qlog (example 3.10)

Average Error: 60.9 → 0.5

Time: 20.9s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r3412943 = 1.0;
        double r3412944 = x;
        double r3412945 = r3412943 - r3412944;
        double r3412946 = log(r3412945);
        double r3412947 = r3412943 + r3412944;
        double r3412948 = log(r3412947);
        double r3412949 = r3412946 / r3412948;
        return r3412949;
}

↓

double f(double x) {
        double r3412950 = -0.5;
        double r3412951 = x;
        double r3412952 = r3412951 * r3412951;
        double r3412953 = r3412950 * r3412952;
        double r3412954 = -1.0;
        double r3412955 = r3412954 - r3412951;
        double r3412956 = r3412953 + r3412955;
        return r3412956;
}

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. Using strategy rm

5. Applied associate--l+ 0.5

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

6. Final simplification 0.5

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

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))))