Average Error: 0.0 → 0.0
Time: 6.8s
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 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))

Details

Time bar (total: 6.5s)Debug log

sample34.0ms

Algorithm
intervals

simplify8.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize24.0ms

Local error

Found 2 expressions with local error:

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

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
2 → 35
Calls

2 calls. Slowest were:

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

series78.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

53.0ms
(/ 2 (+ (exp x) (exp (- x))))
24.0ms
(+ (exp x) (exp (- x)))

simplify405.0ms

Counts
16 → 41
Calls

16 calls. Slowest were:

252.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
78.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
13.0ms
(+ (exp (* -1 x)) (exp x))

prune293.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize5.0ms

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)))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

9.0ms
(+ (exp x) (exp (- x)))
4.0ms
(+ (exp x) (exp (- x)))
3.0ms
(/ 2 (+ (exp x) (exp (- x))))

series174.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

54.0ms
(/ 2 (+ (exp x) (exp (- x))))
42.0ms
(+ (exp x) (exp (- x)))
41.0ms
(+ (exp x) (exp (- x)))
37.0ms
(/ 2 (+ (exp x) (exp (- x))))

simplify901.0ms

Counts
32 → 82
Calls

32 calls. Slowest were:

283.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
242.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
91.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))

prune924.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize6.0ms

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)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 71
Calls

4 calls. Slowest were:

5.0ms
(/ 2 (+ (exp x) (exp (- x))))
4.0ms
(+ (exp x) (exp (- x)))
3.0ms
(log (exp (/ 2 (+ (exp x) (exp (- x))))))

series241.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

110.0ms
(exp (/ 2 (+ (exp x) (exp (- x)))))
50.0ms
(log (exp (/ 2 (+ (exp x) (exp (- x))))))
40.0ms
(/ 2 (+ (exp x) (exp (- x))))
40.0ms
(+ (exp x) (exp (- x)))

simplify1.6s

Counts
38 → 83
Calls

38 calls. Slowest were:

397.0ms
(exp (/ 2 (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))))
259.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
255.0ms
(- (+ (* 1/3 (* (pow x 4) E)) E) (* 1/2 (* (pow x 2) E)))

prune896.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes120.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample661.0ms

Algorithm
intervals