Average Error: 57.8 → 0.6
Time: 8.0s
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 57.8

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

  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 2019016 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))

Details

Time bar (total: 7.6s)Debug log

sample159.0ms

Algorithm
intervals
Results
525×(pre true 80)
280×(body real 80)
105×(body real 1280)
82×(body real 640)
42×(body real 320)
16×(body real 160)

simplify35.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
35.0ms
(/ (- (exp x) (exp (- x))) 2)

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.7b

localize19.0ms

Local error

Found 1 expressions with local error:

6.1b
(- (exp x) (exp (- x)))

rewrite4.0ms

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

series68.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
68.0ms
(- (exp x) (exp (- x)))

simplify95.0ms

Counts
7 → 17
Calls
7 calls:
Slowest
37.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
29.0ms
(/ (exp (exp x)) (exp (exp (- x))))
8.0ms
(- (exp x) (exp (* -1 x)))
5.0ms
(- (exp x) (exp (- x)))
5.0ms
(+ (sqrt (exp x)) (sqrt (exp (- x))))

prune131.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize35.0ms

Local error

Found 4 expressions with local error:

0.1b
(* 1/3 (pow x 3))
0.1b
(* 1/60 (pow x 5))
0.0b
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
0.0b
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))

rewrite12.0ms

Algorithm
rewrite-expression-head
Rules
11×add-log-exp
8×associate-*r*
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
4×add-exp-log
4×add-cbrt-cube
4×sum-log
4×pow1
2×flip-+
2×*-commutative
2×flip3-+
2×+-commutative
1×unpow3
1×associate-+r+
1×cube-mult
Counts
4 → 48
Calls
4 calls:
Slowest
6.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
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))

series156.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
67.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
41.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
31.0ms
(* 1/60 (pow x 5))
16.0ms
(* 1/3 (pow x 3))

simplify502.0ms

Counts
23 → 60
Calls
23 calls:
Slowest
137.0ms
(* (exp (* 2 x)) (* (exp (* 1/3 (pow x 3))) (exp (* 1/60 (pow x 5)))))
136.0ms
(* (exp (* 2 x)) (exp (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))))
37.0ms
(* (exp (* 1/3 (pow x 3))) (exp (* 1/60 (pow x 5))))
28.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
27.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))

prune575.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize0.0ms

Local error

Found 1 expressions with local error:

6.1b
(- (exp x) (exp (- x)))

rewrite4.0ms

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

series47.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
47.0ms
(- (exp x) (exp (- x)))

simplify72.0ms

Counts
7 → 17
Calls
7 calls:
Slowest
28.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
16.0ms
(/ (exp (exp x)) (exp (exp (- x))))
8.0ms
(- (exp x) (exp (* -1 x)))
5.0ms
(- (exp x) (exp (- x)))
5.0ms
(- (exp x) (exp (- x)))

prune95.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes116.0ms

Accuracy

0% (0.6b remaining)

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

bsearch3.0ms

end0.0ms

sample5.5s

Algorithm
intervals
Results
15887×(pre true 80)
8172×(body real 80)
3343×(body real 1280)
2501×(body real 640)
1247×(body real 320)
624×(body real 160)