Average Error: 29.6 → 0.6
Time: 27.2s
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

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

Derivation

  1. Initial program 29.6

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

  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 2019018 
(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: 26.6s)Debug log

sample173.0ms

Algorithm
intervals
Results
112.0ms175×body1280valid
17.0ms43×body640valid
16.0ms263×body80valid
6.0ms24×body320valid
1.0ms6×body160valid

simplify31.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
31.0ms
(+ (- (exp x) 2) (exp (- x)))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 31.0b

localize32.0ms

Local error

Found 2 expressions with local error:

2.8b
(+ (- (exp x) 2) (exp (- x)))
0.0b
(- (exp x) 2)

rewrite18.0ms

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
13.0ms
(+ (- (exp x) 2) (exp (- x)))
5.0ms
(- (exp x) 2)

series96.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
88.0ms
(+ (- (exp x) 2) (exp (- x)))
9.0ms
(- (exp x) 2)

simplify1.5s

Counts
19 → 36
Calls
19 calls:
Slowest
317.0ms
(+ (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) (exp 0)))
304.0ms
(+ (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) 1))
249.0ms
(+ (* (- (* (exp x) (exp x)) (* 2 2)) (exp x)) (* (+ (exp x) 2) (exp 0)))
190.0ms
(+ (* (- (* (exp x) (exp x)) (* 2 2)) (exp x)) (* (+ (exp x) 2) 1))
114.0ms
(* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) (exp x))

prune312.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.1b

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

rewrite12.0ms

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
6.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))
3.0ms
(+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))
1.0ms
(* 1/12 (pow x 4))
1.0ms
(* 1/360 (pow x 6))

series128.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
63.0ms
(+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))
30.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))
19.0ms
(* 1/360 (pow x 6))
16.0ms
(* 1/12 (pow x 4))

simplify249.0ms

Counts
23 → 59
Calls
23 calls:
Slowest
39.0ms
(* (exp (pow x 2)) (* (exp (* 1/12 (pow x 4))) (exp (* 1/360 (pow x 6)))))
32.0ms
(* (exp (pow x 2)) (exp (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6)))))
27.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))
26.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))
25.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))

prune644.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize28.0ms

Local error

Found 4 expressions with local error:

3.0b
(+ (/ (- (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
(/ (- (pow (exp x) 3) (pow 2 3)) (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))))

rewrite152.0ms

Algorithm
rewrite-expression-head
Rules
45×add-sqr-sqrt
31×*-un-lft-identity
24×times-frac
21×add-cube-cbrt
12×difference-of-squares
10×pow1
9×add-cbrt-cube
8×associate-/l*
7×cube-prod
7×unpow-prod-down
6×add-log-exp
6×add-exp-log
4×distribute-lft-out--
4×difference-cubes
3×associate-*l*
3×associate-*r*
3×associate-/r*
2×associate-/l/
2×flip-+
2×associate-/r/
2×frac-add
2×div-sub
2×flip3-+
1×rem-cube-cbrt
1×distribute-lft-out
1×flip--
1×div-inv
1×cbrt-unprod
1×associate-+l-
1×pow-prod-up
1×*-commutative
1×prod-exp
1×pow-exp
1×pow-prod-down
1×div-exp
1×unpow3
1×exp-diff
1×flip3--
1×pow-plus
1×frac-2neg
1×pow-pow
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
1×pow-to-exp
1×cube-mult
1×exp-neg
1×neg-sub0
1×pow2
Counts
4 → 106
Calls
4 calls:
Slowest
109.0ms
(+ (/ (- (pow (exp x) 3) (pow 2 3)) (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2)))) (exp (- x)))
34.0ms
(/ (- (pow (exp x) 3) (pow 2 3)) (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))))
7.0ms
(* (exp x) (exp x))
2.0ms
(pow (exp x) 3)

series391.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
290.0ms
(+ (/ (- (pow (exp x) 3) (pow 2 3)) (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2)))) (exp (- x)))
70.0ms
(/ (- (pow (exp x) 3) (pow 2 3)) (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))))
16.0ms
(pow (exp x) 3)
15.0ms
(* (exp x) (exp x))

simplify4.9s

Counts
103 → 118
Calls
103 calls:
Slowest
273.0ms
(+ (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) 1))
234.0ms
(+ (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) (exp 0)))
216.0ms
(- (* (pow (exp x) 3) (pow (exp x) 3)) (* (pow 2 3) (pow 2 3)))
157.0ms
(/ (* (* (- (pow (exp x) 3) (pow 2 3)) (- (pow (exp x) 3) (pow 2 3))) (- (pow (exp x) 3) (pow 2 3))) (* (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2)))) (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2)))))
139.0ms
(+ (* (* (exp x) (exp x)) (* (exp x) (exp x))) (- (* (+ (* 2 2) (* (exp x) 2)) (+ (* 2 2) (* (exp x) 2))) (* (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2)))))

prune1.4s

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

localize46.0ms

Local error

Found 4 expressions with local error:

3.5b
(+ (+ 4 (+ (pow (exp x) 4) (* (exp x) -8))) (* (exp x) (+ (exp x) 2)))
0.0b
(pow (exp x) 4)
0.0b
(/ (+ (+ 4 (+ (pow (exp x) 4) (* (exp x) -8))) (* (exp x) (+ (exp x) 2))) (+ (* (+ (exp x) 2) (exp (+ x x))) (* 4 (exp x))))
0.0b
(+ 4 (+ (pow (exp x) 4) (* (exp x) -8)))

rewrite166.0ms

Algorithm
rewrite-expression-head
Rules
13×add-cube-cbrt
13×*-un-lft-identity
13×add-sqr-sqrt
12×flip-+
12×flip3-+
9×times-frac
8×associate-*r/
8×frac-add
6×associate-/l/
6×add-log-exp
6×add-exp-log
6×add-cbrt-cube
5×pow1
3×associate-/l*
3×unpow-prod-down
3×associate-/r*
3×associate-+r+
2×associate-/r/
2×+-commutative
1×div-inv
1×pow-exp
1×div-exp
1×distribute-lft-in
1×frac-2neg
1×pow-pow
1×sum-log
1×clear-num
1×associate-+l+
1×cbrt-undiv
1×pow-to-exp
1×distribute-rgt-in
Counts
4 → 77
Calls
4 calls:
Slowest
136.0ms
(/ (+ (+ 4 (+ (pow (exp x) 4) (* (exp x) -8))) (* (exp x) (+ (exp x) 2))) (+ (* (+ (exp x) 2) (exp (+ x x))) (* 4 (exp x))))
22.0ms
(+ (+ 4 (+ (pow (exp x) 4) (* (exp x) -8))) (* (exp x) (+ (exp x) 2)))
5.0ms
(+ 4 (+ (pow (exp x) 4) (* (exp x) -8)))
2.0ms
(pow (exp x) 4)

series430.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
293.0ms
(/ (+ (+ 4 (+ (pow (exp x) 4) (* (exp x) -8))) (* (exp x) (+ (exp x) 2))) (+ (* (+ (exp x) 2) (exp (+ x x))) (* 4 (exp x))))
82.0ms
(+ (+ 4 (+ (pow (exp x) 4) (* (exp x) -8))) (* (exp x) (+ (exp x) 2)))
31.0ms
(+ 4 (+ (pow (exp x) 4) (* (exp x) -8)))
24.0ms
(pow (exp x) 4)

simplify9.0s

Counts
62 → 89
Calls
62 calls:
Slowest
448.0ms
(* (+ (* 4 4) (- (* (+ (pow (exp x) 4) (* (exp x) -8)) (+ (pow (exp x) 4) (* (exp x) -8))) (* 4 (+ (pow (exp x) 4) (* (exp x) -8))))) (- (exp x) 2))
400.0ms
(/ (* (* (+ (+ 4 (+ (pow (exp x) 4) (* (exp x) -8))) (* (exp x) (+ (exp x) 2))) (+ (+ 4 (+ (pow (exp x) 4) (* (exp x) -8))) (* (exp x) (+ (exp x) 2)))) (+ (+ 4 (+ (pow (exp x) 4) (* (exp x) -8))) (* (exp x) (+ (exp x) 2)))) (* (* (+ (* (+ (exp x) 2) (exp (+ x x))) (* 4 (exp x))) (+ (* (+ (exp x) 2) (exp (+ x x))) (* 4 (exp x)))) (+ (* (+ (exp x) 2) (exp (+ x x))) (* 4 (exp x)))))
390.0ms
(/ (- (+ (pow (exp x) 2) (+ (pow (exp x) 4) 4)) (* 6 (exp x))) (+ (* 4 (exp x)) (+ (* (exp x) (exp (* 2 x))) (* 2 (exp (* 2 x))))))
340.0ms
(/ (- (+ (pow (exp x) 2) (+ (pow (exp x) 4) 4)) (* 6 (exp x))) (+ (* 4 (exp x)) (+ (* (exp x) (exp (* 2 x))) (* 2 (exp (* 2 x))))))
288.0ms
(/ (sqrt (+ (+ 4 (+ (pow (exp x) 4) (* (exp x) -8))) (* (exp x) (+ (exp x) 2)))) 1)

prune1.2s

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

regimes35.0ms

Accuracy

0% (0.5b remaining)

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

bsearch2.0ms

end0.0ms

sample5.6s

Algorithm
intervals
Results
4.4s5721×body1280valid
475.0ms1221×body640valid
453.0ms7867×body80valid
129.0ms597×body320valid
41.0ms308×body160valid