Average Error: 0.0 → 0.0
Time: 4.0s
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 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))

Details

Time bar (total: 2.4s)Debug log

start40.0ms

Algorithm
intervals

setup22.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize39.0ms

Local error

Found 2 expressions with local error:

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

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
2 → 35
Calls

2 calls. Slowest were:

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

series56.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

32.0ms
(/ 2 (+ (exp x) (exp (- x))))
24.0ms
(+ (exp x) (exp (- x)))

simplify399.0ms

Counts
16 → 41
Calls

16 calls. Slowest were:

262.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
68.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
13.0ms
(/ 2 (+ (exp (* -1 x)) (exp x)))

prune374.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize0.0ms

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))))

series70.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

37.0ms
(/ 2 (+ (exp x) (exp (- x))))
32.0ms
(+ (exp x) (exp (- x)))

simplify398.0ms

Counts
16 → 41
Calls

16 calls. Slowest were:

271.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
52.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
11.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) 2))

prune436.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 1 expressions with local error:

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

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 10
Calls

1 calls. Slowest were:

0.0ms
(fma 5/24 (pow x 4) (fma (* x x) -1/2 1))

series34.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

34.0ms
(fma 5/24 (pow x 4) (fma (* x x) -1/2 1))

simplify217.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

92.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
73.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
50.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))

prune213.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes86.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms