Average Error: 61.5 → 0.3
Time: 7.2s
Precision: binary64
\[-1 < x \land x < 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[-1 - \left(x + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot 0.4166666666666667\right)\right)\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
-1 - \left(x + \left(x \cdot x\right) \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 (* (* x 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 + ((x * 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

Original61.5
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 61.5

    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
  2. Using strategy rm

  3. Applied flip--_binary6461.1

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

  4. Applied log-div_binary6461.3

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

  5. Applied div-sub_binary6461.3

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

  6. Simplified61.3

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

  7. Simplified61.3

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

  8. Taylor expanded around 0 0.3

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020224 
(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))))