Average Error: 0.0 → 0.0
Time: 13.2s
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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019005 
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))

Details

Time bar (total: 12.8s)Debug log

sample19.0ms

Algorithm
intervals

simplify6.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

6.0ms
(/ 2 (+ (exp x) (exp (- x))))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 1 expressions with local error:

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

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
1 → 16
Calls

1 calls. Slowest were:

3.0ms
(/ 2 (+ (exp x) (exp (- x))))

series43.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

43.0ms
(/ 2 (+ (exp x) (exp (- x))))

simplify246.0ms

Counts
9 → 19
Calls

9 calls. Slowest were:

145.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
56.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
17.0ms
(/ 2 (+ (exp (* -1 x)) (exp x)))

prune140.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize9.0ms

Local error

Found 4 expressions with local error:

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

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 85
Calls

4 calls. Slowest were:

13.0ms
(* (* (cbrt (/ 2 (+ (exp x) (exp (- x))))) (cbrt (/ 2 (+ (exp x) (exp (- x)))))) (cbrt (/ 2 (+ (exp x) (exp (- x))))))
1.0ms
(cbrt (/ 2 (+ (exp x) (exp (- x)))))
1.0ms
(cbrt (/ 2 (+ (exp x) (exp (- x)))))

series898.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

282.0ms
(cbrt (/ 2 (+ (exp x) (exp (- x)))))
226.0ms
(cbrt (/ 2 (+ (exp x) (exp (- x)))))
222.0ms
(cbrt (/ 2 (+ (exp x) (exp (- x)))))
168.0ms
(* (* (cbrt (/ 2 (+ (exp x) (exp (- x))))) (cbrt (/ 2 (+ (exp x) (exp (- x)))))) (cbrt (/ 2 (+ (exp x) (exp (- x))))))

simplify3.0s

Counts
72 → 97
Calls

72 calls. Slowest were:

507.0ms
(* (* (* (* (cbrt (/ 2 (+ (exp x) (exp (- x))))) (cbrt (/ 2 (+ (exp x) (exp (- x)))))) (* (cbrt (/ 2 (+ (exp x) (exp (- x))))) (cbrt (/ 2 (+ (exp x) (exp (- x))))))) (* (cbrt (/ 2 (+ (exp x) (exp (- x))))) (cbrt (/ 2 (+ (exp x) (exp (- x))))))) (/ 2 (+ (exp x) (exp (- x)))))
283.0ms
(cbrt (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))))
226.0ms
(- (+ (* (pow 1/2 1/3) (cbrt 2)) (* 1/24 (* (* (pow x 4) (cbrt 2)) (pow 1/2 1/3)))) (* 1/6 (* (* (pow x 2) (cbrt 2)) (pow 1/2 1/3))))

prune1.0s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize11.0ms

Local error

Found 4 expressions with local error:

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

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 80
Calls

4 calls. Slowest were:

15.0ms
(* (* (cbrt (/ 2 (+ (exp x) (exp (- x))))) (cbrt (/ 2 (+ (exp x) (exp (- x)))))) (pow (/ 2 (+ (exp x) (exp (- x)))) 1/3))
1.0ms
(pow (/ 2 (+ (exp x) (exp (- x)))) 1/3)
1.0ms
(cbrt (/ 2 (+ (exp x) (exp (- x)))))

series1.1s

Counts
4 → 12
Calls

4 calls. Slowest were:

426.0ms
(* (* (cbrt (/ 2 (+ (exp x) (exp (- x))))) (cbrt (/ 2 (+ (exp x) (exp (- x)))))) (pow (/ 2 (+ (exp x) (exp (- x)))) 1/3))
277.0ms
(cbrt (/ 2 (+ (exp x) (exp (- x)))))
253.0ms
(cbrt (/ 2 (+ (exp x) (exp (- x)))))
170.0ms
(pow (/ 2 (+ (exp x) (exp (- x)))) 1/3)

simplify3.9s

Counts
60 → 92
Calls

60 calls. Slowest were:

444.0ms
(* (* (* (* (cbrt (/ 2 (+ (exp x) (exp (- x))))) (cbrt (/ 2 (+ (exp x) (exp (- x)))))) (* (cbrt (/ 2 (+ (exp x) (exp (- x))))) (cbrt (/ 2 (+ (exp x) (exp (- x))))))) (* (cbrt (/ 2 (+ (exp x) (exp (- x))))) (cbrt (/ 2 (+ (exp x) (exp (- x))))))) (/ 2 (+ (exp x) (exp (- x)))))
396.0ms
(- (+ (* 1/2 (* (pow (cbrt 2) 2) (pow 2 1/3))) (* 5/48 (* (* (pow x 4) (pow (cbrt 2) 2)) (pow 2 1/3)))) (* 1/4 (* (* (pow x 2) (pow (cbrt 2) 2)) (pow 2 1/3))))
295.0ms
(* (* (/ 2 (+ (exp x) (exp (- x)))) (/ 2 (+ (exp x) (exp (- x))))) (* (* (pow (/ 2 (+ (exp x) (exp (- x)))) 1/3) (pow (/ 2 (+ (exp x) (exp (- x)))) 1/3)) (pow (/ 2 (+ (exp x) (exp (- x)))) 1/3)))

prune1.2s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes109.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample1.1s

Algorithm
intervals