Average Error: 31.2 → 0.1
Time: 7.7s
Precision: 64
Internal Precision: 128
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.2

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

  3. Applied difference-of-sqr-131.2

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

  4. Applied sqrt-prod0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019008 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))

Details

Time bar (total: 7.4s)Debug log

sample106.0ms

Algorithm
intervals

simplify73.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

73.0ms
(log (+ x (sqrt (- (* x x) 1))))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 31.2b

localize15.0ms

Local error

Found 2 expressions with local error:

30.1b
(sqrt (- (* x x) 1))
0.0b
(+ x (sqrt (- (* x x) 1)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 25
Calls

2 calls. Slowest were:

4.0ms
(sqrt (- (* x x) 1))
3.0ms
(+ x (sqrt (- (* x x) 1)))

series76.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

43.0ms
(+ x (sqrt (- (* x x) 1)))
32.0ms
(sqrt (- (* x x) 1))

simplify836.0ms

Counts
13 → 31
Calls

13 calls. Slowest were:

235.0ms
(- (+ (* 1/2 (/ (pow x 2) (sqrt -1))) (sqrt -1)) (* 1/8 (/ (pow x 4) (pow (sqrt -1) 3))))
87.0ms
(- x (+ (* 1/8 (/ 1 (pow x 3))) (* 1/2 (/ 1 x))))
86.0ms
(sqrt (sqrt (- (* x x) 1)))

prune160.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize17.0ms

Local error

Found 2 expressions with local error:

0.5b
(* (sqrt (+ x 1)) (sqrt (- x 1)))
0.0b
(+ x (* (sqrt (+ x 1)) (sqrt (- x 1))))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
2 → 43
Calls

2 calls. Slowest were:

5.0ms
(* (sqrt (+ x 1)) (sqrt (- x 1)))
1.0ms
(+ x (* (sqrt (+ x 1)) (sqrt (- x 1))))

series58.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

30.0ms
(* (sqrt (+ x 1)) (sqrt (- x 1)))
28.0ms
(+ x (* (sqrt (+ x 1)) (sqrt (- x 1))))

simplify1.3s

Counts
34 → 49
Calls

34 calls. Slowest were:

453.0ms
(* (* (* (sqrt (+ x 1)) (sqrt (+ x 1))) (sqrt (+ x 1))) (* (* (sqrt (- x 1)) (sqrt (- x 1))) (sqrt (- x 1))))
297.0ms
(- (+ (* 1/2 (/ (pow x 2) (sqrt -1))) (sqrt -1)) (* 1/8 (/ (pow x 4) (pow (sqrt -1) 3))))
100.0ms
(- x (+ (* 1/8 (/ 1 (pow x 3))) (* 1/2 (/ 1 x))))

prune423.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample4.3s

Algorithm
intervals