Average Error: 0.1 → 0.1
Time: 9.1s
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 0.1

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

  2. Final simplification0.1

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

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))

Details

Time bar (total: 8.8s)Debug log

sample193.0ms

Algorithm
intervals
Results
1052×(pre true 80)
796×(body nan 80)
256×(body real 80)

simplify18.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
17.0ms
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize39.0ms

Local error

Found 2 expressions with local error:

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

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
6×associate-/l*
6×sqrt-prod
6×add-sqr-sqrt
5×add-cube-cbrt
5×*-un-lft-identity
4×sqrt-div
2×associate-/l/
2×add-log-exp
2×flip--
2×log1p-expm1-u
2×add-exp-log
2×add-cbrt-cube
2×flip3--
2×pow1
2×expm1-log1p-u
1×div-inv
1×pow1/2
1×frac-2neg
1×clear-num
1×rem-sqrt-square
Counts
2 → 36
Calls
2 calls:
Slowest
6.0ms
(/ (sqrt (- 1 (* x x))) x)
2.0ms
(sqrt (- 1 (* x x)))

series109.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
74.0ms
(/ (sqrt (- 1 (* x x))) x)
35.0ms
(sqrt (- 1 (* x x)))

simplify1.8s

Counts
19 → 42
Calls
19 calls:
Slowest
398.0ms
(- (* 1/8 (/ 1 (* (pow x 3) (pow (sqrt -1) 3)))) (+ (* x (sqrt -1)) (* 1/2 (/ 1 (* x (sqrt -1))))))
293.0ms
(- (* 1/8 (/ 1 (* (pow x 4) (pow (sqrt -1) 3)))) (+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (sqrt -1)))
256.0ms
(- (+ (* x (sqrt -1)) (* 1/2 (/ 1 (* x (sqrt -1))))) (* 1/8 (/ 1 (* (pow x 3) (pow (sqrt -1) 3)))))
251.0ms
(- (/ 1 x) (+ (* 1/2 x) (* 1/8 (pow x 3))))
243.0ms
(- (+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (sqrt -1)) (* 1/8 (/ 1 (* (pow x 4) (pow (sqrt -1) 3)))))

prune383.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample6.3s

Algorithm
intervals
Results
32004×(pre true 80)
24004×(body nan 80)
8000×(body real 80)