Average Error: 58.6 → 0.1
Time: 6.6s
Precision: binary64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
↓
\[\left(x + \left({x}^{3} \cdot 0.3333333333333333 + {x}^{5} \cdot 0.2\right)\right) + {x}^{7} \cdot 0.14285714285714285 \]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(x + \left({x}^{3} \cdot 0.3333333333333333 + {x}^{5} \cdot 0.2\right)\right) + {x}^{7} \cdot 0.14285714285714285

(FPCore (x) :precision binary64 (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))
↓
(FPCore (x)
 :precision binary64
 (+
  (+ x (+ (* (pow x 3.0) 0.3333333333333333) (* (pow x 5.0) 0.2)))
  (* (pow x 7.0) 0.14285714285714285)))
double code(double x) {
	return (1.0 / 2.0) * log((1.0 + x) / (1.0 - x));
}
↓
double code(double x) {
	return (x + ((pow(x, 3.0) * 0.3333333333333333) + (pow(x, 5.0) * 0.2))) + (pow(x, 7.0) * 0.14285714285714285);
}

Error

Bits error versus x

Try it out

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 in x around 0 0.1

    \[\leadsto 0.5 \cdot \color{blue}{\left(0.4 \cdot {x}^{5} + \left(0.2857142857142857 \cdot {x}^{7} + \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)\right)\right)} \]

  4. Simplified0.1

    \[\leadsto \color{blue}{\left(x + \left({x}^{3} \cdot 0.3333333333333333 + {x}^{5} \cdot 0.2\right)\right) + {x}^{7} \cdot 0.14285714285714285} \]

  5. Final simplification0.1

    \[\leadsto \left(x + \left({x}^{3} \cdot 0.3333333333333333 + {x}^{5} \cdot 0.2\right)\right) + {x}^{7} \cdot 0.14285714285714285 \]

Reproduce

herbie shell --seed 2021206 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))