Average Error: 57.9 → 0.6
Time: 18.2s
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 0.6

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

  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 2019007 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))

Details

Time bar (total: 17.6s)Debug log

sample153.0ms

Algorithm
intervals

simplify22.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

22.0ms
(/ (- (exp x) (exp (- x))) 2)

prune4.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.4b

localize36.0ms

Local error

Found 1 expressions with local error:

7.5b
(- (exp x) (exp (- x)))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
1 → 14
Calls

1 calls. Slowest were:

5.0ms
(- (exp x) (exp (- x)))

series51.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

51.0ms
(- (exp x) (exp (- x)))

simplify92.0ms

Counts
7 → 17
Calls

7 calls. Slowest were:

29.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
24.0ms
(/ (exp (exp x)) (exp (exp (- x))))
12.0ms
(- (exp x) (exp (* -1 x)))

prune154.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize21.0ms

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)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
3 → 35
Calls

3 calls. Slowest were:

4.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
1.0ms
(* 1/3 (pow x 3))
1.0ms
(* 1/60 (pow x 5))

series88.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

50.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
23.0ms
(* 1/60 (pow x 5))
15.0ms
(* 1/3 (pow x 3))

simplify98.0ms

Counts
17 → 44
Calls

17 calls. Slowest were:

17.0ms
(* (exp (* 1/3 (pow x 3))) (exp (* 1/60 (pow x 5))))
12.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
11.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))

prune414.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize7.0ms

Local error

Found 2 expressions with local error:

7.5b
(- (exp x) (exp (- x)))
7.5b
(log (exp (- (exp x) (exp (- x)))))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
2 → 34
Calls

2 calls. Slowest were:

12.0ms
(log (exp (- (exp x) (exp (- x)))))
4.0ms
(- (exp x) (exp (- x)))

series103.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

53.0ms
(- (exp x) (exp (- x)))
50.0ms
(log (exp (- (exp x) (exp (- x)))))

simplify276.0ms

Counts
22 → 40
Calls

22 calls. Slowest were:

49.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
45.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
27.0ms
(log (* (cbrt (exp (- (exp x) (exp (- x))))) (cbrt (exp (- (exp x) (exp (- x)))))))

prune278.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 4 expressions with local error:

7.5b
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
0.2b
(/ (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (+ (exp x) (exp (- x))))
0.0b
(* (exp (- x)) (exp (- x)))
0.0b
(* (exp x) (exp x))

rewrite48.0ms

Algorithm
rewrite-expression-head
Counts
4 → 111
Calls

4 calls. Slowest were:

26.0ms
(/ (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (+ (exp x) (exp (- x))))
12.0ms
(* (exp (- x)) (exp (- x)))
6.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))

series197.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

111.0ms
(/ (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (+ (exp x) (exp (- x))))
56.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
18.0ms
(* (exp (- x)) (exp (- x)))
12.0ms
(* (exp x) (exp x))

simplify8.9s

Counts
111 → 123
Calls

111 calls. Slowest were:

435.0ms
(/ (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (+ (exp x) (exp (- x))))
372.0ms
(- (* (* (exp x) (exp x)) (* (exp x) (exp x))) (* (* (exp (- x)) (exp (- x))) (* (exp (- x)) (exp (- x)))))
366.0ms
(/ (- (pow (exp x) 2) (pow (exp (* -1 x)) 2)) (+ (exp (* -1 x)) (exp x)))

prune645.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

regimes144.0ms

Accuracy

0% (0.6b remaining)

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

bsearch2.0ms

end0.0ms

sample5.8s

Algorithm
intervals