Average Error: 58.2 → 0.6
Time: 14.9s
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 58.2

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

  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 2019010 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))

Details

Time bar (total: 14.7s)Debug log

sample107.0ms

Algorithm
intervals

simplify20.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
19.0ms
(/ (- (exp x) (exp (- x))) 2)

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.5b

localize41.0ms

Local error

Found 1 expressions with local error:

5.4b
(- (exp x) (exp (- x)))

rewrite9.0ms

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

series42.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
42.0ms
(- (exp x) (exp (- x)))

simplify87.0ms

Counts
7 → 17
Calls
7 calls:
Slowest
41.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
14.0ms
(/ (exp (exp x)) (exp (exp (- x))))
9.0ms
(- (exp x) (exp (* -1 x)))
8.0ms
(- (exp x) (exp (- x)))
6.0ms
(- (sqrt (exp x)) (sqrt (exp (- x))))

prune149.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize20.0ms

Local error

Found 2 expressions with local error:

0.1b
(* 1/3 (pow x 3))
0.0b
(* 1/60 (pow x 5))

rewrite2.0ms

Algorithm
rewrite-expression-head
Rules
8×associate-*r*
4×add-cube-cbrt
4×*-un-lft-identity
4×add-sqr-sqrt
2×add-log-exp
2×*-commutative
2×add-exp-log
2×add-cbrt-cube
2×pow1
1×unpow3
1×cube-mult
Counts
2 → 24
Calls
2 calls:
Slowest
1.0ms
(* 1/3 (pow x 3))
1.0ms
(* 1/60 (pow x 5))

series39.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
19.0ms
(* 1/60 (pow x 5))
19.0ms
(* 1/3 (pow x 3))

simplify49.0ms

Counts
13 → 30
Calls
13 calls:
Slowest
14.0ms
(* 1/3 (pow x 3))
10.0ms
(* 1/3 (pow x 3))
6.0ms
(* 1/3 (pow x 3))
4.0ms
(sqrt (pow x 3))
2.0ms
(cbrt (pow x 3))

prune245.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize17.0ms

Local error

Found 4 expressions with local error:

5.4b
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
0.0b
(* (exp x) (exp x))
0.0b
(* (exp (- x)) (exp (- x)))
0.0b
(/ (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (+ (exp x) (exp (- x))))

rewrite55.0ms

Algorithm
rewrite-expression-head
Rules
32×*-un-lft-identity
22×add-cube-cbrt
22×add-sqr-sqrt
16×times-frac
16×exp-prod
14×pow1
10×add-cbrt-cube
7×exp-neg
6×difference-of-squares
6×add-log-exp
6×pow-prod-up
6×add-exp-log
6×pow-prod-down
6×associate-*l*
6×associate-*r*
6×exp-diff
6×neg-sub0
5×distribute-lft-out
4×associate-/l*
4×neg-mul-1
4×associate-/r*
4×frac-times
2×associate-/l/
2×flip--
2×cbrt-unprod
2×*-commutative
2×associate-/r/
2×associate-*r/
2×prod-exp
2×associate-*l/
2×flip3--
2×pow-plus
2×pow2
1×div-inv
1×flip-+
1×un-div-inv
1×div-exp
1×div-sub
1×diff-log
1×flip3-+
1×frac-2neg
1×sub-neg
1×clear-num
1×cbrt-undiv
Counts
4 → 111
Calls
4 calls:
Slowest
26.0ms
(/ (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (+ (exp x) (exp (- x))))
13.0ms
(* (exp (- x)) (exp (- x)))
7.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
7.0ms
(* (exp x) (exp x))

series218.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
105.0ms
(/ (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (+ (exp x) (exp (- x))))
82.0ms
(- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))
16.0ms
(* (exp (- x)) (exp (- x)))
14.0ms
(* (exp x) (exp x))

simplify8.1s

Counts
111 → 123
Calls
111 calls:
Slowest
520.0ms
(sqrt (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))))
472.0ms
(/ (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (+ (exp x) (exp (- x))))
331.0ms
(- (* (* (exp x) (exp x)) (* (exp x) (exp x))) (* (* (exp (- x)) (exp (- x))) (* (exp (- x)) (exp (- x)))))
319.0ms
(/ (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (+ (exp x) (exp (- x))))
276.0ms
(/ (sqrt (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))) (* (cbrt (+ (exp x) (exp (- x)))) (cbrt (+ (exp x) (exp (- x))))))

prune787.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes18.0ms

Accuracy

0% (0.5b remaining)

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

bsearch2.0ms

end0.0ms

sample4.7s

Algorithm
intervals