Average Error: 58.6 → 0.5
Time: 16.7s
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

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

Derivation

  1. Initial program 58.6

    \[e^{x} - 1\]

  2. Initial simplification58.6

    \[\leadsto e^{x} - 1\]

  3. Taylor expanded around 0 0.5

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

  4. Simplified0.5

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

  5. 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 2018362 
(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: 13.7s)Debug log

start68.0ms

Algorithm
intervals

setup10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.7b

localize5.0ms

Local error

Found 1 expressions with local error:

7.3b
(- (exp x) 1)

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
1 → 12
Calls

1 calls. Slowest were:

3.0ms
(- (exp x) 1)

series21.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

21.0ms
(- (exp x) 1)

simplify139.0ms

Counts
4 → 15
Calls

4 calls. Slowest were:

130.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
3.0ms
(- (exp x) 1)
3.0ms
(- (exp x) 1)

prune114.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.1b

localize14.0ms

Local error

Found 2 expressions with local error:

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

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
2 → 25
Calls

2 calls. Slowest were:

5.0ms
(* (+ (* x 1/6) 1/2) (* x x))
0.0ms
(* x 1/6)

series34.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

27.0ms
(* (+ (* x 1/6) 1/2) (* x x))
7.0ms
(* x 1/6)

simplify1.2s

Counts
13 → 31
Calls

13 calls. Slowest were:

468.0ms
(* (- (* (* x 1/6) (* x 1/6)) (* 1/2 1/2)) (* x x))
412.0ms
(* (+ (pow (* x 1/6) 3) (pow 1/2 3)) (* x x))
126.0ms
(* (* (* (+ (* x 1/6) 1/2) (+ (* x 1/6) 1/2)) (+ (* x 1/6) 1/2)) (* (* (* x x) (* x x)) (* x x)))

prune341.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.1b

localize49.0ms

Local error

Found 4 expressions with local error:

7.2b
(- (pow (exp x) 3) (pow 1 3))
0.0b
(* (exp x) (exp x))
0.0b
(pow (exp x) 3)
0.0b
(+ (* 1 1) (* (exp x) 1))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

13.0ms
(- (pow (exp x) 3) (pow 1 3))
7.0ms
(* (exp x) (exp x))
5.0ms
(+ (* 1 1) (* (exp x) 1))

series68.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

32.0ms
(- (pow (exp x) 3) (pow 1 3))
15.0ms
(pow (exp x) 3)
11.0ms
(+ (* 1 1) (* (exp x) 1))
10.0ms
(* (exp x) (exp x))

simplify868.0ms

Counts
45 → 81
Calls

45 calls. Slowest were:

331.0ms
(+ (* 3 x) (+ (* 9/2 (pow x 3)) (* 9/2 (pow x 2))))
69.0ms
(+ (* 2 x) (+ (* 2 (pow x 2)) 1))
65.0ms
(* (* (* (exp x) (exp x)) (exp x)) (* (* (exp x) (exp x)) (exp x)))

prune633.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize57.0ms

Local error

Found 4 expressions with local error:

7.1b
(- (pow (pow (exp x) 3) 3) (pow (pow 1 3) 3))
0.0b
(* (exp x) (exp x))
0.0b
(pow (pow (exp x) 3) 3)
0.0b
(* (pow (exp x) 3) (pow (exp x) 3))

rewrite41.0ms

Algorithm
rewrite-expression-head
Counts
4 → 145
Calls

4 calls. Slowest were:

27.0ms
(- (pow (pow (exp x) 3) 3) (pow (pow 1 3) 3))
5.0ms
(* (exp x) (exp x))
5.0ms
(* (pow (exp x) 3) (pow (exp x) 3))

series110.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

41.0ms
(- (pow (pow (exp x) 3) 3) (pow (pow 1 3) 3))
35.0ms
(pow (pow (exp x) 3) 3)
18.0ms
(* (pow (exp x) 3) (pow (exp x) 3))
15.0ms
(* (exp x) (exp x))

simplify7.3s

Counts
150 → 157
Calls

150 calls. Slowest were:

901.0ms
(pow (pow (* (cbrt (exp x)) (cbrt (exp x))) 3) 3)
565.0ms
(* (* (* (pow (exp x) 3) (pow (exp x) 3)) (pow (exp x) 3)) (* (* (pow (exp x) 3) (pow (exp x) 3)) (pow (exp x) 3)))
460.0ms
(pow (pow (* (cbrt (exp x)) (cbrt (exp x))) 3) 3)

prune2.4s

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.0b

regimes191.0ms

Accuracy

0% (0.4b remaining)

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

bsearch2.0ms