Average Error: 0.0 → 0.2
Time: 19.6s
Precision: binary64
Cost: 960
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\left(\log \left(1 + \sqrt{1 - x \cdot x}\right) - \log x\right) - \log 1\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\left(\log \left(1 + \sqrt{1 - x \cdot x}\right) - \log x\right) - \log 1
(FPCore (x)
 :precision binary64
 (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))
↓
(FPCore (x)
 :precision binary64
 (- (- (log (+ 1.0 (sqrt (- 1.0 (* x x))))) (log x)) (log 1.0)))
double code(double x) {
	return log((1.0 / x) + (sqrt(1.0 - (x * x)) / x));
}
↓
double code(double x) {
	return (log(1.0 + sqrt(1.0 - (x * x))) - log(x)) - log(1.0);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs
Alternative 1
Accuracy0.0
Cost960
\[\log \left(\frac{1}{x} + \frac{1}{\frac{x}{\sqrt{1 - x \cdot x}}}\right)\]
Alternative 2
Accuracy0.4
Cost1088
\[\log 2 - \left(\left(x \cdot x\right) \cdot 0.25 + \left(\log x + {x}^{4} \cdot 0.09375\right)\right)\]
Alternative 3
Accuracy64.0
Cost1216
\[\left(\frac{1}{x \cdot \sqrt{-1}} + \log \left(\sqrt{-1}\right)\right) - \frac{-0.16666666666666666}{\sqrt{-1} \cdot {x}^{3}}\]

Derivation

  1. Initial program 0.0

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

  3. Applied div-inv_binary64_10980.0

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

  4. Applied div-inv_binary64_10980.0

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

  5. Applied distribute-rgt-out_binary64_10540.0

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

  6. Applied log-prod_binary64_11870.2

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

  7. Simplified0.2

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

  9. Applied *-un-lft-identity_binary64_11010.2

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

  10. Applied log-prod_binary64_11870.2

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

  11. Applied distribute-neg-in_binary64_10620.2

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

  12. Applied associate-+l+_binary64_10340.2

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

  13. Simplified0.2

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

  15. Applied pow1_binary64_11620.2

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

  16. Final simplification0.2

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

Reproduce

herbie shell --seed 2020322 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))