Average Error: 29.2 → 0.0
Time: 6.0s
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

Original29.2
Target0.1
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 29.2

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

  2. Initial simplification0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2018362 +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: 2.7s)Debug log

start102.0ms

Algorithm
intervals

setup12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

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:

1.0ms
(expm1 (* a x))

series40.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

40.0ms
(expm1 (* a x))

simplify225.0ms

Counts
4 → 14
Calls

4 calls. Slowest were:

209.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
7.0ms
(- (exp (* a x)) 1)
7.0ms
(- (exp (* a x)) 1)

prune165.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize17.0ms

Local error

Found 4 expressions with local error:

19.2b
(cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))
0.2b
(* (expm1 (* a x)) (expm1 (* a x)))
0.1b
(* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x)))
0.0b
(expm1 (* a x))

rewrite41.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

19.0ms
(* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x)))
12.0ms
(cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))
7.0ms
(* (expm1 (* a x)) (expm1 (* a x)))

series135.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

46.0ms
(cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))
34.0ms
(* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x)))
29.0ms
(* (expm1 (* a x)) (expm1 (* a x)))
26.0ms
(expm1 (* a x))

simplify1.0s

Counts
46 → 84
Calls

46 calls. Slowest were:

250.0ms
(+ (* 7/12 (* (pow a 4) (pow x 4))) (+ (* (pow a 3) (pow x 3)) (* (pow a 2) (pow x 2))))
210.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
190.0ms
(+ (* 3/2 (* (pow a 4) (pow x 4))) (+ (* 5/4 (* (pow a 5) (pow x 5))) (* (pow a 3) (pow x 3))))

prune885.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes73.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms