Average Error: 58.0 → 0.6
Time: 38.8s
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 58.0

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

  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 2018360 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))

Details

Time bar (total: 30.0s)Debug log

start192.0ms

Algorithm
intervals

setup29.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.1b

localize13.0ms

Local error

Found 1 expressions with local error:

5.7b
(- (exp x) (exp (- x)))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
1 → 14
Calls

1 calls. Slowest were:

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

series47.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

47.0ms
(- (exp x) (exp (- x)))

simplify105.0ms

Counts
7 → 17
Calls

7 calls. Slowest were:

47.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
18.0ms
(- (exp x) (exp (* -1 x)))
13.0ms
(/ (exp (exp x)) (exp (exp (- x))))

prune225.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize44.0ms

Local error

Found 3 expressions with local error:

0.2b
(* 1/3 (pow x 3))
0.0b
(* 1/60 (pow x 5))
0.0b
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
3 → 35
Calls

3 calls. Slowest were:

6.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
2.0ms
(* 1/3 (pow x 3))
2.0ms
(* 1/60 (pow x 5))

series133.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

91.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
26.0ms
(* 1/60 (pow x 5))
15.0ms
(* 1/3 (pow x 3))

simplify136.0ms

Counts
17 → 44
Calls

17 calls. Slowest were:

24.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
24.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
24.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))

prune689.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

5.7b
(- (exp x) (exp (- x)))
5.7b
(- (exp x) (exp (- x)))
5.7b
(- (exp x) (exp (- x)))
0.1b
(cbrt (* (* (- (exp x) (exp (- x))) (- (exp x) (exp (- x)))) (- (exp x) (exp (- x)))))

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
4 → 78
Calls

4 calls. Slowest were:

26.0ms
(cbrt (* (* (- (exp x) (exp (- x))) (- (exp x) (exp (- x)))) (- (exp x) (exp (- x)))))
4.0ms
(- (exp x) (exp (- x)))
4.0ms
(- (exp x) (exp (- x)))

series155.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

40.0ms
(- (exp x) (exp (- x)))
39.0ms
(- (exp x) (exp (- x)))
38.0ms
(cbrt (* (* (- (exp x) (exp (- x))) (- (exp x) (exp (- x)))) (- (exp x) (exp (- x)))))
38.0ms
(- (exp x) (exp (- x)))

simplify7.9s

Counts
52 → 90
Calls

52 calls. Slowest were:

692.0ms
(cbrt (* (* (- (pow (exp x) 3) (pow (exp (- x)) 3)) (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))) (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))))
573.0ms
(cbrt (* (* (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))) (- (exp x) (exp (- x)))))
568.0ms
(cbrt (* (* (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (- (pow (exp x) 3) (pow (exp (- x)) 3))) (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))))

prune765.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize26.0ms

Local error

Found 4 expressions with local error:

5.8b
(- (sqrt (exp x)) (sqrt (exp (- x))))
5.7b
(- (exp x) (exp (- x)))
5.7b
(- (exp x) (exp (- x)))
0.1b
(cbrt (* (* (- (exp x) (exp (- x))) (* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (sqrt (exp x)) (sqrt (exp (- x)))))) (- (exp x) (exp (- x)))))

rewrite104.0ms

Algorithm
rewrite-expression-head
Counts
4 → 135
Calls

4 calls. Slowest were:

82.0ms
(cbrt (* (* (- (exp x) (exp (- x))) (* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (sqrt (exp x)) (sqrt (exp (- x)))))) (- (exp x) (exp (- x)))))
8.0ms
(- (sqrt (exp x)) (sqrt (exp (- x))))
6.0ms
(- (exp x) (exp (- x)))

series974.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

462.0ms
(cbrt (* (* (- (exp x) (exp (- x))) (* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (sqrt (exp x)) (sqrt (exp (- x)))))) (- (exp x) (exp (- x)))))
400.0ms
(- (exp x) (exp (- x)))
60.0ms
(- (sqrt (exp x)) (sqrt (exp (- x))))
52.0ms
(- (exp x) (exp (- x)))

simplify15.9s

Counts
113 → 147
Calls

113 calls. Slowest were:

1.2s
(cbrt (* (* (- (exp x) (exp (- x))) (* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (* (sqrt (exp x)) (sqrt (exp x))) (* (sqrt (exp (- x))) (sqrt (exp (- x))))))) (- (exp x) (exp (- x)))))
550.0ms
(pow (* (- (sqrt (exp x)) (sqrt (exp (- x)))) (* (pow (- (exp x) (exp (- x))) 2) (+ (sqrt (exp x)) (sqrt (exp (- x)))))) 1/3)
523.0ms
(cbrt (* (* (- (exp x) (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))))))) (- (exp x) (exp (- x)))))

prune2.5s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

regimes55.0ms

Accuracy

0% (0.5b remaining)

Error of 0.6b against oracle of 0.1b and baseline of 0.6b

bsearch3.0ms