Average Error: 58.6 → 0.2
Time: 4.8s
Precision: binary64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(2 \cdot x + \left({x}^{3} \cdot 0.66666666666666663 + {x}^{5} \cdot 0.40000000000000002\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(2 \cdot x + \left({x}^{3} \cdot 0.66666666666666663 + {x}^{5} \cdot 0.40000000000000002\right)\right)
double code(double x) {
	return ((double) (((double) (1.0 / 2.0)) * ((double) log(((double) (((double) (1.0 + x)) / ((double) (1.0 - x))))))));
}
↓
double code(double x) {
	return ((double) (((double) (1.0 / 2.0)) * ((double) (((double) (2.0 * x)) + ((double) (((double) (((double) pow(x, 3.0)) * 0.6666666666666666)) + ((double) (((double) pow(x, 5.0)) * 0.4))))))));
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.6

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

  3. Applied log-div58.6

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

  4. Taylor expanded around 0 0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{2}{3} \cdot \frac{{x}^{3}}{{1}^{3}} + \left(2 \cdot x + \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)}\]

  5. Simplified0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{2}{3} \cdot {\left(\frac{x}{1}\right)}^{3} + \left(x \cdot 2 + \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)}\]

  6. Taylor expanded around 0 0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(0.66666666666666663 \cdot {x}^{3} + \left(0.40000000000000002 \cdot {x}^{5} + 2 \cdot x\right)\right)}\]

  7. Simplified0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot 2 + \left({x}^{3} \cdot 0.66666666666666663 + {x}^{5} \cdot 0.40000000000000002\right)\right)}\]

  8. Final simplification0.2

    \[\leadsto \frac{1}{2} \cdot \left(2 \cdot x + \left({x}^{3} \cdot 0.66666666666666663 + {x}^{5} \cdot 0.40000000000000002\right)\right)\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))