Average Error: 58.7 → 0.5
Time: 5.9s
Precision: 64
Internal Precision: 128
\[e^{x} - 1\]
↓
\[x + \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot \left(x \cdot x\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.7
Target0.5
Herbie0.5
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.7

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.5

    \[\leadsto \color{blue}{x + \left(\frac{1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot {x}^{2}\right)}\]

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019004 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

  :herbie-target
  (* x (+ (+ 1 (/ x 2)) (/ (* x x) 6)))

  (- (exp x) 1))

Details

Time bar (total: 5.5s)Debug log

sample67.0ms

Algorithm
intervals

simplify3.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

3.0ms
(- (exp x) 1)

prune4.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.0b

localize8.0ms

Local error

Found 1 expressions with local error:

5.7b
(- (exp x) 1)

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
1 → 12
Calls

1 calls. Slowest were:

5.0ms
(- (exp x) 1)

series19.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

19.0ms
(- (exp x) 1)

simplify94.0ms

Counts
4 → 15
Calls

4 calls. Slowest were:

83.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
4.0ms
(+ (sqrt (exp x)) 1)
3.0ms
(- (exp x) 1)

prune152.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.1b

localize12.0ms

Local error

Found 3 expressions with local error:

0.4b
(* x 1/6)
0.0b
(* (+ (* x 1/6) 1/2) (* x x))
0.0b
(+ (* x 1/6) 1/2)

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
3 → 35
Calls

3 calls. Slowest were:

6.0ms
(* (+ (* x 1/6) 1/2) (* x x))
1.0ms
(+ (* x 1/6) 1/2)
0.0ms
(* x 1/6)

series56.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

29.0ms
(* (+ (* x 1/6) 1/2) (* x x))
17.0ms
(+ (* x 1/6) 1/2)
10.0ms
(* x 1/6)

simplify1.0s

Counts
16 → 44
Calls

16 calls. Slowest were:

427.0ms
(* (- (* (* x 1/6) (* x 1/6)) (* 1/2 1/2)) (* x x))
351.0ms
(* (+ (pow (* x 1/6) 3) (pow 1/2 3)) (* x x))
76.0ms
(* (* (* (+ (* x 1/6) 1/2) (+ (* x 1/6) 1/2)) (+ (* x 1/6) 1/2)) (* (* (* x x) (* x x)) (* x x)))

prune277.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.1b

localize0.0ms

Local error

Found 1 expressions with local error:

5.7b
(- (exp x) 1)

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
1 → 12
Calls

1 calls. Slowest were:

3.0ms
(- (exp x) 1)

series11.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

11.0ms
(- (exp x) 1)

simplify119.0ms

Counts
4 → 15
Calls

4 calls. Slowest were:

110.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
3.0ms
(- (exp x) 1)
3.0ms
(- (exp x) 1)

prune110.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.1b

localize11.0ms

Local error

Found 3 expressions with local error:

5.9b
(log (exp (- (exp x) 1)))
5.7b
(- (exp x) 1)
0.0b
(exp (- (exp x) 1))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
3 → 46
Calls

3 calls. Slowest were:

13.0ms
(exp (- (exp x) 1))
13.0ms
(log (exp (- (exp x) 1)))
5.0ms
(- (exp x) 1)

series97.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

60.0ms
(exp (- (exp x) 1))
19.0ms
(log (exp (- (exp x) 1)))
18.0ms
(- (exp x) 1)

simplify417.0ms

Counts
29 → 55
Calls

29 calls. Slowest were:

142.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
100.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
20.0ms
(+ x (+ (pow x 2) 1))

prune338.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.1b

regimes170.0ms

Accuracy

0% (0.5b remaining)

Error of 0.5b against oracle of 0.0b and baseline of 0.5b

bsearch1.0ms

end0.0ms

sample2.5s

Algorithm
intervals