Average Error: 58.1 → 0.6
Time: 12.0s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{2 \cdot x + \left({x}^{5} \cdot \frac{1}{60} + \frac{1}{3} \cdot {x}^{3}\right)}{2}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.1

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

  2. Taylor expanded around 0 0.6

    \[\leadsto \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2}\]

  3. Final simplification0.6

    \[\leadsto \frac{2 \cdot x + \left({x}^{5} \cdot \frac{1}{60} + \frac{1}{3} \cdot {x}^{3}\right)}{2}\]

Reproduce

herbie shell --seed 2019002 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))

Details

Time bar (total: 11.5s)Debug log

sample196.0ms

Algorithm
intervals

simplify18.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

18.0ms
(/ (- (exp x) (exp (- x))) 2)

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.8b

localize13.0ms

Local error

Found 1 expressions with local error:

5.3b
(- (exp x) (exp (- x)))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
1 → 14
Calls

1 calls. Slowest were:

4.0ms
(- (exp x) (exp (- x)))

series44.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

44.0ms
(- (exp x) (exp (- x)))

simplify70.0ms

Counts
7 → 17
Calls

7 calls. Slowest were:

25.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
13.0ms
(- (exp x) (exp (* -1 x)))
13.0ms
(/ (exp (exp x)) (exp (exp (- x))))

prune140.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize25.0ms

Local error

Found 4 expressions with local error:

0.1b
(* 1/3 (pow x 3))
0.0b
(* 1/60 (pow x 5))
0.0b
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
0.0b
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

6.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
3.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
1.0ms
(* 1/3 (pow x 3))

series141.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

73.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
34.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
19.0ms
(* 1/60 (pow x 5))
15.0ms
(* 1/3 (pow x 3))

simplify528.0ms

Counts
23 → 60
Calls

23 calls. Slowest were:

157.0ms
(* (exp (* 2 x)) (* (exp (* 1/3 (pow x 3))) (exp (* 1/60 (pow x 5)))))
136.0ms
(* (exp (* 2 x)) (exp (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))))
44.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))

prune439.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

5.3b
(- (sqrt (exp x)) (sqrt (exp (- x))))
0.0b
(+ (sqrt (exp x)) (sqrt (exp (- x))))
0.0b
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (sqrt (exp x)) (sqrt (exp (- x)))))
0.0b
(sqrt (exp (- x)))

rewrite35.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

21.0ms
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (sqrt (exp x)) (sqrt (exp (- x)))))
6.0ms
(- (sqrt (exp x)) (sqrt (exp (- x))))
5.0ms
(+ (sqrt (exp x)) (sqrt (exp (- x))))

series217.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

110.0ms
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (sqrt (exp x)) (sqrt (exp (- x)))))
56.0ms
(- (sqrt (exp x)) (sqrt (exp (- x))))
39.0ms
(+ (sqrt (exp x)) (sqrt (exp (- x))))
12.0ms
(sqrt (exp (- x)))

simplify2.8s

Counts
65 → 94
Calls

65 calls. Slowest were:

333.0ms
(* (+ (* (sqrt (exp x)) (sqrt (exp x))) (- (* (sqrt (exp (- x))) (sqrt (exp (- x)))) (* (sqrt (exp x)) (sqrt (exp (- x)))))) (+ (* (sqrt (exp x)) (sqrt (exp x))) (+ (* (sqrt (exp (- x))) (sqrt (exp (- x)))) (* (sqrt (exp x)) (sqrt (exp (- x)))))))
274.0ms
(* (+ (pow (sqrt (exp x)) 3) (pow (sqrt (exp (- x))) 3)) (- (pow (sqrt (exp x)) 3) (pow (sqrt (exp (- x))) 3)))
189.0ms
(* (+ (pow (sqrt (exp x)) 3) (pow (sqrt (exp (- x))) 3)) (- (sqrt (exp x)) (sqrt (exp (- x)))))

prune828.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes124.0ms

Accuracy

0% (0.6b remaining)

Error of 0.6b against oracle of 0.0b and baseline of 0.6b

bsearch3.0ms

end0.0ms

sample5.9s

Algorithm
intervals