Average Error: 0.0 → 0.0
Time: 27.7s
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. Initial simplification0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2018365 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))

Details

Time bar (total: 18.9s)Debug log

start289.0ms

Algorithm
intervals

setup28.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize31.0ms

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)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

4.0ms
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))
3.0ms
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
3.0ms
(/ (sqrt (- 1 (* x x))) x)

series244.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

111.0ms
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
58.0ms
(/ (sqrt (- 1 (* x x))) x)
49.0ms
(+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))
26.0ms
(sqrt (- 1 (* x x)))

simplify3.8s

Counts
47 → 81
Calls

47 calls. Slowest were:

547.0ms
(- (+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (sqrt -1)) (* 1/8 (/ 1 (* (pow x 4) (pow (sqrt -1) 3)))))
521.0ms
(- (* 1/8 (/ 1 (* (pow x 4) (pow (sqrt -1) 3)))) (+ (* 1/2 (/ 1 (* (pow x 2) (sqrt -1)))) (sqrt -1)))
362.0ms
(- (+ (log (sqrt -1)) (/ 1 (* x (sqrt -1)))) (* 1/6 (/ 1 (* (pow x 3) (pow (sqrt -1) 3)))))

prune1.1s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
0.6b
(cbrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
0.6b
(cbrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
0.5b
(* (cbrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))) (cbrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))

rewrite42.0ms

Algorithm
rewrite-expression-head
Counts
4 → 110
Calls

4 calls. Slowest were:

25.0ms
(* (cbrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))) (cbrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))
5.0ms
(cbrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
5.0ms
(cbrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))

series584.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

165.0ms
(cbrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
149.0ms
(cbrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
141.0ms
(* (cbrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))) (cbrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))
128.0ms
(cbrt (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))

simplify11.2s

Counts
99 → 122
Calls

99 calls. Slowest were:

491.0ms
(- (pow -1 1/3) (+ (* 2/3 (* (/ 1 (pow x 2)) (pow -1 1/3))) (+ (* 2/3 (* (/ (sqrt -1) x) (pow -1 1/3))) (* 4/9 (* (/ (pow (sqrt -1) 2) (pow x 2)) (pow -1 1/3))))))
412.0ms
(+ (* 1/3 (* (/ 1 x) (pow (/ 1 (pow (sqrt -1) 2)) 1/3))) (+ (* 1/18 (* (/ 1 (pow x 2)) (pow (/ 1 (pow (sqrt -1) 5)) 1/3))) (pow (sqrt -1) 1/3)))
396.0ms
(+ (* 1/3 (* (/ 1 x) (pow (/ 1 (pow (sqrt -1) 2)) 1/3))) (+ (* 1/18 (* (/ 1 (pow x 2)) (pow (/ 1 (pow (sqrt -1) 5)) 1/3))) (pow (sqrt -1) 1/3)))

prune1.5s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes149.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms