Average Error: 29.8 → 0.6
Time: 16.1s
Precision: 64
Internal Precision: 128
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[(\frac{1}{12} \cdot \left({x}^{4}\right) + \left((\frac{1}{360} \cdot \left({x}^{6}\right) + \left(x \cdot x\right))_*\right))_*\]

Error

Bits error versus x

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. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(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: 15.5s)Debug log

sample197.0ms

Algorithm
intervals

simplify20.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

20.0ms
(+ (- (exp x) 2) (exp (- x)))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 28.9b

localize37.0ms

Local error

Found 2 expressions with local error:

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

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
2 → 40
Calls

2 calls. Slowest were:

15.0ms
(+ (- (exp x) 2) (exp (- x)))
3.0ms
(- (exp x) 2)

series76.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

63.0ms
(+ (- (exp x) 2) (exp (- x)))
13.0ms
(- (exp x) 2)

simplify1.7s

Counts
23 → 46
Calls

23 calls. Slowest were:

518.0ms
(+ (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) (exp 0)))
306.0ms
(+ (* (- (* (exp x) (exp x)) (* 2 2)) (exp x)) (* (+ (exp x) 2) (exp 0)))
205.0ms
(+ (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) 1))

prune324.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 1 expressions with local error:

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

rewrite0.0ms

Algorithm
rewrite-expression-head
Counts
1 → 10
Calls

1 calls. Slowest were:

0.0ms
(fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x)))

series29.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

29.0ms
(fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x)))

simplify93.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

33.0ms
(+ (pow x 2) (+ (* 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))))
29.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))

prune187.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize32.0ms

Local error

Found 4 expressions with local error:

4.1b
(fma (fma (exp x) (exp x) -4) (exp x) (+ (exp x) 2))
0.0b
(fma (exp x) (exp x) -4)
0.0b
(/ (fma (fma (exp x) (exp x) -4) (exp x) (+ (exp x) 2)) (* (+ (exp x) 2) (exp x)))
0.0b
(* (+ (exp x) 2) (exp x))

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

15.0ms
(* (+ (exp x) 2) (exp x))
13.0ms
(/ (fma (fma (exp x) (exp x) -4) (exp x) (+ (exp x) 2)) (* (+ (exp x) 2) (exp x)))
0.0ms
(fma (fma (exp x) (exp x) -4) (exp x) (+ (exp x) 2))

series351.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

232.0ms
(/ (fma (fma (exp x) (exp x) -4) (exp x) (+ (exp x) 2)) (* (+ (exp x) 2) (exp x)))
84.0ms
(fma (fma (exp x) (exp x) -4) (exp x) (+ (exp x) 2))
18.0ms
(fma (exp x) (exp x) -4)
16.0ms
(* (+ (exp x) 2) (exp x))

simplify3.8s

Counts
37 → 78
Calls

37 calls. Slowest were:

572.0ms
(* (* (* (+ (exp x) 2) (+ (exp x) 2)) (+ (exp x) 2)) (* (* (exp x) (exp x)) (exp x)))
419.0ms
(* (- (* (exp x) (exp x)) (* 2 2)) (exp x))
346.0ms
(/ (- (+ (pow (exp x) 3) 2) (* 3 (exp x))) (* (exp x) (+ (exp x) 2)))

prune780.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize28.0ms

Local error

Found 4 expressions with local error:

4.9b
(+ (fma (* (cbrt (exp x)) (cbrt (exp x))) (cbrt (exp x)) (- 2)) (exp (- x)))
0.0b
(cbrt (exp x))
0.0b
(cbrt (exp x))
0.0b
(cbrt (exp x))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 57
Calls

4 calls. Slowest were:

9.0ms
(+ (fma (* (cbrt (exp x)) (cbrt (exp x))) (cbrt (exp x)) (- 2)) (exp (- x)))
1.0ms
(cbrt (exp x))
1.0ms
(cbrt (exp x))

series963.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

531.0ms
(+ (fma (* (cbrt (exp x)) (cbrt (exp x))) (cbrt (exp x)) (- 2)) (exp (- x)))
157.0ms
(cbrt (exp x))
156.0ms
(cbrt (exp x))
119.0ms
(cbrt (exp x))

simplify473.0ms

Counts
26 → 69
Calls

26 calls. Slowest were:

78.0ms
(- (+ (exp (* -1 x)) (* (pow (pow (exp x) 2) 1/3) (exp (* 1/3 x)))) 2)
72.0ms
(- (+ (* (pow (pow (exp x) 2) 1/3) (exp (* 1/3 x))) (exp (- x))) 2)
57.0ms
(+ (* 1/3 x) (+ (* 1/18 (pow x 2)) 1))

prune731.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

regimes23.0ms

Accuracy

0% (0.6b remaining)

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

bsearch1.0ms

end0.0ms

sample5.5s

Algorithm
intervals