Average Error: 0.0 → 0.0
Time: 15.0s
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 2019005 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))

Details

Time bar (total: 14.7s)Debug log

sample200.0ms

Algorithm
intervals

simplify24.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

24.0ms
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize38.0ms

Local error

Found 4 expressions with local error:

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

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 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)

series252.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

105.0ms
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
81.0ms
(/ (sqrt (- 1 (* x x))) x)
36.0ms
(sqrt (- 1 (* x x)))
30.0ms
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))

simplify3.7s

Counts
47 → 81
Calls

47 calls. Slowest were:

488.0ms
(- (+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (sqrt -1)) (* 1/8 (/ 1 (* (pow x 4) (pow (sqrt -1) 3)))))
475.0ms
(- (* 1/8 (/ 1 (* (pow x 4) (pow (sqrt -1) 3)))) (+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (sqrt -1)))
341.0ms
(log (- (* (/ 1 x) (/ 1 x)) (* (/ (sqrt (- 1 (* x x))) x) (/ (sqrt (- 1 (* x x))) x))))

prune770.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize21.0ms

Local error

Found 4 expressions with local error:

0.2b
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
0.2b
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
0.0b
(sqrt (- 1 (* x x)))
0.0b
(sqrt (- 1 (* x x)))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

10.0ms
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
10.0ms
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
1.0ms
(sqrt (- 1 (* x x)))

series225.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

80.0ms
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
74.0ms
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
37.0ms
(sqrt (- 1 (* x x)))
35.0ms
(sqrt (- 1 (* x x)))

simplify2.9s

Counts
48 → 80
Calls

48 calls. Slowest were:

287.0ms
(+ (* 1/8 (* (/ 1 (pow x 2)) (sqrt (/ 1 (pow (sqrt -1) 3))))) (+ (sqrt (sqrt -1)) (* 1/2 (* (/ 1 x) (sqrt (/ 1 (sqrt -1)))))))
284.0ms
(+ (* 1/8 (* (/ 1 (pow x 2)) (sqrt (/ 1 (pow (sqrt -1) 3))))) (+ (sqrt (sqrt -1)) (* 1/2 (* (/ 1 x) (sqrt (/ 1 (sqrt -1)))))))
279.0ms
(sqrt (- (* (/ 1 x) (/ 1 x)) (* (/ (sqrt (- 1 (* x x))) x) (/ (sqrt (- 1 (* x x))) x))))

prune874.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes86.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample5.5s

Algorithm
intervals