Average Error: 0.0 → 0.0
Time: 14.6s
Precision: 64
Internal Precision: 128
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right) + \log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)\]

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. Using strategy rm

  3. Applied add-sqr-sqrt0.0

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

  4. Applied log-prod0.0

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

  5. Final simplification0.0

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

Reproduce

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

Details

Time bar (total: 14.3s)Debug log

sample218.0ms

Algorithm
intervals

simplify35.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

35.0ms
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize34.0ms

Local error

Found 4 expressions with local error:

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

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 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)

series256.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

111.0ms
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
75.0ms
(/ (sqrt (- 1 (* x x))) x)
47.0ms
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))
22.0ms
(sqrt (- 1 (* x x)))

simplify3.8s

Counts
47 → 81
Calls

47 calls. Slowest were:

543.0ms
(- (* 1/8 (/ 1 (* (pow x 4) (pow (sqrt -1) 3)))) (+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (sqrt -1)))
510.0ms
(- (+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (sqrt -1)) (* 1/8 (/ 1 (* (pow x 4) (pow (sqrt -1) 3)))))
292.0ms
(- (+ (log (sqrt -1)) (/ 1 (* x (sqrt -1)))) (* 1/6 (/ 1 (* (pow x 3) (pow (sqrt -1) 3)))))

prune802.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize17.0ms

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

rewrite38.0ms

Algorithm
rewrite-expression-head
Counts
4 → 74
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)))
9.0ms
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))

series182.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

57.0ms
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
55.0ms
(sqrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
42.0ms
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))
28.0ms
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))

simplify3.5s

Counts
56 → 86
Calls

56 calls. Slowest were:

618.0ms
(- (/ 1 x) (+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (sqrt -1)))
314.0ms
(+ (* 1/8 (* (/ 1 (pow x 2)) (sqrt (/ 1 (pow (sqrt -1) 3))))) (+ (sqrt (sqrt -1)) (* 1/2 (* (/ 1 x) (sqrt (/ 1 (sqrt -1)))))))
308.0ms
(+ (* 1/8 (* (/ 1 (pow x 2)) (sqrt (/ 1 (pow (sqrt -1) 3))))) (+ (sqrt (sqrt -1)) (* 1/2 (* (/ 1 x) (sqrt (/ 1 (sqrt -1)))))))

prune938.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample4.5s

Algorithm
intervals