Average Error: 29.3 → 0.6
Time: 11.6s
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

Original29.3
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.3

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

  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 2019004 
(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: 10.9s)Debug log

sample168.0ms

Algorithm
intervals

simplify34.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

34.0ms
(+ (- (exp x) 2) (exp (- x)))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 27.7b

localize22.0ms

Local error

Found 2 expressions with local error:

4.3b
(+ (- (exp x) 2) (exp (- x)))
0.0b
(- (exp x) 2)

rewrite42.0ms

Algorithm
rewrite-expression-head
Counts
2 → 30
Calls

2 calls. Slowest were:

36.0ms
(+ (- (exp x) 2) (exp (- x)))
5.0ms
(- (exp x) 2)

series111.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

98.0ms
(+ (- (exp x) 2) (exp (- x)))
13.0ms
(- (exp x) 2)

simplify1.4s

Counts
19 → 36
Calls

19 calls. Slowest were:

319.0ms
(+ (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) (exp 0)))
298.0ms
(+ (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) 1))
212.0ms
(+ (* (- (* (exp x) (exp x)) (* 2 2)) (exp x)) (* (+ (exp x) 2) (exp 0)))

prune332.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize36.0ms

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))

series206.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

94.0ms
(+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))
45.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))
35.0ms
(* 1/360 (pow x 6))
31.0ms
(* 1/12 (pow x 4))

simplify199.0ms

Counts
23 → 59
Calls

23 calls. Slowest were:

41.0ms
(* (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)))))
25.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))

prune713.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize5.0ms

Local error

Found 3 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)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
3 → 33
Calls

3 calls. Slowest were:

3.0ms
(+ (pow x 2) (* 1/12 (pow x 4)))
1.0ms
(* 1/12 (pow x 4))
1.0ms
(* 1/360 (pow x 6))

series87.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

51.0ms
(+ (pow x 2) (* 1/12 (pow x 4)))
19.0ms
(* 1/360 (pow x 6))
17.0ms
(* 1/12 (pow x 4))

simplify41.0ms

Counts
16 → 42
Calls

16 calls. Slowest were:

10.0ms
(* (exp (pow x 2)) (exp (* 1/12 (pow x 4))))
3.0ms
(+ (pow x 2) (* 1/12 (pow x 4)))
3.0ms
(+ (pow x 2) (* 1/12 (pow x 4)))

prune519.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize7.0ms

Local error

Found 4 expressions with local error:

1.4b
(log (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))))
0.1b
(* 1/12 (pow x 4))
0.1b
(* 1/360 (pow x 6))
0.0b
(+ (pow x 2) (log (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

15.0ms
(+ (pow x 2) (log (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))))
6.0ms
(log (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))))
1.0ms
(* 1/12 (pow x 4))

series151.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

52.0ms
(log (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))))
41.0ms
(+ (pow x 2) (log (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))))
35.0ms
(* 1/360 (pow x 6))
23.0ms
(* 1/12 (pow x 4))

simplify410.0ms

Counts
33 → 67
Calls

33 calls. Slowest were:

54.0ms
(* (exp (pow x 2)) (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))))
53.0ms
(+ (pow x 2) (log (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))))
49.0ms
(+ (pow x 2) (log (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))))

prune801.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

regimes125.0ms

Accuracy

0% (0.6b remaining)

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

bsearch3.0ms

end0.0ms

sample5.4s

Algorithm
intervals