Average Error: 58.6 → 0.5
Time: 12.6s
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 2019021 
(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: 12.2s)Debug log

sample95.0ms

Algorithm
intervals
Results
38.0ms107×body1280valid
21.0ms90×body640valid
9.0ms383×pre80true
5.0ms132×body80valid
5.0ms35×body320valid
2.0ms19×body160valid

simplify3.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
3.0ms
(- (exp x) 1)

prune4.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.5b

localize6.0ms

Local error

Found 1 expressions with local error:

4.2b
(- (exp x) 1)

rewrite3.0ms

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
3.0ms
(- (exp x) 1)

series12.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
12.0ms
(- (exp x) 1)

simplify133.0ms

Counts
4 → 15
Calls
4 calls:
Slowest
123.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
4.0ms
(- (exp x) 1)
4.0ms
(- (exp x) 1)
2.0ms
(+ (sqrt (exp x)) 1)

prune120.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize26.0ms

Local error

Found 2 expressions with local error:

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

rewrite12.0ms

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
11.0ms
(* (+ (* x 1/6) 1/2) (* x x))
0.0ms
(* x 1/6)

series61.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
49.0ms
(* (+ (* x 1/6) 1/2) (* x x))
12.0ms
(* x 1/6)

simplify1.1s

Counts
13 → 31
Calls
13 calls:
Slowest
407.0ms
(* (- (* (* x 1/6) (* x 1/6)) (* 1/2 1/2)) (* x x))
368.0ms
(* (+ (pow (* x 1/6) 3) (pow 1/2 3)) (* x x))
131.0ms
(* (* (* (+ (* x 1/6) 1/2) (+ (* x 1/6) 1/2)) (+ (* x 1/6) 1/2)) (* (* (* x x) (* x x)) (* x x)))
55.0ms
(+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))
36.0ms
(+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))

prune207.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

4.3b
(- (sqrt (exp x)) 1)
0.0b
(* (+ (sqrt (exp x)) 1) (- (sqrt (exp x)) 1))
0.0b
(+ (sqrt (exp x)) 1)
0.0b
(sqrt (exp x))

rewrite36.0ms

Algorithm
rewrite-expression-head
Rules
12×add-sqr-sqrt
7×add-cube-cbrt
7×*-un-lft-identity
6×add-exp-log
6×add-cbrt-cube
6×pow1
5×sqrt-prod
5×associate-*r*
4×add-log-exp
4×flip--
4×flip-+
4×difference-of-sqr-1
4×flip3--
4×flip3-+
4×frac-times
3×associate-*l*
3×sub-neg
2×associate-*r/
2×associate-*l/
2×distribute-lft-in
2×distribute-rgt-in
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×pow1/2
1×rem-sqrt-square
1×+-commutative
Counts
4 → 66
Calls
4 calls:
Slowest
19.0ms
(* (+ (sqrt (exp x)) 1) (- (sqrt (exp x)) 1))
9.0ms
(- (sqrt (exp x)) 1)
6.0ms
(+ (sqrt (exp x)) 1)
1.0ms
(sqrt (exp x))

series104.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
47.0ms
(* (+ (sqrt (exp x)) 1) (- (sqrt (exp x)) 1))
22.0ms
(- (sqrt (exp x)) 1)
18.0ms
(+ (sqrt (exp x)) 1)
16.0ms
(sqrt (exp x))

simplify3.8s

Counts
44 → 78
Calls
44 calls:
Slowest
355.0ms
(* (+ (* (sqrt (exp x)) (sqrt (exp x))) (- (* 1 1) (* (sqrt (exp x)) 1))) (+ (* (sqrt (exp x)) (sqrt (exp x))) (+ (* 1 1) (* (sqrt (exp x)) 1))))
352.0ms
(* (- (* (sqrt (exp x)) (sqrt (exp x))) (* 1 1)) (- (sqrt (exp x)) 1))
302.0ms
(* (- (sqrt (exp x)) 1) (+ (sqrt (exp x)) 1))
276.0ms
(* (- (sqrt (exp x)) 1) (+ (sqrt (exp x)) 1))
265.0ms
(* (+ (pow (sqrt (exp x)) 3) (pow 1 3)) (- (* (sqrt (exp x)) (sqrt (exp x))) (* 1 1)))

prune330.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 4 expressions with local error:

4.3b
(log (exp (- (sqrt (exp x)) 1)))
4.3b
(- (sqrt (exp x)) 1)
0.0b
(* (+ (sqrt (exp x)) 1) (log (exp (- (sqrt (exp x)) 1))))
0.0b
(+ (sqrt (exp x)) 1)

rewrite53.0ms

Algorithm
rewrite-expression-head
Rules
17×add-sqr-sqrt
15×log-prod
12×log-pow
11×add-cube-cbrt
11×*-un-lft-identity
10×exp-prod
9×associate-*r*
8×pow1
6×add-exp-log
6×exp-sum
6×difference-of-sqr-1
6×add-cbrt-cube
5×distribute-lft-in
5×distribute-rgt-in
4×add-log-exp
4×sub-neg
3×sqrt-prod
3×associate-*l*
2×flip-+
2×associate-*l/
2×flip3-+
1×flip--
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×rem-log-exp
1×exp-diff
1×flip3--
1×log-div
1×+-commutative
Counts
4 → 78
Calls
4 calls:
Slowest
31.0ms
(* (+ (sqrt (exp x)) 1) (log (exp (- (sqrt (exp x)) 1))))
9.0ms
(log (exp (- (sqrt (exp x)) 1)))
6.0ms
(+ (sqrt (exp x)) 1)
4.0ms
(- (sqrt (exp x)) 1)

series112.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
54.0ms
(* (+ (sqrt (exp x)) 1) (log (exp (- (sqrt (exp x)) 1))))
21.0ms
(+ (sqrt (exp x)) 1)
19.0ms
(log (exp (- (sqrt (exp x)) 1)))
18.0ms
(- (sqrt (exp x)) 1)

simplify3.1s

Counts
52 → 90
Calls
52 calls:
Slowest
647.0ms
(* (* (* (+ (sqrt (exp x)) 1) (+ (sqrt (exp x)) 1)) (+ (sqrt (exp x)) 1)) (* (* (log (exp (- (sqrt (exp x)) 1))) (log (exp (- (sqrt (exp x)) 1)))) (log (exp (- (sqrt (exp x)) 1)))))
336.0ms
(* (- (sqrt (exp x)) 1) (+ (sqrt (exp x)) 1))
330.0ms
(* (log (exp (- (sqrt (exp x)) 1))) (+ (sqrt (exp x)) 1))
291.0ms
(* (- (sqrt (exp x)) 1) (+ (sqrt (exp x)) 1))
213.0ms
(* (- (* (sqrt (exp x)) (sqrt (exp x))) (* 1 1)) (log (exp (- (sqrt (exp x)) 1))))

prune531.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

regimes91.0ms

Accuracy

0% (0.4b remaining)

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

bsearch1.0ms

end0.0ms

sample2.3s

Algorithm
intervals
Results
853.0ms2523×body640valid
813.0ms3418×body1280valid
232.0ms12106×pre80true
121.0ms1235×body320valid
108.0ms4293×body80valid
42.0ms637×body160valid