Average Error: 58.6 → 0.5
Time: 12.8s
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. 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 2019007 
(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: 12.4s)Debug log

sample101.0ms

Algorithm
intervals

simplify6.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

6.0ms
(- (exp x) 1)

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.3b

localize11.0ms

Local error

Found 1 expressions with local error:

4.6b
(- (exp x) 1)

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
1 → 12
Calls

1 calls. Slowest were:

6.0ms
(- (exp x) 1)

series30.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

30.0ms
(- (exp x) 1)

simplify141.0ms

Counts
4 → 15
Calls

4 calls. Slowest were:

124.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
7.0ms
(- (exp x) 1)
6.0ms
(- (exp x) 1)

prune117.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 3 expressions with local error:

0.3b
(* 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))
2.0ms
(+ (* x 1/6) 1/2)
0.0ms
(* x 1/6)

series52.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

26.0ms
(* (+ (* x 1/6) 1/2) (* x x))
16.0ms
(+ (* x 1/6) 1/2)
9.0ms
(* x 1/6)

simplify1.1s

Counts
16 → 44
Calls

16 calls. Slowest were:

487.0ms
(* (- (* (* x 1/6) (* x 1/6)) (* 1/2 1/2)) (* x x))
323.0ms
(* (+ (pow (* x 1/6) 3) (pow 1/2 3)) (* x x))
109.0ms
(* (* (* (+ (* x 1/6) 1/2) (+ (* x 1/6) 1/2)) (+ (* x 1/6) 1/2)) (* (* (* x x) (* x x)) (* x x)))

prune284.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize28.0ms

Local error

Found 4 expressions with local error:

4.5b
(- (pow (exp x) 3) (pow 1 3))
0.3b
(/ (- (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (+ (* 1 1) (* (exp x) 1))))
0.0b
(pow (exp x) 3)
0.0b
(* (exp x) (exp x))

rewrite50.0ms

Algorithm
rewrite-expression-head
Counts
4 → 113
Calls

4 calls. Slowest were:

28.0ms
(/ (- (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (+ (* 1 1) (* (exp x) 1))))
10.0ms
(- (pow (exp x) 3) (pow 1 3))
7.0ms
(* (exp x) (exp x))

series115.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

59.0ms
(/ (- (pow (exp x) 3) (pow 1 3)) (+ (* (exp x) (exp x)) (+ (* 1 1) (* (exp x) 1))))
32.0ms
(- (pow (exp x) 3) (pow 1 3))
14.0ms
(pow (exp x) 3)
10.0ms
(* (exp x) (exp x))

simplify5.6s

Counts
113 → 125
Calls

113 calls. Slowest were:

284.0ms
(+ (* 3 x) (+ (* 9/2 (pow x 3)) (* 9/2 (pow x 2))))
186.0ms
(/ (* (* (- (pow (exp x) 3) (pow 1 3)) (- (pow (exp x) 3) (pow 1 3))) (- (pow (exp x) 3) (pow 1 3))) (* (* (+ (* (exp x) (exp x)) (+ (* 1 1) (* (exp x) 1))) (+ (* (exp x) (exp x)) (+ (* 1 1) (* (exp x) 1)))) (+ (* (exp x) (exp x)) (+ (* 1 1) (* (exp x) 1)))))
179.0ms
(+ (* (* (exp x) (exp x)) (* (exp x) (exp x))) (- (* (+ (* 1 1) (* (exp x) 1)) (+ (* 1 1) (* (exp x) 1))) (* (* (exp x) (exp x)) (+ (* 1 1) (* (exp x) 1)))))

prune1.0s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize7.0ms

Local error

Found 2 expressions with local error:

4.6b
(- (exp x) 1)
4.5b
(log (exp (- (exp x) 1)))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
2 → 31
Calls

2 calls. Slowest were:

13.0ms
(log (exp (- (exp x) 1)))
6.0ms
(- (exp x) 1)

series37.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

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

simplify342.0ms

Counts
18 → 37
Calls

18 calls. Slowest were:

126.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
120.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
21.0ms
(log (* (cbrt (exp (- (exp x) 1))) (cbrt (exp (- (exp x) 1)))))

prune275.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.0b

regimes152.0ms

Accuracy

0% (0.4b remaining)

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

bsearch2.0ms

end0.0ms

sample2.8s

Algorithm
intervals