Average Error: 57.8 → 0.7
Time: 7.4s
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 57.8

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

  2. Taylor expanded around 0 0.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2019008 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))

Details

Time bar (total: 7.2s)Debug log

sample160.0ms

Algorithm
intervals

simplify35.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

35.0ms
(/ (- (exp x) (exp (- x))) 2)

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.3b

localize15.0ms

Local error

Found 1 expressions with local error:

5.2b
(- (exp x) (exp (- x)))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
1 → 14
Calls

1 calls. Slowest were:

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

series75.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

75.0ms
(- (exp x) (exp (- x)))

simplify75.0ms

Counts
7 → 17
Calls

7 calls. Slowest were:

27.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
23.0ms
(/ (exp (exp x)) (exp (exp (- x))))
8.0ms
(- (exp x) (exp (* -1 x)))

prune115.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize40.0ms

Local error

Found 2 expressions with local error:

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

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
2 → 24
Calls

2 calls. Slowest were:

2.0ms
(* 1/3 (pow x 3))
1.0ms
(* 1/60 (pow x 5))

series35.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

19.0ms
(* 1/60 (pow x 5))
16.0ms
(* 1/3 (pow x 3))

simplify30.0ms

Counts
13 → 30
Calls

13 calls. Slowest were:

8.0ms
(* 1/3 (pow x 3))
5.0ms
(* 1/3 (pow x 3))
5.0ms
(* 1/3 (pow x 3))

prune237.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 4 expressions with local error:

5.2b
(- (exp x) (exp (- x)))
5.2b
(- (exp x) (exp (- x)))
5.2b
(- (exp x) (exp (- x)))
0.1b
(cbrt (- (exp x) (exp (- x))))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 57
Calls

4 calls. Slowest were:

7.0ms
(- (exp x) (exp (- x)))
4.0ms
(- (exp x) (exp (- x)))
4.0ms
(- (exp x) (exp (- x)))

series328.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

208.0ms
(cbrt (- (exp x) (exp (- x))))
43.0ms
(- (exp x) (exp (- x)))
39.0ms
(- (exp x) (exp (- x)))
38.0ms
(- (exp x) (exp (- x)))

simplify909.0ms

Counts
31 → 69
Calls

31 calls. Slowest were:

258.0ms
(- (+ (exp (* 1/3 (+ (log 2) (log x)))) (* 1/18 (* (pow x 2) (exp (* 1/3 (+ (log 2) (log x))))))) (* 1/3240 (* (pow x 4) (exp (* 1/3 (+ (log 2) (log x)))))))
224.0ms
(cbrt (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))))
117.0ms
(cbrt (- (pow (exp x) 3) (pow (exp (- x)) 3)))

prune592.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes65.0ms

Accuracy

0% (0.7b remaining)

Error of 0.7b against oracle of 0.0b and baseline of 0.7b

bsearch3.0ms

end0.0ms

sample4.5s

Algorithm
intervals