Average Error: 29.8 → 0.7
Time: 10.9s
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

Original29.8
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.8

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

  2. Initial simplification29.8

    \[\leadsto \left(e^{x} - 2\right) - \frac{-1}{e^{x}}\]

  3. Taylor expanded around 0 0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2018362 
(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: 4.3s)Debug log

start177.0ms

Algorithm
intervals

setup28.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 29.8b

localize37.0ms

Local error

Found 3 expressions with local error:

3.5b
(- (- (exp x) 2) (/ -1 (exp x)))
0.0b
(- (exp x) 2)
0.0b
(/ -1 (exp x))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
3 → 43
Calls

3 calls. Slowest were:

16.0ms
(- (- (exp x) 2) (/ -1 (exp x)))
3.0ms
(- (exp x) 2)
0.0ms
(/ -1 (exp x))

series89.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

69.0ms
(- (- (exp x) 2) (/ -1 (exp x)))
11.0ms
(- (exp x) 2)
9.0ms
(/ -1 (exp x))

simplify799.0ms

Counts
23 → 52
Calls

23 calls. Slowest were:

228.0ms
(- (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) -1))
128.0ms
(- (* (- (* (exp x) (exp x)) (* 2 2)) (exp x)) (* (+ (exp x) 2) -1))
83.0ms
(* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) (exp x))

prune416.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize20.0ms

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)) (* 1/360 (pow x 6))))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
3 → 36
Calls

3 calls. Slowest were:

12.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))
1.0ms
(* 1/12 (pow x 4))
1.0ms
(* 1/360 (pow x 6))

series81.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

34.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))
27.0ms
(* 1/12 (pow x 4))
20.0ms
(* 1/360 (pow x 6))

simplify227.0ms

Counts
19 → 45
Calls

19 calls. Slowest were:

58.0ms
(* (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))))

prune783.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize1.0ms

Local error

Found 3 expressions with local error:

3.5b
(- (- (exp x) 2) (/ -1 (exp x)))
0.0b
(- (exp x) 2)
0.0b
(/ -1 (exp x))

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
3 → 43
Calls

3 calls. Slowest were:

26.0ms
(- (- (exp x) 2) (/ -1 (exp x)))
3.0ms
(- (exp x) 2)
0.0ms
(/ -1 (exp x))

series114.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

87.0ms
(- (- (exp x) 2) (/ -1 (exp x)))
14.0ms
(- (exp x) 2)
13.0ms
(/ -1 (exp x))

simplify865.0ms

Counts
23 → 52
Calls

23 calls. Slowest were:

293.0ms
(- (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) -1))
127.0ms
(- (* (- (* (exp x) (exp x)) (* 2 2)) (exp x)) (* (+ (exp x) 2) -1))
127.0ms
(* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) (exp x))

prune528.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes37.0ms

Accuracy

0% (0.6b remaining)

Error of 0.7b against oracle of 0.0b and baseline of 0.7b

bsearch1.0ms