Average Error: 0.0 → 0.0
Time: 14.5s
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 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))

Details

Time bar (total: 14.1s)Debug log

sample17.0ms

Algorithm
intervals

simplify4.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 1 expressions with local error:

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

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
1 → 18
Calls

1 calls. Slowest were:

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

series31.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

31.0ms
(/ 2 (+ (exp x) (exp (- x))))

simplify349.0ms

Counts
9 → 21
Calls

9 calls. Slowest were:

198.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
90.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
17.0ms
(/ 2 (+ (exp (* -1 x)) (exp x)))

prune185.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize14.0ms

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))))))

rewrite37.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

28.0ms
(* (* (cbrt (/ 2 (+ (exp x) (exp (- x))))) (cbrt (/ 2 (+ (exp x) (exp (- x)))))) (cbrt (/ 2 (+ (exp x) (exp (- x))))))
4.0ms
(cbrt (/ 2 (+ (exp x) (exp (- x)))))
2.0ms
(cbrt (/ 2 (+ (exp x) (exp (- x)))))

series902.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

290.0ms
(cbrt (/ 2 (+ (exp x) (exp (- x)))))
254.0ms
(cbrt (/ 2 (+ (exp x) (exp (- x)))))
234.0ms
(cbrt (/ 2 (+ (exp x) (exp (- x)))))
124.0ms
(* (* (cbrt (/ 2 (+ (exp x) (exp (- x))))) (cbrt (/ 2 (+ (exp x) (exp (- x)))))) (cbrt (/ 2 (+ (exp x) (exp (- x))))))

simplify3.4s

Counts
72 → 105
Calls

72 calls. Slowest were:

455.0ms
(* (* (* (* (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)))))
442.0ms
(cbrt (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))))
369.0ms
(cbrt (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))))

prune1.2s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize13.0ms

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))

rewrite35.0ms

Algorithm
rewrite-expression-head
Counts
4 → 88
Calls

4 calls. Slowest were:

27.0ms
(* (* (cbrt (/ 2 (+ (exp x) (exp (- x))))) (cbrt (/ 2 (+ (exp x) (exp (- x)))))) (pow (/ 2 (+ (exp x) (exp (- x)))) 1/3))
3.0ms
(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:

364.0ms
(* (* (cbrt (/ 2 (+ (exp x) (exp (- x))))) (cbrt (/ 2 (+ (exp x) (exp (- x)))))) (pow (/ 2 (+ (exp x) (exp (- x)))) 1/3))
253.0ms
(cbrt (/ 2 (+ (exp x) (exp (- x)))))
233.0ms
(pow (/ 2 (+ (exp x) (exp (- x)))) 1/3)
231.0ms
(cbrt (/ 2 (+ (exp x) (exp (- x)))))

simplify4.7s

Counts
60 → 100
Calls

60 calls. Slowest were:

507.0ms
(- (+ (* 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))))
474.0ms
(* (* (* (* (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)))))
408.0ms
(cbrt (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))))

prune1.3s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes77.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample697.0ms

Algorithm
intervals