Average Error: 0.0 → 0.0
Time: 19.9s
Precision: 64
Internal Precision: 128
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  2. Final simplification0.0

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

Reproduce

herbie shell --seed 2019007 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))

Details

Time bar (total: 19.6s)Debug log

sample197.0ms

Algorithm
intervals

simplify13.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

12.0ms
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize38.0ms

Local error

Found 3 expressions with local error:

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

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
3 → 55
Calls

3 calls. Slowest were:

9.0ms
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))
6.0ms
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
6.0ms
(/ (sqrt (- 1 (* x x))) x)

series235.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

133.0ms
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
52.0ms
(/ (sqrt (- 1 (* x x))) x)
50.0ms
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))

simplify3.2s

Counts
38 → 64
Calls

38 calls. Slowest were:

528.0ms
(- (* 1/8 (/ 1 (* (pow x 4) (pow (sqrt -1) 3)))) (+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (sqrt -1)))
474.0ms
(- (+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (sqrt -1)) (* 1/8 (/ 1 (* (pow x 4) (pow (sqrt -1) 3)))))
355.0ms
(- (+ (log (sqrt -1)) (/ 1 (* x (sqrt -1)))) (* 1/6 (/ 1 (* (pow x 3) (pow (sqrt -1) 3)))))

prune576.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize9.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))) (sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))
0.1b
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
0.1b
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
0.0b
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))

rewrite35.0ms

Algorithm
rewrite-expression-head
Counts
4 → 112
Calls

4 calls. Slowest were:

14.0ms
(* (sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))) (sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))
7.0ms
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))
5.0ms
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))

series239.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

76.0ms
(* (sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))) (sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))
66.0ms
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
52.0ms
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
45.0ms
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))

simplify8.5s

Counts
103 → 124
Calls

103 calls. Slowest were:

391.0ms
(* (sqrt (- (/ 1 x) (/ (sqrt (- 1 (* x x))) x))) (sqrt (+ (* (/ 1 x) (/ 1 x)) (- (* (/ (sqrt (- 1 (* x x))) x) (/ (sqrt (- 1 (* x x))) x)) (* (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))))
387.0ms
(* (sqrt (- (* (/ 1 x) (/ 1 x)) (* (/ (sqrt (- 1 (* x x))) x) (/ (sqrt (- 1 (* x x))) x)))) (sqrt (+ (* 1 x) (* x (sqrt (- 1 (* x x)))))))
347.0ms
(* (sqrt (+ (* (/ 1 x) (/ 1 x)) (- (* (/ (sqrt (- 1 (* x x))) x) (/ (sqrt (- 1 (* x x))) x)) (* (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))) (sqrt (+ (* (/ 1 x) (/ 1 x)) (- (* (/ (sqrt (- 1 (* x x))) x) (/ (sqrt (- 1 (* x x))) x)) (* (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))))

prune1.3s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes92.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample5.2s

Algorithm
intervals