Average Error: 31.8 → 0.0
Time: 3.8s
Precision: binary64
Cost: 19776
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{x + 1} \cdot \sqrt{x - 1}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{x + 1} \cdot \sqrt{x - 1}\right)
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (log (+ x (* (sqrt (+ x 1.0)) (sqrt (- x 1.0))))))
double code(double x) {
	return log(x + sqrt((x * x) - 1.0));
}
↓
double code(double x) {
	return log(x + (sqrt(x + 1.0) * sqrt(x - 1.0)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error31.8
Cost13248
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
Alternative 2
Error0.4
Cost6848
\[\log \left(x + \left(x - \frac{0.5}{x}\right)\right)\]
Alternative 3
Error0.7
Cost6592
\[\log \left(x + x\right)\]
Alternative 4
Error55.0
Cost64
\[1\]
Alternative 5
Error62.0
Cost64
\[0\]
Alternative 6
Error63.0
Cost64
\[-1\]

Error

Derivation

  1. Initial program 31.8

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

  3. Applied difference-of-sqr-1_binary64_209431.8

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

  4. Applied sqrt-prod_binary64_21400.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2021022 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))