Average Error: 29.1 → 0.0
Time: 4.9s
Precision: 64
Internal Precision: 128
\[e^{a \cdot x} - 1\]
↓
\[(e^{a \cdot x} - 1)^*\]

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.1
Target0.2
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Initial program 29.1

    \[e^{a \cdot x} - 1\]

  2. Initial simplification0.0

    \[\leadsto (e^{a \cdot x} - 1)^*\]

  3. Final simplification0.0

    \[\leadsto (e^{a \cdot x} - 1)^*\]

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"

  :herbie-target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))

Details

Time bar (total: 1.7s)Debug log

start99.0ms

Algorithm
intervals

setup21.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 1 expressions with local error:

0.0b
(expm1 (* a x))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 11
Calls

1 calls. Slowest were:

0.0ms
(expm1 (* a x))

series39.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

39.0ms
(expm1 (* a x))

simplify154.0ms

Counts
4 → 14
Calls

4 calls. Slowest were:

144.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
4.0ms
(- (exp (* a x)) 1)
3.0ms
(- (exp (* a x)) 1)

prune104.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 3 expressions with local error:

0.0b
(expm1 (* a x))
0.0b
(log1p (expm1 (expm1 (* a x))))
0.0b
(expm1 (expm1 (* a x)))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
3 → 33
Calls

3 calls. Slowest were:

0.0ms
(expm1 (* a x))
0.0ms
(expm1 (expm1 (* a x)))
0.0ms
(log1p (expm1 (expm1 (* a x))))

series186.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

82.0ms
(expm1 (expm1 (* a x)))
77.0ms
(log1p (expm1 (expm1 (* a x))))
26.0ms
(expm1 (* a x))

simplify534.0ms

Counts
12 → 42
Calls

12 calls. Slowest were:

190.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
144.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
140.0ms
(+ (* (pow a 2) (pow x 2)) (+ (* a x) (* 5/6 (* (pow a 3) (pow x 3)))))

prune398.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes112.0ms

Accuracy

100% (0.0b remaining)

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

bsearch1.0ms