Average Error: 29.1 → 0.6
Time: 12.3s
Precision: 64
Internal Precision: 128
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[{x}^{2} + \left({x}^{6} \cdot \frac{1}{360} + \frac{1}{12} \cdot {x}^{4}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.1
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.1

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

  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. Final simplification0.6

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

Reproduce

herbie shell --seed 2019021 
(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: 11.8s)Debug log

sample171.0ms

Algorithm
intervals
Results
115.0ms179×body1280valid
15.0ms40×body640valid
15.0ms243×body80valid
5.0ms18×body320valid
2.0ms14×body160valid

simplify37.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
37.0ms
(+ (- (exp x) 2) (exp (- x)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 31.1b

localize38.0ms

Local error

Found 2 expressions with local error:

4.2b
(+ (- (exp x) 2) (exp (- x)))
0.0b
(- (exp x) 2)

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×frac-add
4×*-un-lft-identity
3×flip--
3×flip3--
2×add-cube-cbrt
2×add-exp-log
2×exp-diff
2×add-cbrt-cube
2×sub-neg
2×associate-+l+
2×pow1
2×exp-neg
2×add-sqr-sqrt
2×neg-sub0
1×distribute-lft-out
1×flip-+
1×associate-+l-
1×flip3-+
1×sum-log
1×+-commutative
Counts
2 → 30
Calls
2 calls:
Slowest
15.0ms
(+ (- (exp x) 2) (exp (- x)))
5.0ms
(- (exp x) 2)

series90.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
81.0ms
(+ (- (exp x) 2) (exp (- x)))
9.0ms
(- (exp x) 2)

simplify1.3s

Counts
19 → 36
Calls
19 calls:
Slowest
285.0ms
(+ (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) 1))
282.0ms
(+ (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) (exp 0)))
212.0ms
(+ (* (- (* (exp x) (exp x)) (* 2 2)) (exp x)) (* (+ (exp x) 2) (exp 0)))
145.0ms
(+ (* (- (* (exp x) (exp x)) (* 2 2)) (exp x)) (* (+ (exp x) 2) 1))
90.0ms
(* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) (exp x))

prune313.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize41.0ms

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))))

rewrite24.0ms

Algorithm
rewrite-expression-head
Rules
11×add-log-exp
8×*-un-lft-identity
6×add-cube-cbrt
6×associate-*r*
6×add-sqr-sqrt
4×add-exp-log
4×add-cbrt-cube
4×sum-log
4×pow1
2×flip-+
2×*-commutative
2×flip3-+
2×+-commutative
1×distribute-lft-out
1×associate-+r+
Counts
4 → 47
Calls
4 calls:
Slowest
12.0ms
(+ (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))
2.0ms
(* 1/360 (pow x 6))

series151.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
68.0ms
(+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))
32.0ms
(* 1/360 (pow x 6))
27.0ms
(* 1/12 (pow x 4))
24.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))

simplify212.0ms

Counts
23 → 59
Calls
23 calls:
Slowest
42.0ms
(* (exp (pow x 2)) (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))))
33.0ms
(* (exp (pow x 2)) (* (exp (* 1/12 (pow x 4))) (exp (* 1/360 (pow x 6)))))
24.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))
24.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))
12.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))

prune669.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize19.0ms

Local error

Found 4 expressions with local error:

2.1b
(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
(+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))

rewrite22.0ms

Algorithm
rewrite-expression-head
Rules
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×add-log-exp
6×associate-*r*
5×pow1
4×log-pow
4×add-exp-log
4×add-cbrt-cube
4×log-prod
3×exp-prod
2×*-commutative
1×flip-+
1×exp-sum
1×rem-log-exp
1×flip3-+
1×sum-log
1×+-commutative
Counts
4 → 49
Calls
4 calls:
Slowest
11.0ms
(log (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))))
5.0ms
(+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))
2.0ms
(* 1/12 (pow x 4))
2.0ms
(* 1/360 (pow x 6))

series180.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
63.0ms
(log (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))))
57.0ms
(+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))
31.0ms
(* 1/360 (pow x 6))
30.0ms
(* 1/12 (pow x 4))

simplify114.0ms

Counts
27 → 61
Calls
27 calls:
Slowest
12.0ms
(log (* (cbrt (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))) (cbrt (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))))))
11.0ms
(* (exp (* 1/12 (pow x 4))) (exp (* 1/360 (pow x 6))))
10.0ms
(log (sqrt (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))))
7.0ms
(+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))
7.0ms
(+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))

prune836.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize33.0ms

Local error

Found 4 expressions with local error:

6.4b
(+ (pow (exp x) 3) (+ (+ (exp x) 2) (* (exp x) -4)))
0.1b
(+ (+ (exp x) 2) (* (exp x) -4))
0.0b
(pow (exp x) 3)
0.0b
(* (+ (exp x) 2) (exp x))

rewrite56.0ms

Algorithm
rewrite-expression-head
Rules
11×add-log-exp
10×*-un-lft-identity
8×add-cube-cbrt
8×add-sqr-sqrt
7×add-cbrt-cube
7×pow1
5×add-exp-log
4×sum-log
3×flip-+
3×associate-*l*
3×associate-*r*
3×cube-prod
3×flip3-+
3×unpow-prod-down
2×associate-*l/
2×+-commutative
1×rem-cube-cbrt
1×distribute-lft-out
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-exp
1×pow-prod-down
1×unpow3
1×pow-pow
1×associate-+r+
1×associate-+l+
1×pow-to-exp
1×cube-mult
Counts
4 → 64
Calls
4 calls:
Slowest
37.0ms
(+ (pow (exp x) 3) (+ (+ (exp x) 2) (* (exp x) -4)))
11.0ms
(+ (+ (exp x) 2) (* (exp x) -4))
7.0ms
(* (+ (exp x) 2) (exp x))
1.0ms
(pow (exp x) 3)

series107.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
62.0ms
(+ (pow (exp x) 3) (+ (+ (exp x) 2) (* (exp x) -4)))
18.0ms
(* (+ (exp x) 2) (exp x))
14.0ms
(+ (+ (exp x) 2) (* (exp x) -4))
14.0ms
(pow (exp x) 3)

simplify1.7s

Counts
37 → 76
Calls
37 calls:
Slowest
421.0ms
(* (* (* (+ (exp x) 2) (+ (exp x) 2)) (+ (exp x) 2)) (* (* (exp x) (exp x)) (exp x)))
201.0ms
(* (- (* (exp x) (exp x)) (* 2 2)) (exp x))
166.0ms
(* (exp (pow (exp x) 3)) (exp (+ (+ (exp x) 2) (* (exp x) -4))))
154.0ms
(* (exp (pow (exp x) 3)) (* (exp (+ (exp x) 2)) (exp (* (exp x) -4))))
127.0ms
(- (+ (* 3 x) (+ (* 3/2 (pow x 2)) 1)))

prune678.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes28.0ms

Accuracy

0% (0.5b remaining)

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

bsearch3.0ms

end0.0ms

sample5.0s

Algorithm
intervals
Results
3.7s5689×body1280valid
469.0ms7894×body80valid
415.0ms1215×body640valid
171.0ms613×body320valid
41.0ms325×body160valid