Average Error: 61.4 → 0.3
Time: 9.5s
Precision: binary64
\[-1 < x \land x < 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\left(\left(-1 - 0.4166666666666667 \cdot {x}^{3}\right) - 0.5 \cdot {x}^{2}\right) - x\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\left(\left(-1 - 0.4166666666666667 \cdot {x}^{3}\right) - 0.5 \cdot {x}^{2}\right) - x
(FPCore (x) :precision binary64 (/ (log (- 1.0 x)) (log (+ 1.0 x))))
↓
(FPCore (x)
 :precision binary64
 (- (- (- -1.0 (* 0.4166666666666667 (pow x 3.0))) (* 0.5 (pow x 2.0))) x))
double code(double x) {
	return log(1.0 - x) / log(1.0 + x);
}
↓
double code(double x) {
	return ((-1.0 - (0.4166666666666667 * pow(x, 3.0))) - (0.5 * pow(x, 2.0))) - x;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.4
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 61.4

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

  3. Applied flip3-+_binary64_42261.2

    \[\leadsto \frac{\log \left(1 - x\right)}{\log \color{blue}{\left(\frac{{1}^{3} + {x}^{3}}{1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)}\right)}}\]

  4. Applied log-div_binary64_50661.4

    \[\leadsto \frac{\log \left(1 - x\right)}{\color{blue}{\log \left({1}^{3} + {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)}}\]

  5. Simplified61.4

    \[\leadsto \frac{\log \left(1 - x\right)}{\color{blue}{\log \left({x}^{3} + 1\right)} - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)}\]

  6. Simplified61.4

    \[\leadsto \frac{\log \left(1 - x\right)}{\log \left({x}^{3} + 1\right) - \color{blue}{\log \left(x \cdot x + \left(1 - x\right)\right)}}\]
  7. Using strategy rm

  8. Applied diff-log_binary64_51161.2

    \[\leadsto \frac{\log \left(1 - x\right)}{\color{blue}{\log \left(\frac{{x}^{3} + 1}{x \cdot x + \left(1 - x\right)}\right)}}\]

  9. Taylor expanded around 0 0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2021032 
(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))))