Average Error: 61.4 → 0.3
Time: 7.7s
Precision: binary64
\[-1 < x \land x < 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[-1 - \left(x + \left(0.5 \cdot {x}^{2} + \left(0.4166666666666667 \cdot {x}^{3} + 0.2916666666666667 \cdot {x}^{4}\right)\right)\right)\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
-1 - \left(x + \left(0.5 \cdot {x}^{2} + \left(0.4166666666666667 \cdot {x}^{3} + 0.2916666666666667 \cdot {x}^{4}\right)\right)\right)
(FPCore (x) :precision binary64 (/ (log (- 1.0 x)) (log (+ 1.0 x))))
↓
(FPCore (x)
 :precision binary64
 (-
  -1.0
  (+
   x
   (+
    (* 0.5 (pow x 2.0))
    (+
     (* 0.4166666666666667 (pow x 3.0))
     (* 0.2916666666666667 (pow x 4.0)))))))
double code(double x) {
	return log(1.0 - x) / log(1.0 + x);
}
↓
double code(double x) {
	return -1.0 - (x + ((0.5 * pow(x, 2.0)) + ((0.4166666666666667 * pow(x, 3.0)) + (0.2916666666666667 * pow(x, 4.0)))));
}

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 flip--_binary6461.0

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

  4. Applied log-div_binary6461.2

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

  5. Applied div-sub_binary6461.2

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

  6. Simplified61.2

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

  7. Simplified61.2

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

  8. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{\left(-\left(x + \left(0.5 \cdot {x}^{2} + \left(0.4166666666666667 \cdot {x}^{3} + 0.2916666666666667 \cdot {x}^{4}\right)\right)\right)\right)} - 1\]

  9. Final simplification0.3

    \[\leadsto -1 - \left(x + \left(0.5 \cdot {x}^{2} + \left(0.4166666666666667 \cdot {x}^{3} + 0.2916666666666667 \cdot {x}^{4}\right)\right)\right)\]

Reproduce

herbie shell --seed 2021173 
(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))))