Average Error: 0.0 → 0.0
Time: 20.9s
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 2018360 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))

Details

Time bar (total: 11.3s)Debug log

start253.0ms

Algorithm
intervals

setup31.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize39.0ms

Local error

Found 2 expressions with local error:

0.2b
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
0.0b
(- 1 (* x x))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
2 → 31
Calls

2 calls. Slowest were:

7.0ms
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
1.0ms
(- 1 (* x x))

series155.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

126.0ms
(log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x)))
29.0ms
(- 1 (* x x))

simplify1.2s

Counts
18 → 37
Calls

18 calls. Slowest were:

259.0ms
(log (+ (pow (/ 1 x) 3) (pow (/ (sqrt (- 1 (* x x))) x) 3)))
244.0ms
(- (log 2) (+ (* 1/4 (pow x 2)) (+ (* 3/32 (pow x 4)) (log x))))
220.0ms
(- (+ (log (sqrt -1)) (/ 1 (* x (sqrt -1)))) (* 1/6 (/ 1 (* (pow x 3) (pow (sqrt -1) 3)))))

prune356.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize19.0ms

Local error

Found 3 expressions with local error:

0.2b
(+ (- (log x)) (log (+ 1 (sqrt (- 1 (* x x))))))
0.0b
(+ 1 (sqrt (- 1 (* x x))))
0.0b
(- 1 (* x x))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
3 → 43
Calls

3 calls. Slowest were:

7.0ms
(+ (- (log x)) (log (+ 1 (sqrt (- 1 (* x x))))))
1.0ms
(+ 1 (sqrt (- 1 (* x x))))
1.0ms
(- 1 (* x x))

series161.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

125.0ms
(+ (- (log x)) (log (+ 1 (sqrt (- 1 (* x x))))))
19.0ms
(- 1 (* x x))
17.0ms
(+ 1 (sqrt (- 1 (* x x))))

simplify1.2s

Counts
20 → 52
Calls

20 calls. Slowest were:

242.0ms
(- (log 2) (+ (* 1/4 (pow x 2)) (+ (* 3/32 (pow x 4)) (log x))))
241.0ms
(log (+ (* 1 1) (- (* (sqrt (- 1 (* x x))) (sqrt (- 1 (* x x)))) (* 1 (sqrt (- 1 (* x x)))))))
226.0ms
(- (+ (* 1/6 (/ 1 (* (pow x 3) (pow (sqrt -1) 3)))) (log (sqrt -1))) (+ (/ 1 (* x (sqrt -1))) (log -1)))

prune596.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize27.0ms

Local error

Found 4 expressions with local error:

1.0b
(* (log (+ 1 (sqrt (- 1 (* x x))))) (log (+ 1 (sqrt (- 1 (* x x))))))
1.0b
(pow (log (+ 1 (sqrt (- 1 (* x x))))) 3)
0.7b
(pow (- (log x)) 3)
0.5b
(* (- (log x)) (- (log x)))

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 120
Calls

4 calls. Slowest were:

11.0ms
(* (- (log x)) (- (log x)))
9.0ms
(* (log (+ 1 (sqrt (- 1 (* x x))))) (log (+ 1 (sqrt (- 1 (* x x))))))
3.0ms
(pow (- (log x)) 3)

series641.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

272.0ms
(pow (- (log x)) 3)
161.0ms
(* (- (log x)) (- (log x)))
124.0ms
(pow (log (+ 1 (sqrt (- 1 (* x x))))) 3)
84.0ms
(* (log (+ 1 (sqrt (- 1 (* x x))))) (log (+ 1 (sqrt (- 1 (* x x))))))

simplify4.0s

Counts
91 → 132
Calls

91 calls. Slowest were:

576.0ms
(pow (- (log (/ -1 x)) (log -1)) 3)
446.0ms
(- (+ (pow (log (/ -1 x)) 2) (+ (* 2 (/ (log (/ -1 x)) (* x (sqrt -1)))) (pow (log (sqrt -1)) 2))) (+ (/ 1 (pow x 2)) (+ (* 2 (/ (log (sqrt -1)) (* x (sqrt -1)))) (* 2 (* (log (sqrt -1)) (log (/ -1 x)))))))
420.0ms
(- (+ (* 1/16 (pow x 4)) (pow (log 2) 2)) (+ (* 1/2 (* (log 2) (pow x 2))) (* 3/16 (* (log 2) (pow x 4)))))

prune2.4s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes141.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms