Average Error: 31.0 → 0.2
Time: 9.8s
Precision: 64
Internal Precision: 128
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\left(x + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.0

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

  2. Initial simplification31.0

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

  3. Taylor expanded around inf 0.2

    \[\leadsto \log \left(x + \color{blue}{\left(x - \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\]

  4. Simplified0.2

    \[\leadsto \log \left(x + \color{blue}{\left(\left(\frac{\frac{-1}{2}}{x} + x\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)}\right)\]

  5. Final simplification0.2

    \[\leadsto \log \left(x + \left(\left(x + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\]

Reproduce

herbie shell --seed 2018362 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))

Details

Time bar (total: 4.8s)Debug log

start170.0ms

Algorithm
intervals

setup101.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 31.0b

localize14.0ms

Local error

Found 3 expressions with local error:

29.9b
(sqrt (+ -1 (* x x)))
0.0b
(+ x (sqrt (+ -1 (* x x))))
0.0b
(log (+ x (sqrt (+ -1 (* x x)))))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
3 → 38
Calls

3 calls. Slowest were:

2.0ms
(log (+ x (sqrt (+ -1 (* x x)))))
2.0ms
(+ x (sqrt (+ -1 (* x x))))
1.0ms
(sqrt (+ -1 (* x x)))

series178.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

110.0ms
(log (+ x (sqrt (+ -1 (* x x)))))
34.0ms
(+ x (sqrt (+ -1 (* x x))))
34.0ms
(sqrt (+ -1 (* x x)))

simplify2.8s

Counts
21 → 47
Calls

21 calls. Slowest were:

371.0ms
(- (+ (* 1/2 (/ (pow x 2) (sqrt -1))) (sqrt -1)) (* 1/8 (/ (pow x 4) (pow (sqrt -1) 3))))
348.0ms
(sqrt (- (* -1 -1) (* (* x x) (* x x))))
325.0ms
(+ (log -1/2) (+ (log (/ -1 x)) (+ (* 3/32 (/ 1 (pow x 4))) (* 1/4 (/ 1 (pow x 2))))))

prune439.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize23.0ms

Local error

Found 1 expressions with local error:

0.1b
(/ (/ 1/8 x) (* x x))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
1 → 22
Calls

1 calls. Slowest were:

6.0ms
(/ (/ 1/8 x) (* x x))

series29.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

28.0ms
(/ (/ 1/8 x) (* x x))

simplify782.0ms

Counts
15 → 25
Calls

15 calls. Slowest were:

716.0ms
(/ (* (* (/ 1/8 x) (/ 1/8 x)) (/ 1/8 x)) (* (* (* x x) (* x x)) (* x x)))
30.0ms
(- (log (/ 1/8 x)) (log (* x x)))
11.0ms
(/ (* (cbrt (/ 1/8 x)) (cbrt (/ 1/8 x))) x)

prune309.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes35.0ms

Accuracy

0% (0.2b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 0.2b

bsearch3.0ms