Average Error: 58.6 → 0.1
Time: 11.0s
Precision: 64
Internal Precision: 128
\[e^{x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.00015865222814556896:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} - 1\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.4
Herbie0.1
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 0.00015865222814556896

    1. Initial program 59.2

      \[e^{x} - 1\]

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

    if 0.00015865222814556896 < x

    1. Initial program 2.9

      \[e^{x} - 1\]

    2. Taylor expanded around inf 2.9

      \[\leadsto \color{blue}{e^{x} - 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.00015865222814556896:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 1137770271 
(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: 10.3s)Debug log

sample150.0ms

Algorithm
intervals
Results
112×53.0ms(body real 1280)
80×26.0ms(body real 640)
45×20.0ms(body real 320)
387×14.0ms(pre true 80)
137×8.0ms(body real 80)
13×2.0ms(body real 160)

simplify7.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
7.0ms
(- (exp x) 1)

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.3b

localize23.0ms

Local error

Found 1 expressions with local error:

4.0b
(- (exp x) 1)

rewrite8.0ms

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
8.0ms
(- (exp x) 1)

series26.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
26.0ms
(- (exp x) 1)

simplify209.0ms

Counts
4 → 15
Calls
4 calls:
Slowest
187.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
8.0ms
(- (exp x) 1)
8.0ms
(- (exp x) 1)
4.0ms
(+ (sqrt (exp x)) 1)

prune231.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0b

localize40.0ms

Local error

Found 4 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)
0.0b
(+ x (* (+ (* x 1/6) 1/2) (* x x)))

rewrite24.0ms

Algorithm
rewrite-expression-head
Rules
6×add-exp-log
6×add-cbrt-cube
6×pow1
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-log-exp
3×flip-+
3×associate-*l*
3×flip3-+
2×*-commutative
2×associate-*l/
2×+-commutative
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×associate-*r*
Counts
4 → 45
Calls
4 calls:
Slowest
14.0ms
(* (+ (* x 1/6) 1/2) (* x x))
4.0ms
(+ x (* (+ (* x 1/6) 1/2) (* x x)))
4.0ms
(+ (* x 1/6) 1/2)
0.0ms
(* x 1/6)

series166.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
62.0ms
(* (+ (* x 1/6) 1/2) (* x x))
45.0ms
(+ x (* (+ (* x 1/6) 1/2) (* x x)))
43.0ms
(+ (* x 1/6) 1/2)
15.0ms
(* x 1/6)

simplify2.4s

Counts
19 → 57
Calls
19 calls:
Slowest
705.0ms
(* (- (* (* x 1/6) (* x 1/6)) (* 1/2 1/2)) (* x x))
655.0ms
(* (+ (pow (* x 1/6) 3) (pow 1/2 3)) (* x x))
178.0ms
(* (* (* (+ (* x 1/6) 1/2) (+ (* x 1/6) 1/2)) (+ (* x 1/6) 1/2)) (* (* (* x x) (* x x)) (* x x)))
172.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
162.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))

prune731.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize1.0ms

Local error

Found 1 expressions with local error:

4.0b
(- (exp x) 1)

rewrite7.0ms

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
7.0ms
(- (exp x) 1)

series26.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
26.0ms
(- (exp x) 1)

simplify196.0ms

Counts
4 → 15
Calls
4 calls:
Slowest
174.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
9.0ms
(- (exp x) 1)
7.0ms
(- (exp x) 1)
5.0ms
(+ (sqrt (exp x)) 1)

prune203.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize25.0ms

Local error

Found 4 expressions with local error:

4.0b
(- (exp x) 1)
4.0b
(- (exp x) 1)
4.0b
(- (exp x) 1)
0.1b
(cbrt (- (exp x) 1))

rewrite43.0ms

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
14.0ms
(- (exp x) 1)
10.0ms
(cbrt (- (exp x) 1))
9.0ms
(- (exp x) 1)
8.0ms
(- (exp x) 1)

series316.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
239.0ms
(cbrt (- (exp x) 1))
32.0ms
(- (exp x) 1)
23.0ms
(- (exp x) 1)
23.0ms
(- (exp x) 1)

simplify743.0ms

Counts
21 → 62
Calls
21 calls:
Slowest
173.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
166.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
162.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
72.0ms
(cbrt (- (* (exp x) (exp x)) (* 1 1)))
31.0ms
(+ (* 1/36 (pow (pow x 7) 1/3)) (+ (* 1/6 (pow (pow x 4) 1/3)) (pow x 1/3)))

prune893.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

regimes252.0ms

Accuracy

99.1% (0.0b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 0.4b

bsearch55.0ms

end0.0ms

sample3.5s

Algorithm
intervals
Results
3339×1.5s(body real 1280)
2566×830.0ms(body real 640)
11907×369.0ms(pre true 80)
4091×286.0ms(body real 80)
1222×245.0ms(body real 320)
689×151.0ms(body real 160)