Average Error: 58.3 → 58.3
Time: 12.8s
Precision: binary64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[0.5 \cdot \left(\sqrt[3]{\log \left(1 + x\right)} \cdot \left(\sqrt[3]{\log \left(1 + x\right)} \cdot \sqrt[3]{\log \left(1 + x\right)}\right) - \log \left(1 - x\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
0.5 \cdot \left(\sqrt[3]{\log \left(1 + x\right)} \cdot \left(\sqrt[3]{\log \left(1 + x\right)} \cdot \sqrt[3]{\log \left(1 + x\right)}\right) - \log \left(1 - x\right)\right)
(FPCore (x) :precision binary64 (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))
↓
(FPCore (x)
 :precision binary64
 (*
  0.5
  (-
   (* (cbrt (log (+ 1.0 x))) (* (cbrt (log (+ 1.0 x))) (cbrt (log (+ 1.0 x)))))
   (log (- 1.0 x)))))
double code(double x) {
	return (1.0 / 2.0) * log((1.0 + x) / (1.0 - x));
}
↓
double code(double x) {
	return 0.5 * ((cbrt(log(1.0 + x)) * (cbrt(log(1.0 + x)) * cbrt(log(1.0 + x)))) - log(1.0 - x));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.3

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

  2. Simplified58.3

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

  4. Applied add-exp-log_binary6458.3

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

  5. Applied add-exp-log_binary6458.3

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

  6. Applied div-exp_binary6458.3

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

  7. Applied rem-log-exp_binary6458.3

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

  9. Applied add-cube-cbrt_binary6458.3

    \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\sqrt[3]{\log \left(1 + x\right)} \cdot \sqrt[3]{\log \left(1 + x\right)}\right) \cdot \sqrt[3]{\log \left(1 + x\right)}} - \log \left(1 - x\right)\right)\]

  10. Final simplification58.3

    \[\leadsto 0.5 \cdot \left(\sqrt[3]{\log \left(1 + x\right)} \cdot \left(\sqrt[3]{\log \left(1 + x\right)} \cdot \sqrt[3]{\log \left(1 + x\right)}\right) - \log \left(1 - x\right)\right)\]

Reproduce

herbie shell --seed 2020288 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))