 0 0.4

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

3. Simplified 0.4

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

5. Applied associate--l+ 0.4

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

6. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019129 
(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))))