Average Error: 0.0 → 0.0
Time: 9.2s
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 2019010 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))

Details

Time bar (total: 9.0s)Debug log

sample184.0ms

Algorithm
intervals

simplify6.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
6.0ms
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize25.0ms

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
(sqrt (- 1 (* x x)))

rewrite38.0ms

Algorithm
rewrite-expression-head
Rules
10×*-un-lft-identity
7×add-sqr-sqrt
6×associate-/l*
6×div-inv
6×sqrt-prod
6×add-cube-cbrt
5×add-log-exp
4×sqrt-div
3×add-exp-log
3×add-cbrt-cube
3×pow1
2×associate-/l/
2×distribute-lft-out
2×flip--
2×flip3--
2×distribute-rgt-out
1×flip-+
1×distribute-rgt1-in
1×frac-add
1×pow1/2
1×flip3-+
1×frac-2neg
1×sum-log
1×clear-num
1×rem-sqrt-square
1×+-commutative
Counts
3 → 49
Calls
3 calls:
Slowest
29.0ms
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))
7.0ms
(/ (sqrt (- 1 (* x x))) x)
1.0ms
(sqrt (- 1 (* x x)))

series149.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
76.0ms
(/ (sqrt (- 1 (* x x))) x)
45.0ms
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))
29.0ms
(sqrt (- 1 (* x x)))

simplify2.6s

Counts
32 → 58
Calls
32 calls:
Slowest
540.0ms
(- (* 1/8 (/ 1 (* (pow x 4) (pow (sqrt -1) 3)))) (+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (sqrt -1)))
523.0ms
(- (+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (sqrt -1)) (* 1/8 (/ 1 (* (pow x 4) (pow (sqrt -1) 3)))))
285.0ms
(- (* 1/8 (/ 1 (* (pow x 3) (pow (sqrt -1) 3)))) (+ (* x (sqrt -1)) (* 1/2 (/ 1 (* x (sqrt -1))))))
239.0ms
(- (+ (* x (sqrt -1)) (* 1/2 (/ 1 (* x (sqrt -1))))) (* 1/8 (/ 1 (* (pow x 3) (pow (sqrt -1) 3)))))
210.0ms
(- (/ 1 x) (+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (sqrt -1)))

prune575.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample5.4s

Algorithm
intervals