Average Error: 0.0 → 0.0
Time: 5.4s
Precision: 64
Internal Precision: 128
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}\]

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. Using strategy rm

  3. Applied add-sqr-sqrt0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019004 
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))

Details

Time bar (total: 5.0s)Debug log

sample33.0ms

Algorithm
intervals

simplify8.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize21.0ms

Local error

Found 2 expressions with local error:

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

rewrite20.0ms

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))))

series68.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

40.0ms
(/ 2 (+ (exp x) (exp (- x))))
28.0ms
(+ (exp x) (exp (- x)))

simplify309.0ms

Counts
14 → 34
Calls

14 calls. Slowest were:

163.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
50.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
19.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) 2))

prune324.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize10.0ms

Local error

Found 4 expressions with local error:

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

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

8.0ms
(+ (exp x) (exp (- x)))
8.0ms
(+ (exp x) (exp (- x)))
5.0ms
(/ 2 (+ (exp x) (exp (- x))))

series150.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

51.0ms
(/ 2 (+ (exp x) (exp (- x))))
42.0ms
(/ 2 (+ (exp x) (exp (- x))))
31.0ms
(+ (exp x) (exp (- x)))
25.0ms
(+ (exp x) (exp (- x)))

simplify478.0ms

Counts
28 → 68
Calls

28 calls. Slowest were:

131.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
123.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
61.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))

prune563.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize21.0ms

Local error

Found 4 expressions with local error:

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

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

8.0ms
(+ (exp x) (exp (- x)))
5.0ms
(/ 2 (+ (exp x) (exp (- x))))
5.0ms
(log (exp (/ 2 (+ (exp x) (exp (- x))))))

series236.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

110.0ms
(exp (/ 2 (+ (exp x) (exp (- x)))))
52.0ms
(/ 2 (+ (exp x) (exp (- x))))
40.0ms
(+ (exp x) (exp (- x)))
33.0ms
(log (exp (/ 2 (+ (exp x) (exp (- x))))))

simplify1.0s

Counts
36 → 72
Calls

36 calls. Slowest were:

203.0ms
(exp (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))))
165.0ms
(- (+ (* 1/3 (* (pow x 4) E)) E) (* 1/2 (* (pow x 2) E)))
160.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))

prune636.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes91.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample988.0ms

Algorithm
intervals