Average Error: 58.1 → 0.6
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 58.1

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

  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 2019022 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))

Details

Time bar (total: 19.8s)Debug log

sample142.0ms

Algorithm
intervals
Results
60.0ms113×body1280valid
31.0ms249×body80valid
28.0ms80×body640valid
9.0ms38×body320valid
2.0ms15×body160valid

simplify27.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
27.0ms
(/ (- (exp x) (exp (- x))) 2)

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.8b

localize17.0ms

Local error

Found 1 expressions with local error:

6.4b
(- (exp x) (exp (- x)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
3×add-log-exp
3×*-un-lft-identity
3×add-sqr-sqrt
1×difference-of-squares
1×distribute-lft-out--
1×flip--
1×add-cube-cbrt
1×add-exp-log
1×diff-log
1×add-cbrt-cube
1×flip3--
1×sub-neg
1×pow1
Counts
1 → 14
Calls
1 calls:
Slowest
6.0ms
(- (exp x) (exp (- x)))

series49.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
49.0ms
(- (exp x) (exp (- x)))

simplify65.0ms

Counts
7 → 17
Calls
7 calls:
Slowest
26.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
14.0ms
(/ (exp (exp x)) (exp (exp (- x))))
8.0ms
(- (exp x) (exp (* -1 x)))
5.0ms
(- (sqrt (exp x)) (sqrt (exp (- x))))
4.0ms
(- (exp x) (exp (- x)))

prune150.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize36.0ms

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)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
8×associate-*r*
5×add-log-exp
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
3×add-exp-log
3×add-cbrt-cube
3×pow1
2×*-commutative
1×flip-+
1×unpow3
1×flip3-+
1×sum-log
1×+-commutative
1×cube-mult
Counts
3 → 35
Calls
3 calls:
Slowest
4.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
2.0ms
(* 1/3 (pow x 3))
2.0ms
(* 1/60 (pow x 5))

series107.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
53.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
32.0ms
(* 1/60 (pow x 5))
22.0ms
(* 1/3 (pow x 3))

simplify98.0ms

Counts
17 → 44
Calls
17 calls:
Slowest
19.0ms
(* (exp (* 1/3 (pow x 3))) (exp (* 1/60 (pow x 5))))
15.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
13.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
12.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
8.0ms
(* 1/3 (pow x 3))

prune390.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 4 expressions with local error:

6.4b
(- (sqrt (exp x)) (sqrt (exp (- x))))
0.0b
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (sqrt (exp x)) (sqrt (exp (- x)))))
0.0b
(+ (sqrt (exp x)) (sqrt (exp (- x))))
0.0b
(sqrt (exp x))

rewrite45.0ms

Algorithm
rewrite-expression-head
Rules
24×add-sqr-sqrt
23×*-un-lft-identity
19×sqrt-prod
9×associate-*r*
8×difference-of-squares
8×add-log-exp
7×add-cube-cbrt
6×add-exp-log
6×add-cbrt-cube
6×pow1
5×associate-*l*
4×distribute-lft-out--
4×distribute-lft-out
4×flip--
4×flip-+
4×flip3--
4×flip3-+
4×frac-times
3×sub-neg
2×associate-*r/
2×associate-*l/
2×distribute-lft-in
2×distribute-rgt-in
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×diff-log
1×pow1/2
1×sum-log
1×rem-sqrt-square
1×+-commutative
Counts
4 → 80
Calls
4 calls:
Slowest
30.0ms
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (sqrt (exp x)) (sqrt (exp (- x)))))
8.0ms
(+ (sqrt (exp x)) (sqrt (exp (- x))))
5.0ms
(- (sqrt (exp x)) (sqrt (exp (- x))))
1.0ms
(sqrt (exp x))

series285.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
138.0ms
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (sqrt (exp x)) (sqrt (exp (- x)))))
74.0ms
(- (sqrt (exp x)) (sqrt (exp (- x))))
57.0ms
(+ (sqrt (exp x)) (sqrt (exp (- x))))
15.0ms
(sqrt (exp x))

simplify2.9s

Counts
63 → 92
Calls
63 calls:
Slowest
308.0ms
(* (+ (* (sqrt (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))) (sqrt (exp (- x)))) (* (sqrt (exp x)) (sqrt (exp (- x)))))))
296.0ms
(+ (sqrt (exp x)) (sqrt (exp (- x))))
252.0ms
(* (+ (pow (sqrt (exp x)) 3) (pow (sqrt (exp (- x))) 3)) (- (pow (sqrt (exp x)) 3) (pow (sqrt (exp (- x))) 3)))
156.0ms
(* (- (sqrt (exp x)) (sqrt (exp (- x)))) (+ (* (sqrt (exp x)) (sqrt (exp x))) (+ (* (sqrt (exp (- x))) (sqrt (exp (- x)))) (* (sqrt (exp x)) (sqrt (exp (- x)))))))
154.0ms
(* (+ (pow (sqrt (exp x)) 3) (pow (sqrt (exp (- x))) 3)) (- (* (sqrt (exp x)) (sqrt (exp x))) (* (sqrt (exp (- x))) (sqrt (exp (- x))))))

prune840.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize30.0ms

Local error

Found 4 expressions with local error:

6.4b
(- (sqrt (exp x)) (sqrt (exp (- x))))
0.1b
(cbrt (* (pow (- (sqrt (exp x)) (sqrt (exp (- x)))) 3) (pow (+ (sqrt (exp (- x))) (sqrt (exp x))) 3)))
0.0b
(pow (+ (sqrt (exp (- x))) (sqrt (exp x))) 3)
0.0b
(pow (- (sqrt (exp x)) (sqrt (exp (- x)))) 3)

rewrite56.0ms

Algorithm
rewrite-expression-head
Rules
32×add-sqr-sqrt
28×*-un-lft-identity
22×sqrt-prod
16×cube-div
14×cube-prod
14×unpow-prod-down
12×difference-of-squares
8×add-cube-cbrt
8×cbrt-div
6×add-log-exp
6×distribute-lft-out--
6×add-exp-log
6×add-cbrt-cube
6×pow1
5×flip--
5×flip3--
4×distribute-lft-out
4×flip-+
4×flip3-+
4×frac-times
2×rem-cube-cbrt
2×associate-*r/
2×pow-exp
2×associate-*l/
2×unpow3
2×pow-pow
2×pow-to-exp
2×cube-mult
1×rem-cbrt-cube
1×pow-prod-down
1×diff-log
1×pow1/3
1×sub-neg
1×cbrt-prod
Counts
4 → 94
Calls
4 calls:
Slowest
27.0ms
(cbrt (* (pow (- (sqrt (exp x)) (sqrt (exp (- x)))) 3) (pow (+ (sqrt (exp (- x))) (sqrt (exp x))) 3)))
10.0ms
(- (sqrt (exp x)) (sqrt (exp (- x))))
10.0ms
(pow (+ (sqrt (exp (- x))) (sqrt (exp x))) 3)
7.0ms
(pow (- (sqrt (exp x)) (sqrt (exp (- x)))) 3)

series346.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
141.0ms
(cbrt (* (pow (- (sqrt (exp x)) (sqrt (exp (- x)))) 3) (pow (+ (sqrt (exp (- x))) (sqrt (exp x))) 3)))
90.0ms
(pow (- (sqrt (exp x)) (sqrt (exp (- x)))) 3)
58.0ms
(pow (+ (sqrt (exp (- x))) (sqrt (exp x))) 3)
56.0ms
(- (sqrt (exp x)) (sqrt (exp (- x))))

simplify7.8s

Counts
72 → 106
Calls
72 calls:
Slowest
633.0ms
(cbrt (* (pow (- (pow (sqrt (exp x)) 3) (pow (sqrt (exp (- x))) 3)) 3) (pow (+ (pow (sqrt (exp (- x))) 3) (pow (sqrt (exp x)) 3)) 3)))
619.0ms
(cbrt (* (pow (- (* (sqrt (exp x)) (sqrt (exp x))) (* (sqrt (exp (- x))) (sqrt (exp (- x))))) 3) (pow (+ (sqrt (exp (- x))) (sqrt (exp x))) 3)))
555.0ms
(cbrt (* (pow (- (pow (sqrt (exp x)) 3) (pow (sqrt (exp (- x))) 3)) 3) (pow (+ (sqrt (exp (- x))) (sqrt (exp x))) 3)))
432.0ms
(cbrt (* (pow (- (sqrt (exp x)) (sqrt (exp (- x)))) 3) (pow (+ (pow (sqrt (exp (- x))) 3) (pow (sqrt (exp x)) 3)) 3)))
428.0ms
(pow (- (* (sqrt (exp (- x))) (sqrt (exp (- x)))) (* (sqrt (exp x)) (sqrt (exp x)))) 3)

prune1.4s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

regimes41.0ms

Accuracy

0% (0.5b remaining)

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

bsearch2.0ms

end0.0ms

sample4.9s

Algorithm
intervals
Results
2.6s3384×body1280valid
985.0ms2457×body640valid
717.0ms8087×body80valid
353.0ms1296×body320valid
96.0ms598×body160valid