Average Error: 30.9 → 0.4
Time: 16.7s
Precision: 64
Internal Precision: 128
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\frac{\frac{-1}{4}}{x \cdot x} + \left(\frac{\frac{-3}{32}}{{x}^{4}} + \left(\log 2 + \log x\right)\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.9

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

  2. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{\log 2 - \left(\log \left(\frac{1}{x}\right) + \left(\frac{3}{32} \cdot \frac{1}{{x}^{4}} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{\left(\left(\log 2 + \log x\right) + \frac{\frac{-3}{32}}{{x}^{4}}\right) + \frac{\frac{-1}{4}}{x \cdot x}}\]

  4. Final simplification0.4

    \[\leadsto \frac{\frac{-1}{4}}{x \cdot x} + \left(\frac{\frac{-3}{32}}{{x}^{4}} + \left(\log 2 + \log x\right)\right)\]

Reproduce

herbie shell --seed 2019004 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))

Details

Time bar (total: 16.3s)Debug log

sample184.0ms

Algorithm
intervals

simplify95.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

95.0ms
(log (+ x (sqrt (- (* x x) 1))))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 32.0b

localize15.0ms

Local error

Found 3 expressions with local error:

30.7b
(sqrt (- (* x x) 1))
0.2b
(log (+ x (sqrt (- (* x x) 1))))
0.0b
(+ x (sqrt (- (* x x) 1)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
3 → 39
Calls

3 calls. Slowest were:

7.0ms
(log (+ x (sqrt (- (* x x) 1))))
2.0ms
(sqrt (- (* x x) 1))
1.0ms
(+ x (sqrt (- (* x x) 1)))

series140.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

91.0ms
(log (+ x (sqrt (- (* x x) 1))))
28.0ms
(+ x (sqrt (- (* x x) 1)))
20.0ms
(sqrt (- (* x x) 1))

simplify2.7s

Counts
22 → 48
Calls

22 calls. Slowest were:

395.0ms
(- (+ (* 1/2 (/ (pow x 2) (sqrt -1))) (sqrt -1)) (* 1/8 (/ (pow x 4) (pow (sqrt -1) 3))))
306.0ms
(+ (log -1/2) (+ (log (/ -1 x)) (+ (* 3/32 (/ 1 (pow x 4))) (* 1/4 (/ 1 (pow x 2))))))
284.0ms
(log (- (* x x) (* (sqrt (- (* x x) 1)) (sqrt (- (* x x) 1)))))

prune341.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize29.0ms

Local error

Found 4 expressions with local error:

0.6b
(/ -1/4 (* x x))
0.2b
(/ -3/32 (pow x 4))
0.2b
(+ (log 2) (log x))
0.0b
(+ (+ (+ (log 2) (log x)) (/ -3/32 (pow x 4))) (/ -1/4 (* x x)))

rewrite136.0ms

Algorithm
rewrite-expression-head
Counts
4 → 57
Calls

4 calls. Slowest were:

126.0ms
(+ (+ (+ (log 2) (log x)) (/ -3/32 (pow x 4))) (/ -1/4 (* x x)))
6.0ms
(+ (log 2) (log x))
1.0ms
(/ -3/32 (pow x 4))

series538.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

380.0ms
(+ (log 2) (log x))
114.0ms
(+ (+ (+ (log 2) (log x)) (/ -3/32 (pow x 4))) (/ -1/4 (* x x)))
26.0ms
(/ -3/32 (pow x 4))
18.0ms
(/ -1/4 (* x x))

simplify2.7s

Counts
32 → 69
Calls

32 calls. Slowest were:

258.0ms
(+ (* (+ (* (+ (pow (log 2) 3) (pow (log x) 3)) (pow x 4)) (* (+ (* (log 2) (log 2)) (- (* (log x) (log x)) (* (log 2) (log x)))) -3/32)) (* x x)) (* (* (+ (* (log 2) (log 2)) (- (* (log x) (log x)) (* (log 2) (log x)))) (pow x 4)) -1/4))
255.0ms
(* (+ (* (+ (log 2) (log x)) (+ (log 2) (log x))) (- (* (/ -3/32 (pow x 4)) (/ -3/32 (pow x 4))) (* (+ (log 2) (log x)) (/ -3/32 (pow x 4))))) (* x x))
250.0ms
(+ (* (+ (* (- (* (log 2) (log 2)) (* (log x) (log x))) (pow x 4)) (* (- (log 2) (log x)) -3/32)) (* x x)) (* (* (- (log 2) (log x)) (pow x 4)) -1/4))

prune742.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize30.0ms

Local error

Found 3 expressions with local error:

0.5b
(* (sqrt (+ x 1)) (sqrt (- x 1)))
0.2b
(log (+ x (* (sqrt (+ x 1)) (sqrt (- x 1)))))
0.0b
(+ x (* (sqrt (+ x 1)) (sqrt (- x 1))))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
3 → 57
Calls

3 calls. Slowest were:

5.0ms
(* (sqrt (+ x 1)) (sqrt (- x 1)))
2.0ms
(+ x (* (sqrt (+ x 1)) (sqrt (- x 1))))
1.0ms
(log (+ x (* (sqrt (+ x 1)) (sqrt (- x 1)))))

series135.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

73.0ms
(log (+ x (* (sqrt (+ x 1)) (sqrt (- x 1)))))
31.0ms
(* (sqrt (+ x 1)) (sqrt (- x 1)))
30.0ms
(+ x (* (sqrt (+ x 1)) (sqrt (- x 1))))

simplify3.4s

Counts
43 → 66
Calls

43 calls. Slowest were:

692.0ms
(log (- (* x x) (* (* (sqrt (+ x 1)) (sqrt (- x 1))) (* (sqrt (+ x 1)) (sqrt (- x 1))))))
642.0ms
(* (* (* (sqrt (+ x 1)) (sqrt (+ x 1))) (sqrt (+ x 1))) (* (* (sqrt (- x 1)) (sqrt (- x 1))) (sqrt (- x 1))))
343.0ms
(- (+ (* 1/2 (/ (pow x 2) (sqrt -1))) (sqrt -1)) (* 1/8 (/ (pow x 4) (pow (sqrt -1) 3))))

prune581.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes18.0ms

Accuracy

0% (0.4b remaining)

Error of 0.4b against oracle of 0.0b and baseline of 0.4b

bsearch2.0ms

end0.0ms

sample4.4s

Algorithm
intervals