Average Error: 57.9 → 0.6
Time: 1.3m
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 57.9

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

  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. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))

Details

Time bar (total: 1.2m)Debug log

start190.0ms

Algorithm
intervals

setup82.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.6b

localize27.0ms

Local error

Found 1 expressions with local error:

7.8b
(- (exp x) (exp (- x)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
1 → 28
Calls

1 calls. Slowest were:

11.0ms
(- (exp x) (exp (- x)))

series67.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

66.0ms
(- (exp x) (exp (- x)))

simplify613.0ms

Counts
27 → 31
Calls

27 calls. Slowest were:

74.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
51.0ms
(fma 1 (exp x) (- (* (exp (- x)) 1)))
51.0ms
(fma 1 (exp x) (- (* (cbrt (exp (- x))) (* (cbrt (exp (- x))) (cbrt (exp (- x)))))))

prune415.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize37.0ms

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)

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
3 → 33
Calls

3 calls. Slowest were:

2.0ms
(* (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)

series90.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

37.0ms
(fma 1/3 (* x x) 2)
29.0ms
(fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60))
24.0ms
(* (pow x 5) 1/60)

simplify332.0ms

Counts
11 → 42
Calls

11 calls. Slowest were:

94.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
92.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
90.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))

prune639.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

7.8b
(- (exp x) (exp (- x)))
7.8b
(- (exp x) (exp (- x)))
7.8b
(- (exp x) (exp (- x)))
0.1b
(cbrt (- (exp x) (exp (- x))))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 101
Calls

4 calls. Slowest were:

5.0ms
(- (exp x) (exp (- x)))
5.0ms
(- (exp x) (exp (- x)))
5.0ms
(- (exp x) (exp (- x)))

series333.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

194.0ms
(cbrt (- (exp x) (exp (- x))))
50.0ms
(- (exp x) (exp (- x)))
47.0ms
(- (exp x) (exp (- x)))
41.0ms
(- (exp x) (exp (- x)))

simplify2.4s

Counts
91 → 113
Calls

91 calls. Slowest were:

296.0ms
(cbrt (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))))
170.0ms
(cbrt (- (pow (exp x) 3) (pow (exp (- x)) 3)))
132.0ms
(- (+ (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

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize26.0ms

Local error

Found 4 expressions with local error:

7.8b
(- (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))
0.0b
(sqrt (exp x))

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
4 → 201
Calls

4 calls. Slowest were:

26.0ms
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (sqrt (exp x)) (sqrt (exp (- x)))))
8.0ms
(- (sqrt (exp x)) (sqrt (exp (- x))))
1.0ms
(sqrt (exp x))

series252.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

142.0ms
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (sqrt (exp x)) (sqrt (exp (- x)))))
67.0ms
(- (sqrt (exp x)) (sqrt (exp (- x))))
23.0ms
(sqrt (exp x))
20.0ms
(sqrt (exp x))

simplify59.0s

Counts
212 → 213
Calls

212 calls. Slowest were:

1.9s
(* (fma (- (sqrt (cbrt (exp (- x))))) (sqrt (* (cbrt (exp (- x))) (cbrt (exp (- x))))) (* (sqrt (cbrt (exp (- x)))) (sqrt (* (cbrt (exp (- x))) (cbrt (exp (- x))))))) (+ (sqrt (exp x)) (sqrt (exp (- x)))))
1.7s
(* (fma (- (sqrt (cbrt (exp (- x))))) (sqrt (* (cbrt (exp (- x))) (cbrt (exp (- x))))) (* (sqrt (cbrt (exp (- x)))) (sqrt (* (cbrt (exp (- x))) (cbrt (exp (- x))))))) (+ (sqrt (exp x)) (sqrt (exp (- x)))))
1.6s
(* (fma (- (sqrt (cbrt (exp (- x))))) (sqrt (* (cbrt (exp (- x))) (cbrt (exp (- x))))) (* (sqrt (cbrt (exp (- x)))) (sqrt (* (cbrt (exp (- x))) (cbrt (exp (- x))))))) (+ (sqrt (exp x)) (sqrt (exp (- x)))))

prune3.2s

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.0b

regimes88.0ms

Accuracy

0% (0.6b remaining)

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

bsearch2.0ms