Average Error: 29.8 → 0.6
Time: 14.2s
Precision: 64
Internal Precision: 128
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\frac{1}{360} \cdot {x}^{6} + \left({x}^{2} + \frac{1}{12} \cdot {x}^{4}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.8

    \[\left(e^{x} - 2\right) + e^{-x}\]

  2. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + \frac{1}{360} \cdot {x}^{6}\right)}\]
  3. Using strategy rm

  4. Applied associate-+r+0.6

    \[\leadsto \color{blue}{\left({x}^{2} + \frac{1}{12} \cdot {x}^{4}\right) + \frac{1}{360} \cdot {x}^{6}}\]

  5. Final simplification0.6

    \[\leadsto \frac{1}{360} \cdot {x}^{6} + \left({x}^{2} + \frac{1}{12} \cdot {x}^{4}\right)\]

Reproduce

herbie shell --seed 2019002 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4 (pow (sinh (/ x 2)) 2))

  (+ (- (exp x) 2) (exp (- x))))

Details

Time bar (total: 13.7s)Debug log

sample234.0ms

Algorithm
intervals

simplify32.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

32.0ms
(+ (- (exp x) 2) (exp (- x)))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 28.9b

localize12.0ms

Local error

Found 2 expressions with local error:

4.4b
(+ (- (exp x) 2) (exp (- x)))
0.0b
(- (exp x) 2)

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
2 → 30
Calls

2 calls. Slowest were:

27.0ms
(+ (- (exp x) 2) (exp (- x)))
3.0ms
(- (exp x) 2)

series71.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

62.0ms
(+ (- (exp x) 2) (exp (- x)))
9.0ms
(- (exp x) 2)

simplify1.4s

Counts
19 → 36
Calls

19 calls. Slowest were:

340.0ms
(+ (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) (exp 0)))
282.0ms
(+ (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) 1))
232.0ms
(+ (* (- (* (exp x) (exp x)) (* 2 2)) (exp x)) (* (+ (exp x) 2) (exp 0)))

prune380.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize39.0ms

Local error

Found 4 expressions with local error:

0.1b
(* 1/12 (pow x 4))
0.1b
(* 1/360 (pow x 6))
0.0b
(+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))
0.0b
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 47
Calls

4 calls. Slowest were:

13.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))
6.0ms
(+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))
2.0ms
(* 1/12 (pow x 4))

series157.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

76.0ms
(+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))
33.0ms
(* 1/12 (pow x 4))
24.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))
24.0ms
(* 1/360 (pow x 6))

simplify192.0ms

Counts
23 → 59
Calls

23 calls. Slowest were:

31.0ms
(* (exp (pow x 2)) (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))))
29.0ms
(* (exp (pow x 2)) (* (exp (* 1/12 (pow x 4))) (exp (* 1/360 (pow x 6)))))
26.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))

prune699.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.1b
(* 1/12 (pow x 4))
0.1b
(* 1/360 (pow x 6))
0.0b
(+ (pow x 2) (* 1/12 (pow x 4)))
0.0b
(+ (+ (pow x 2) (* 1/12 (pow x 4))) (* 1/360 (pow x 6)))

rewrite37.0ms

Algorithm
rewrite-expression-head
Counts
4 → 46
Calls

4 calls. Slowest were:

26.0ms
(+ (+ (pow x 2) (* 1/12 (pow x 4))) (* 1/360 (pow x 6)))
6.0ms
(+ (pow x 2) (* 1/12 (pow x 4)))
2.0ms
(* 1/12 (pow x 4))

series107.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

36.0ms
(+ (pow x 2) (* 1/12 (pow x 4)))
25.0ms
(+ (+ (pow x 2) (* 1/12 (pow x 4))) (* 1/360 (pow x 6)))
25.0ms
(* 1/12 (pow x 4))
20.0ms
(* 1/360 (pow x 6))

simplify136.0ms

Counts
22 → 58
Calls

22 calls. Slowest were:

32.0ms
(* (exp (+ (pow x 2) (* 1/12 (pow x 4)))) (exp (* 1/360 (pow x 6))))
31.0ms
(* (* (exp (pow x 2)) (exp (* 1/12 (pow x 4)))) (exp (* 1/360 (pow x 6))))
13.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))

prune574.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize28.0ms

Local error

Found 4 expressions with local error:

4.7b
(+ (/ (- (pow (exp x) 3) (pow 2 3)) (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2)))) (exp (- x)))
0.0b
(pow (exp x) 3)
0.0b
(* (exp x) (exp x))
0.0b
(+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2)))

rewrite109.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

83.0ms
(+ (/ (- (pow (exp x) 3) (pow 2 3)) (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2)))) (exp (- x)))
16.0ms
(+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2)))
7.0ms
(* (exp x) (exp x))

series393.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

329.0ms
(+ (/ (- (pow (exp x) 3) (pow 2 3)) (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2)))) (exp (- x)))
35.0ms
(+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2)))
14.0ms
(pow (exp x) 3)
13.0ms
(* (exp x) (exp x))

simplify2.5s

Counts
42 → 79
Calls

42 calls. Slowest were:

449.0ms
(+ (/ (- (pow (exp x) 3) (pow 2 3)) (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2)))) (exp (- x)))
342.0ms
(+ (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) 1))
327.0ms
(+ (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) (exp 0)))

prune924.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

regimes25.0ms

Accuracy

0% (0.5b remaining)

Error of 0.6b against oracle of 0.1b and baseline of 0.6b

bsearch3.0ms

end0.0ms

sample5.6s

Algorithm
intervals