qlog (example 3.10)

Average Error: 60.8 → 0.5

Time: 20.8s

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 r1654193 = 1.0;
        double r1654194 = x;
        double r1654195 = r1654193 - r1654194;
        double r1654196 = log(r1654195);
        double r1654197 = r1654193 + r1654194;
        double r1654198 = log(r1654197);
        double r1654199 = r1654196 / r1654198;
        return r1654199;
}

↓

double f(double x) {
        double r1654200 = x;
        double r1654201 = r1654200 * r1654200;
        double r1654202 = -0.5;
        double r1654203 = r1654201 * r1654202;
        double r1654204 = r1654203 - r1654200;
        double r1654205 = -1.0;
        double r1654206 = r1654204 + r1654205;
        return r1654206;
}

Error

Bits error versus x

Target

Original	60.8
Target	0.4
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 60.8

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

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

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

5. Applied associate--l+ 0.5

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

6. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019138 
(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))))