Average Error: 61.6 → 0.3
Time: 7.6s
Precision: binary64
\[-1 < x \land x < 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[-1 + x \cdot \left(-1 - x \cdot \left(0.5 + x \cdot 0.4166666666666667\right)\right)\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
-1 + x \cdot \left(-1 - x \cdot \left(0.5 + x \cdot 0.4166666666666667\right)\right)
(FPCore (x) :precision binary64 (/ (log (- 1.0 x)) (log (+ 1.0 x))))
↓
(FPCore (x)
 :precision binary64
 (+ -1.0 (* x (- -1.0 (* x (+ 0.5 (* x 0.4166666666666667)))))))
double code(double x) {
	return log(1.0 - x) / log(1.0 + x);
}
↓
double code(double x) {
	return -1.0 + (x * (-1.0 - (x * (0.5 + (x * 0.4166666666666667)))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.6
Target0.3
Herbie0.3
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666667 \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.6

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

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{-\left(x + \left(0.5 \cdot {x}^{2} + \left(0.4166666666666667 \cdot {x}^{3} + 1\right)\right)\right)}\]

  3. Simplified0.3

    \[\leadsto \color{blue}{-1 + x \cdot \left(-1 - x \cdot \left(0.5 + x \cdot 0.4166666666666667\right)\right)}\]

  4. Final simplification0.3

    \[\leadsto -1 + x \cdot \left(-1 - x \cdot \left(0.5 + x \cdot 0.4166666666666667\right)\right)\]

Reproduce

herbie shell --seed 2021175 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :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))))