Average Error: 58.5 → 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

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

Derivation

  1. Initial program 58.5

    \[e^{x} - 1\]

  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 2019018 
(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: 5.7s)Debug log

sample70.0ms

Algorithm
intervals
Results
32.0ms113×body1280valid
13.0ms78×body640valid
6.0ms360×pre80true
5.0ms47×body320valid
3.0ms107×body80valid
1.0ms15×body160valid

simplify5.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
5.0ms
(- (exp x) 1)

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 59.1b

localize11.0ms

Local error

Found 1 expressions with local error:

4.8b
(- (exp x) 1)

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
2×add-sqr-sqrt
1×add-log-exp
1×flip--
1×add-cube-cbrt
1×add-exp-log
1×difference-of-sqr-1
1×add-cbrt-cube
1×flip3--
1×*-un-lft-identity
1×sub-neg
1×pow1
Counts
1 → 12
Calls
1 calls:
Slowest
6.0ms
(- (exp x) 1)

series20.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
20.0ms
(- (exp x) 1)

simplify144.0ms

Counts
4 → 15
Calls
4 calls:
Slowest
132.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
4.0ms
(+ (sqrt (exp x)) 1)
4.0ms
(- (exp x) 1)
4.0ms
(- (exp x) 1)

prune134.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize28.0ms

Local error

Found 2 expressions with local error:

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

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×add-cube-cbrt
3×associate-*l*
3×*-un-lft-identity
3×add-sqr-sqrt
2×add-log-exp
2×*-commutative
2×associate-*l/
1×flip-+
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×associate-*r*
1×flip3-+
Counts
2 → 25
Calls
2 calls:
Slowest
10.0ms
(* (+ (* x 1/6) 1/2) (* x x))
0.0ms
(* x 1/6)

series58.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
46.0ms
(* (+ (* x 1/6) 1/2) (* x x))
11.0ms
(* x 1/6)

simplify1.1s

Counts
13 → 31
Calls
13 calls:
Slowest
416.0ms
(* (- (* (* x 1/6) (* x 1/6)) (* 1/2 1/2)) (* x x))
383.0ms
(* (+ (pow (* x 1/6) 3) (pow 1/2 3)) (* x x))
104.0ms
(* (* (* (+ (* x 1/6) 1/2) (+ (* x 1/6) 1/2)) (+ (* x 1/6) 1/2)) (* (* (* x x) (* x x)) (* x x)))
60.0ms
(+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))
57.0ms
(+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))

prune232.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize17.0ms

Local error

Found 4 expressions with local error:

4.8b
(- (exp x) 1)
4.8b
(- (exp x) 1)
4.8b
(- (exp x) 1)
0.1b
(cbrt (- (exp x) 1))

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
9×add-sqr-sqrt
5×add-cube-cbrt
5×*-un-lft-identity
4×add-log-exp
4×flip--
4×add-exp-log
4×difference-of-sqr-1
4×add-cbrt-cube
4×flip3--
4×pow1
4×cbrt-prod
3×sub-neg
2×cbrt-div
1×pow1/3
Counts
4 → 50
Calls
4 calls:
Slowest
6.0ms
(- (exp x) 1)
5.0ms
(- (exp x) 1)
5.0ms
(- (exp x) 1)
4.0ms
(cbrt (- (exp x) 1))

series197.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
146.0ms
(cbrt (- (exp x) 1))
18.0ms
(- (exp x) 1)
18.0ms
(- (exp x) 1)
15.0ms
(- (exp x) 1)

simplify518.0ms

Counts
21 → 62
Calls
21 calls:
Slowest
144.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
138.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
101.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
38.0ms
(cbrt (- (* (exp x) (exp x)) (* 1 1)))
19.0ms
(cbrt (- (pow (exp x) 3) (pow 1 3)))

prune468.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes54.0ms

Accuracy

0% (0.4b remaining)

Error of 0.5b against oracle of 0.1b and baseline of 0.5b

bsearch1.0ms

end0.0ms

sample2.5s

Algorithm
intervals
Results
1.4s3378×body1280valid
478.0ms2537×body640valid
234.0ms12023×pre80true
144.0ms4234×body80valid
138.0ms1266×body320valid
77.0ms608×body160valid