Average Error: 57.8 → 0.7
Time: 11.7s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{(\left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right) \cdot x + \left(\frac{1}{60} \cdot {x}^{5}\right))_*}{2}\]

Error

Bits error versus x

Derivation

  1. Initial program 57.8

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))

Details

Time bar (total: 11.4s)Debug log

sample124.0ms

Algorithm
intervals

simplify68.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

68.0ms
(/ (- (exp x) (exp (- x))) 2)

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.3b

localize23.0ms

Local error

Found 1 expressions with local error:

5.2b
(- (exp x) (exp (- x)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
1 → 28
Calls

1 calls. Slowest were:

10.0ms
(- (exp x) (exp (- x)))

series68.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

68.0ms
(- (exp x) (exp (- x)))

simplify557.0ms

Counts
27 → 31
Calls

27 calls. Slowest were:

63.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
55.0ms
(fma 1 (exp x) (- (* (exp (- x)) 1)))
38.0ms
(fma (* (cbrt (exp x)) (cbrt (exp x))) (cbrt (exp x)) (- (* (exp (- x)) 1)))

prune268.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize38.0ms

Local error

Found 3 expressions with local error:

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

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 33
Calls

3 calls. Slowest were:

9.0ms
(* (pow x 5) 1/60)
0.0ms
(fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60))
0.0ms
(fma 1/3 (* x x) 2)

series70.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

24.0ms
(fma 1/3 (* x x) 2)
23.0ms
(fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60))
23.0ms
(* (pow x 5) 1/60)

simplify244.0ms

Counts
11 → 42
Calls

11 calls. Slowest were:

76.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
71.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
53.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))

prune364.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 4 expressions with local error:

5.2b
(- (exp x) (exp (- x)))
5.2b
(- (exp x) (exp (- x)))
5.2b
(- (exp x) (exp (- x)))
0.1b
(cbrt (- (exp x) (exp (- x))))

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 101
Calls

4 calls. Slowest were:

9.0ms
(- (exp x) (exp (- x)))
8.0ms
(- (exp x) (exp (- x)))
5.0ms
(- (exp x) (exp (- x)))

series423.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

265.0ms
(cbrt (- (exp x) (exp (- x))))
65.0ms
(- (exp x) (exp (- x)))
48.0ms
(- (exp x) (exp (- x)))
44.0ms
(- (exp x) (exp (- x)))

simplify2.6s

Counts
91 → 113
Calls

91 calls. Slowest were:

272.0ms
(cbrt (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))))
158.0ms
(cbrt (- (pow (exp x) 3) (pow (exp (- x)) 3)))
128.0ms
(- (+ (exp (* 1/3 (+ (log 2) (log x)))) (* 1/18 (* (pow x 2) (exp (* 1/3 (+ (log 2) (log x))))))) (* 1/3240 (* (pow x 4) (exp (* 1/3 (+ (log 2) (log x)))))))

prune1.4s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes59.0ms

Accuracy

0% (0.7b remaining)

Error of 0.7b against oracle of 0.0b and baseline of 0.7b

bsearch2.0ms

end0.0ms

sample5.1s

Algorithm
intervals