Average Error: 0.0 → 0.0
Time: 16.5s
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 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))

Details

Time bar (total: 16.1s)Debug log

sample191.0ms

Algorithm
intervals

simplify29.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

29.0ms
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize24.0ms

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)))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
3 → 65
Calls

3 calls. Slowest were:

9.0ms
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))
7.0ms
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
6.0ms
(/ (sqrt (- 1 (* x x))) x)

series578.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

439.0ms
(/ (sqrt (- 1 (* x x))) x)
92.0ms
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
47.0ms
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))

simplify3.1s

Counts
40 → 74
Calls

40 calls. Slowest were:

417.0ms
(- (log 2) (+ (* 1/4 (pow x 2)) (+ (* 3/32 (pow x 4)) (log x))))
356.0ms
(log (- (* (/ 1 x) (/ 1 x)) (* (/ (sqrt (- 1 (* x x))) x) (/ (sqrt (- 1 (* x x))) x))))
283.0ms
(- (* 1/8 (/ 1 (* (pow x 4) (pow (sqrt -1) 3)))) (+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (sqrt -1)))

prune652.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize18.0ms

Local error

Found 4 expressions with local error:

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

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 98
Calls

4 calls. Slowest were:

7.0ms
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
6.0ms
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
4.0ms
(log (sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))

series337.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

124.0ms
(log (sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))
109.0ms
(log (sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))
54.0ms
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
49.0ms
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))

simplify5.1s

Counts
68 → 110
Calls

68 calls. Slowest were:

432.0ms
(log (sqrt (- (* (/ 1 x) (/ 1 x)) (* (/ (sqrt (- 1 (* x x))) x) (/ (sqrt (- 1 (* x x))) x)))))
407.0ms
(log (sqrt (- (* (/ 1 x) (/ 1 x)) (* (/ (sqrt (- 1 (* x x))) x) (/ (sqrt (- 1 (* x x))) x)))))
396.0ms
(+ (* 1/8 (* (/ 1 (pow x 2)) (sqrt (/ 1 (pow (sqrt -1) 3))))) (+ (sqrt (sqrt -1)) (* 1/2 (* (/ 1 x) (sqrt (/ 1 (sqrt -1)))))))

prune1.2s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes126.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample4.8s

Algorithm
intervals