Average Error: 57.9 → 0.5
Time: 16.9s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\left(\sqrt{e^{-x}} + \sqrt{e^{x}}\right) \cdot \left(\left({x}^{3} \cdot \frac{1}{24} + \frac{1}{1920} \cdot {x}^{5}\right) + x\right)}{2}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.9

    \[\frac{e^{x} - e^{-x}}{2}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt58.0

    \[\leadsto \frac{e^{x} - \color{blue}{\sqrt{e^{-x}} \cdot \sqrt{e^{-x}}}}{2}\]

  4. Applied add-sqr-sqrt58.0

    \[\leadsto \frac{\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}} - \sqrt{e^{-x}} \cdot \sqrt{e^{-x}}}{2}\]

  5. Applied difference-of-squares58.0

    \[\leadsto \frac{\color{blue}{\left(\sqrt{e^{x}} + \sqrt{e^{-x}}\right) \cdot \left(\sqrt{e^{x}} - \sqrt{e^{-x}}\right)}}{2}\]

  6. Taylor expanded around 0 0.5

    \[\leadsto \frac{\left(\sqrt{e^{x}} + \sqrt{e^{-x}}\right) \cdot \color{blue}{\left(x + \left(\frac{1}{24} \cdot {x}^{3} + \frac{1}{1920} \cdot {x}^{5}\right)\right)}}{2}\]

  7. Final simplification0.5

    \[\leadsto \frac{\left(\sqrt{e^{-x}} + \sqrt{e^{x}}\right) \cdot \left(\left({x}^{3} \cdot \frac{1}{24} + \frac{1}{1920} \cdot {x}^{5}\right) + x\right)}{2}\]

Reproduce

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

Details

Time bar (total: 7.7s)Debug log

start256.0ms

Algorithm
intervals

setup46.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.6b

localize50.0ms

Local error

Found 1 expressions with local error:

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

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
1 → 14
Calls

1 calls. Slowest were:

8.0ms
(- (exp x) (exp (- x)))

series48.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

48.0ms
(- (exp x) (exp (- x)))

simplify81.0ms

Counts
7 → 17
Calls

7 calls. Slowest were:

30.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
24.0ms
(/ (exp (exp x)) (exp (exp (- x))))
8.0ms
(- (exp x) (exp (* -1 x)))

prune186.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize21.0ms

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)))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
3 → 35
Calls

3 calls. Slowest were:

4.0ms
(* 1/60 (pow x 5))
3.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
1.0ms
(* 1/3 (pow x 3))

series145.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

90.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
35.0ms
(* 1/60 (pow x 5))
20.0ms
(* 1/3 (pow x 3))

simplify113.0ms

Counts
17 → 44
Calls

17 calls. Slowest were:

26.0ms
(* (exp (* 1/3 (pow x 3))) (exp (* 1/60 (pow x 5))))
22.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))
15.0ms
(+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))

prune813.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize20.0ms

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))))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 57
Calls

4 calls. Slowest were:

7.0ms
(- (exp x) (exp (- x)))
7.0ms
(- (exp x) (exp (- x)))
7.0ms
(- (exp x) (exp (- x)))

series412.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

235.0ms
(cbrt (- (exp x) (exp (- x))))
62.0ms
(- (exp x) (exp (- x)))
60.0ms
(- (exp x) (exp (- x)))
55.0ms
(- (exp x) (exp (- x)))

simplify1.1s

Counts
31 → 69
Calls

31 calls. Slowest were:

305.0ms
(cbrt (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))))
272.0ms
(- (+ (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)))))))
133.0ms
(cbrt (- (pow (exp x) 3) (pow (exp (- x)) 3)))

prune527.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize14.0ms

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))

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
4 → 79
Calls

4 calls. Slowest were:

23.0ms
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (sqrt (exp x)) (sqrt (exp (- x)))))
6.0ms
(- (sqrt (exp x)) (sqrt (exp (- x))))
1.0ms
(sqrt (exp x))

series211.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

127.0ms
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (sqrt (exp x)) (sqrt (exp (- x)))))
54.0ms
(- (sqrt (exp x)) (sqrt (exp (- x))))
18.0ms
(sqrt (exp x))
12.0ms
(sqrt (exp x))

simplify2.7s

Counts
63 → 91
Calls

63 calls. Slowest were:

329.0ms
(* (+ (* (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)))))))
317.0ms
(* (+ (pow (sqrt (exp x)) 3) (pow (sqrt (exp (- x))) 3)) (- (pow (sqrt (exp x)) 3) (pow (sqrt (exp (- x))) 3)))
197.0ms
(* (+ (pow (sqrt (exp x)) 3) (pow (sqrt (exp (- x))) 3)) (- (* (sqrt (exp x)) (sqrt (exp x))) (* (sqrt (exp (- x))) (sqrt (exp (- x))))))

prune755.0ms

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0.0b

regimes69.0ms

Accuracy

0% (0.5b remaining)

Error of 0.5b against oracle of 0.0b and baseline of 0.5b

bsearch2.0ms