Average Error: 58.6 → 0.2
Time: 6.4s
Precision: binary64
Cost: 13760
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[0.5 \cdot \left(\left(x + x\right) + \left(0.4 \cdot {x}^{5} + 0.6666666666666666 \cdot {x}^{3}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
0.5 \cdot \left(\left(x + x\right) + \left(0.4 \cdot {x}^{5} + 0.6666666666666666 \cdot {x}^{3}\right)\right)
(FPCore (x) :precision binary64 (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))
↓
(FPCore (x)
 :precision binary64
 (*
  0.5
  (+ (+ x x) (+ (* 0.4 (pow x 5.0)) (* 0.6666666666666666 (pow x 3.0))))))
double code(double x) {
	return (1.0 / 2.0) * log((1.0 + x) / (1.0 - x));
}
↓
double code(double x) {
	return 0.5 * ((x + x) + ((0.4 * pow(x, 5.0)) + (0.6666666666666666 * pow(x, 3.0))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error61.3
Cost46016
\[0.5 \cdot \left(\left(2 \cdot \log \left(\sqrt[3]{1 + x}\right) - \log \left(1 + \sqrt{x}\right)\right) + \log \left(\frac{\sqrt[3]{1 + x}}{1 - \sqrt{x}}\right)\right)\]
Alternative 2
Error58.6
Cost40000
\[0.5 \cdot \left(\sqrt[3]{\log \left(\frac{1 + x}{1 - x}\right)} \cdot \left(\sqrt[3]{\log \left(\frac{1 + x}{1 - x}\right)} \cdot \sqrt[3]{\log \left(\frac{1 + x}{1 - x}\right)}\right)\right)\]
Alternative 3
Error61.3
Cost39488
\[0.5 \cdot \left(\log \left(\frac{\sqrt{1 + x}}{1 + \sqrt{x}}\right) + \log \left(\frac{\sqrt{1 + x}}{1 - \sqrt{x}}\right)\right)\]
Alternative 4
Error58.7
Cost26816
\[0.5 \cdot \left(\log \left(\sqrt[3]{\frac{1 + x}{1 - x}}\right) + 2 \cdot \log \left(\sqrt[3]{\frac{1 + x}{1 - x}}\right)\right)\]
Alternative 5
Error58.6
Cost26688
\[0.5 \cdot \left(\log \left(\sqrt{\frac{1 + x}{1 - x}}\right) + \log \left(\sqrt{\frac{1 + x}{1 - x}}\right)\right)\]
Alternative 6
Error58.7
Cost26560
\[0.5 \cdot \left(\log \left(\sqrt[3]{1 - x}\right) \cdot -2 + \log \left(\frac{1 + x}{\sqrt[3]{1 - x}}\right)\right)\]
Alternative 7
Error61.3
Cost26432
\[0.5 \cdot \left(\log \left(\frac{1 + x}{1 - \sqrt{x}}\right) - \log \left(1 + \sqrt{x}\right)\right)\]
Alternative 8
Error58.6
Cost26432
\[0.5 \cdot \left(\log \left(\sqrt{1 + x}\right) + \log \left(\frac{\sqrt{1 + x}}{1 - x}\right)\right)\]
Alternative 9
Error58.6
Cost13376
\[0.5 \cdot \left(\log \left(1 + x\right) - \log \left(1 - x\right)\right)\]
Alternative 10
Error0.2
Cost7552
\[0.5 \cdot \left(\left(x + x\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right) + 0.4 \cdot {x}^{5}\right)\right)\]
Alternative 11
Error0.3
Cost7040
\[0.5 \cdot \left(\left(x + x\right) + 0.6666666666666666 \cdot {x}^{3}\right)\]
Alternative 12
Error58.6
Cost6976
\[0.5 \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
Alternative 13
Error0.7
Cost320
\[0.5 \cdot \left(x + x\right)\]
Alternative 14
Error61.5
Cost64
\[1\]
Alternative 15
Error60.6
Cost64
\[0\]
Alternative 16
Error61.6
Cost64
\[-1\]

Error

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 0.2

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

    \[\leadsto 0.5 \cdot \left(\left(x + x\right) + \left(0.4 \cdot {x}^{5} + 0.6666666666666666 \cdot {x}^{3}\right)\right)\]

Reproduce

herbie shell --seed 2021022 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))