Average Error: 57.9 → 0.7
Time: 9.3s
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 2019005 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))

Details

Time bar (total: 8.6s)Debug log

sample152.0ms

Algorithm
intervals

simplify26.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

26.0ms
(/ (- (exp x) (exp (- x))) 2)

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.5b

localize37.0ms

Local error

Found 1 expressions with local error:

6.7b
(- (exp x) (exp (- x)))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
1 → 14
Calls

1 calls. Slowest were:

9.0ms
(- (exp x) (exp (- x)))

series45.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

45.0ms
(- (exp x) (exp (- x)))

simplify89.0ms

Counts
7 → 17
Calls

7 calls. Slowest were:

34.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
17.0ms
(/ (exp (exp x)) (exp (exp (- x))))
15.0ms
(- (exp x) (exp (* -1 x)))

prune161.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize36.0ms

Local error

Found 3 expressions with local error:

0.2b
(* 1/3 (pow x 3))
0.1b
(* 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))

series104.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

52.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
37.0ms
(* 1/60 (pow x 5))
15.0ms
(* 1/3 (pow x 3))

simplify119.0ms

Counts
17 → 44
Calls

17 calls. Slowest were:

22.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
21.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
18.0ms
(* (exp (* 1/3 (pow x 3))) (exp (* 1/60 (pow x 5))))

prune744.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 4 expressions with local error:

2.0b
(log (exp (* 1/3 (pow x 3))))
0.2b
(* 1/3 (pow x 3))
0.1b
(* 1/60 (pow x 5))
0.0b
(exp (* 1/3 (pow x 3)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 46
Calls

4 calls. Slowest were:

3.0ms
(log (exp (* 1/3 (pow x 3))))
3.0ms
(exp (* 1/3 (pow x 3)))
2.0ms
(* 1/3 (pow x 3))

series132.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

50.0ms
(exp (* 1/3 (pow x 3)))
31.0ms
(* 1/60 (pow x 5))
25.0ms
(* 1/3 (pow x 3))
25.0ms
(log (exp (* 1/3 (pow x 3))))

simplify186.0ms

Counts
25 → 58
Calls

25 calls. Slowest were:

27.0ms
(+ (* 1/3 (pow x 3)) (+ (* 1/18 (pow x 6)) 1))
21.0ms
(log (* (cbrt (exp (* 1/3 (pow x 3)))) (cbrt (exp (* 1/3 (pow x 3))))))
19.0ms
(log (sqrt (exp (* 1/3 (pow x 3)))))

prune561.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 3 expressions with local error:

7.0b
(log (exp (- (exp x) (exp (- x)))))
6.7b
(- (exp x) (exp (- x)))
0.0b
(exp (- (exp x) (exp (- x))))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
3 → 51
Calls

3 calls. Slowest were:

13.0ms
(exp (- (exp x) (exp (- x))))
10.0ms
(log (exp (- (exp x) (exp (- x)))))
4.0ms
(- (exp x) (exp (- x)))

series164.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

74.0ms
(exp (- (exp x) (exp (- x))))
48.0ms
(log (exp (- (exp x) (exp (- x)))))
42.0ms
(- (exp x) (exp (- x)))

simplify284.0ms

Counts
35 → 60
Calls

35 calls. Slowest were:

35.0ms
(+ (* 2 x) (+ (* 2 (pow x 2)) 1))
31.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
31.0ms
(exp (- (exp x) (exp (* -1 x))))

prune450.0ms

Pruning

5 alts after pruning (1 fresh and 4 done)

Merged error: 0.0b

regimes122.0ms

Accuracy

0% (0.7b remaining)

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

bsearch3.0ms

end0.0ms

sample5.1s

Algorithm
intervals