Average Error: 57.9 → 0.7
Time: 20.2s
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 57.9

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

  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 2019004 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))

Details

Time bar (total: 19.6s)Debug log

sample189.0ms

Algorithm
intervals

simplify33.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

33.0ms
(/ (- (exp x) (exp (- x))) 2)

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.0b

localize19.0ms

Local error

Found 1 expressions with local error:

5.1b
(- (exp x) (exp (- x)))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
1 → 14
Calls

1 calls. Slowest were:

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

series56.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

56.0ms
(- (exp x) (exp (- x)))

simplify112.0ms

Counts
7 → 17
Calls

7 calls. Slowest were:

45.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
20.0ms
(/ (exp (exp x)) (exp (exp (- x))))
17.0ms
(- (exp x) (exp (* -1 x)))

prune188.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.0b

localize30.0ms

Local error

Found 3 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)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
3 → 35
Calls

3 calls. Slowest were:

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

series115.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

68.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
31.0ms
(* 1/60 (pow x 5))
16.0ms
(* 1/3 (pow x 3))

simplify103.0ms

Counts
17 → 44
Calls

17 calls. Slowest were:

21.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
17.0ms
(* (exp (* 1/3 (pow x 3))) (exp (* 1/60 (pow x 5))))
13.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))

prune389.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 4 expressions with local error:

5.0b
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
0.0b
(* (exp x) (exp x))
0.0b
(/ (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (+ (exp x) (exp (- x))))
0.0b
(* (exp (- x)) (exp (- x)))

rewrite35.0ms

Algorithm
rewrite-expression-head
Counts
4 → 111
Calls

4 calls. Slowest were:

15.0ms
(/ (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (+ (exp x) (exp (- x))))
8.0ms
(* (exp (- x)) (exp (- x)))
6.0ms
(* (exp x) (exp x))

series157.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

85.0ms
(/ (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (+ (exp x) (exp (- x))))
51.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
11.0ms
(* (exp (- x)) (exp (- x)))
11.0ms
(* (exp x) (exp x))

simplify7.7s

Counts
111 → 123
Calls

111 calls. Slowest were:

317.0ms
(/ (sqrt (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))) (+ (exp x) (exp (- x))))
311.0ms
(/ (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (+ (exp x) (exp (- x))))
310.0ms
(/ (sqrt (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))) 1)

prune821.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize20.0ms

Local error

Found 4 expressions with local error:

5.0b
(+ (/ -1 (exp (+ x x))) (exp (+ x x)))
5.0b
(log (exp (+ (/ -1 (exp (+ x x))) (exp (+ x x)))))
0.0b
(/ (log (exp (+ (/ -1 (exp (+ x x))) (exp (+ x x))))) (+ (exp x) (exp (- x))))
0.0b
(/ -1 (exp (+ x x)))

rewrite105.0ms

Algorithm
rewrite-expression-head
Counts
4 → 101
Calls

4 calls. Slowest were:

75.0ms
(/ (log (exp (+ (/ -1 (exp (+ x x))) (exp (+ x x))))) (+ (exp x) (exp (- x))))
20.0ms
(log (exp (+ (/ -1 (exp (+ x x))) (exp (+ x x)))))
6.0ms
(+ (/ -1 (exp (+ x x))) (exp (+ x x)))

series181.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

79.0ms
(/ (log (exp (+ (/ -1 (exp (+ x x))) (exp (+ x x))))) (+ (exp x) (exp (- x))))
46.0ms
(log (exp (+ (/ -1 (exp (+ x x))) (exp (+ x x)))))
45.0ms
(+ (/ -1 (exp (+ x x))) (exp (+ x x)))
10.0ms
(/ -1 (exp (+ x x)))

simplify2.3s

Counts
105 → 113
Calls

105 calls. Slowest were:

554.0ms
(/ (* (* (log (exp (+ (/ -1 (exp (+ x x))) (exp (+ x x))))) (log (exp (+ (/ -1 (exp (+ x x))) (exp (+ x x)))))) (log (exp (+ (/ -1 (exp (+ x x))) (exp (+ x x)))))) (* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x)))))
158.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
102.0ms
(/ (- (exp (* 2 x)) (/ 1 (exp (* 2 x)))) (+ (exp (* -1 x)) (exp x)))

prune1.0s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

regimes68.0ms

Accuracy

0% (0.6b remaining)

Error of 0.7b against oracle of 0.1b and baseline of 0.7b

bsearch4.0ms

end0.0ms

sample5.9s

Algorithm
intervals