Average Error: 58.6 → 0.6
Time: 10.2s
Precision: binary64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[0.5 \cdot \left(\log 1 + \left(x + x\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
0.5 \cdot \left(\log 1 + \left(x + x\right)\right)
(FPCore (x) :precision binary64 (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))
↓
(FPCore (x) :precision binary64 (* 0.5 (+ (log 1.0) (+ x x))))
double code(double x) {
	return (1.0 / 2.0) * log((1.0 + x) / (1.0 - x));
}
↓
double code(double x) {
	return 0.5 * (log(1.0) + (x + x));
}

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. Simplified58.6

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

  3. Taylor expanded around 0 0.6

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2020277 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))