Average Error: 61.5 → 0.5
Time: 8.4s
Precision: binary64
\[-1 < x \land x < 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\log 1 + x \cdot \left(\frac{-0.5}{1} \cdot \frac{x}{1} - 1\right)}{1 \cdot x + \left(\log 1 + -0.5 \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)\right)}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\log 1 + x \cdot \left(\frac{-0.5}{1} \cdot \frac{x}{1} - 1\right)}{1 \cdot x + \left(\log 1 + -0.5 \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)\right)}
(FPCore (x) :precision binary64 (/ (log (- 1.0 x)) (log (+ 1.0 x))))
↓
(FPCore (x)
 :precision binary64
 (/
  (+ (log 1.0) (* x (- (* (/ -0.5 1.0) (/ x 1.0)) 1.0)))
  (+ (* 1.0 x) (+ (log 1.0) (* -0.5 (* (/ x 1.0) (/ x 1.0)))))))
double code(double x) {
	return (((double) log(((double) (1.0 - x)))) / ((double) log(((double) (1.0 + x)))));
}
↓
double code(double x) {
	return (((double) (((double) log(1.0)) + ((double) (x * ((double) (((double) ((-0.5 / 1.0) * (x / 1.0))) - 1.0)))))) / ((double) (((double) (1.0 * x)) + ((double) (((double) log(1.0)) + ((double) (-0.5 * ((double) ((x / 1.0) * (x / 1.0))))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.5
Target0.4
Herbie0.5
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666667 \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.5

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

  2. Taylor expanded around 0 60.7

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

  3. Simplified60.7

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

  4. Taylor expanded around 0 0.5

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

  5. Simplified0.5

    \[\leadsto \frac{\color{blue}{\log 1 + \left(x \cdot \frac{x \cdot \frac{-0.5}{1}}{1} - 1 \cdot x\right)}}{1 \cdot x + \left(\log 1 + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot -0.5\right)}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity0.5

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

  8. Applied associate-/r*0.5

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

  9. Simplified0.5

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

  10. Final simplification0.5

    \[\leadsto \frac{\log 1 + x \cdot \left(\frac{-0.5}{1} \cdot \frac{x}{1} - 1\right)}{1 \cdot x + \left(\log 1 + -0.5 \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)\right)}\]

Reproduce

herbie shell --seed 2020198 
(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))))