Average Error: 0.0 → 0.0
Time: 15.9s
Precision: 64
Internal Precision: 128
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\frac{2}{e^{x} + e^{-x}}\]

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

    \[\frac{2}{e^{x} + e^{-x}}\]

  2. Initial simplification0.0

    \[\leadsto \frac{2}{e^{x} + e^{-x}}\]

  3. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{\frac{2}{e^{x} + e^{-x}}}\]

  4. Final simplification0.0

    \[\leadsto \frac{2}{e^{x} + e^{-x}}\]

Reproduce

herbie shell --seed 2018362 
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))

Details

Time bar (total: 13.6s)Debug log

start45.0ms

Algorithm
intervals

setup17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize19.0ms

Local error

Found 2 expressions with local error:

0.0b
(/ 2 (+ (exp x) (exp (- x))))
0.0b
(+ (exp x) (exp (- x)))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
2 → 28
Calls

2 calls. Slowest were:

8.0ms
(+ (exp x) (exp (- x)))
4.0ms
(/ 2 (+ (exp x) (exp (- x))))

series58.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

34.0ms
(/ 2 (+ (exp x) (exp (- x))))
24.0ms
(+ (exp x) (exp (- x)))

simplify310.0ms

Counts
14 → 34
Calls

14 calls. Slowest were:

150.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
66.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
19.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) 2))

prune362.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize1.0ms

Local error

Found 2 expressions with local error:

0.0b
(/ 2 (+ (exp x) (exp (- x))))
0.0b
(+ (exp x) (exp (- x)))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
2 → 28
Calls

2 calls. Slowest were:

7.0ms
(+ (exp x) (exp (- x)))
5.0ms
(/ 2 (+ (exp x) (exp (- x))))

series67.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

35.0ms
(+ (exp x) (exp (- x)))
32.0ms
(/ 2 (+ (exp x) (exp (- x))))

simplify325.0ms

Counts
14 → 34
Calls

14 calls. Slowest were:

191.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
46.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
19.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) 2))

prune322.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize28.0ms

Local error

Found 4 expressions with local error:

4.8b
(+ -1 (* (exp x) (exp x)))
0.7b
(* (/ 2 (+ (pow (exp x) 3) (pow (exp (- x)) 3))) (+ (exp (- (+ x x))) (+ -1 (* (exp x) (exp x)))))
0.0b
(pow (exp (- x)) 3)
0.0b
(pow (exp x) 3)

rewrite34.0ms

Algorithm
rewrite-expression-head
Counts
4 → 90
Calls

4 calls. Slowest were:

25.0ms
(* (/ 2 (+ (pow (exp x) 3) (pow (exp (- x)) 3))) (+ (exp (- (+ x x))) (+ -1 (* (exp x) (exp x)))))
4.0ms
(pow (exp (- x)) 3)
2.0ms
(pow (exp x) 3)

series216.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

146.0ms
(* (/ 2 (+ (pow (exp x) 3) (pow (exp (- x)) 3))) (+ (exp (- (+ x x))) (+ -1 (* (exp x) (exp x)))))
26.0ms
(+ -1 (* (exp x) (exp x)))
22.0ms
(pow (exp (- x)) 3)
21.0ms
(pow (exp x) 3)

simplify9.0s

Counts
68 → 102
Calls

68 calls. Slowest were:

711.0ms
(* 2 (+ (* 1 (+ (* -1 -1) (- (* (* (exp x) (exp x)) (* (exp x) (exp x))) (* -1 (* (exp x) (exp x)))))) (* (exp (+ x x)) (+ (pow -1 3) (pow (* (exp x) (exp x)) 3)))))
507.0ms
(* 2 (+ (pow (exp (- (+ x x))) 3) (pow (+ -1 (* (exp x) (exp x))) 3)))
476.0ms
(/ 2 (+ (pow (pow (exp x) 3) 3) (pow (pow (exp (- x)) 3) 3)))

prune1.9s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize16.0ms

Local error

Found 2 expressions with local error:

11.2b
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
0.1b
(* 5/24 (pow x 4))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
2 → 24
Calls

2 calls. Slowest were:

11.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
1.0ms
(* 5/24 (pow x 4))

series40.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

22.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
19.0ms
(* 5/24 (pow x 4))

simplify231.0ms

Counts
11 → 30
Calls

11 calls. Slowest were:

89.0ms
(/ (exp (+ (* 5/24 (pow x 4)) 1)) (exp (* 1/2 (pow x 2))))
54.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
37.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))

prune504.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes84.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms