Average Error: 0.0 → 0.0
Time: 12.6s
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

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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. Initial simplification0.0

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

  3. Final simplification0.0

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

Runtime

Time bar (total: 12.6s)Debug logProfile

start259.0ms

Algorithm
intervals

setup453.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize20.0ms

Local error

Found 1 expressions with local error:

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

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
1 → 23
Calls

1 calls. Slowest were:

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

series34.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

34.0ms
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))

simplify891.0ms

Counts
15 → 26
Calls

15 calls. Slowest were:

323.0ms
(- (/ 1 x) (+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (sqrt -1)))
309.0ms
(- (* 2 (/ 1 x)) (+ (* 1/2 x) (* 1/8 (pow x 3))))
112.0ms
(+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (+ (sqrt -1) (/ 1 x)))

prune302.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

herbie shell --seed 2018357 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))