Average Error: 28.9 → 0.0
Time: 10.8s
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

Original28.9
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 28.9

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019007 +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: 10.0s)Debug log

sample81.0ms

Algorithm
intervals

simplify7.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

7.0ms
(- (exp (* a x)) 1)

prune16.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize12.0ms

Local error

Found 1 expressions with local error:

0.1b
(expm1 (* a x))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 11
Calls

1 calls. Slowest were:

1.0ms
(expm1 (* a x))

series43.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

43.0ms
(expm1 (* a x))

simplify156.0ms

Counts
4 → 14
Calls

4 calls. Slowest were:

145.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
3.0ms
(- (exp (* a x)) 1)
3.0ms
(- (exp (* a x)) 1)

prune137.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.1b

localize15.0ms

Local error

Found 4 expressions with local error:

10.1b
(cbrt (expm1 (* a x)))
10.1b
(cbrt (expm1 (* a x)))
10.1b
(cbrt (expm1 (* a x)))
0.3b
(* (cbrt (expm1 (* a x))) (cbrt (expm1 (* a x))))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

5.0ms
(* (cbrt (expm1 (* a x))) (cbrt (expm1 (* a x))))
1.0ms
(cbrt (expm1 (* a x)))
1.0ms
(cbrt (expm1 (* a x)))

series406.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

111.0ms
(* (cbrt (expm1 (* a x))) (cbrt (expm1 (* a x))))
106.0ms
(cbrt (expm1 (* a x)))
97.0ms
(cbrt (expm1 (* a x)))
92.0ms
(cbrt (expm1 (* a x)))

simplify1.0s

Counts
41 → 81
Calls

41 calls. Slowest were:

294.0ms
(+ (exp (* 1/3 (+ (* 2 (log a)) (* 2 (log x))))) (+ (* 1/3 (* a (* x (exp (* 1/3 (+ (* 2 (log a)) (* 2 (log x)))))))) (* 1/12 (* (pow a 2) (* (pow x 2) (exp (* 1/3 (+ (* 2 (log a)) (* 2 (log x))))))))))
234.0ms
(+ (* 1/6 (* a (* x (exp (* 1/3 (+ (log a) (log x))))))) (+ (exp (* 1/3 (+ (log a) (log x)))) (* 1/36 (* (pow a 2) (* (pow x 2) (exp (* 1/3 (+ (log a) (log x)))))))))
210.0ms
(+ (* 1/6 (* a (* x (exp (* 1/3 (+ (log a) (log x))))))) (+ (exp (* 1/3 (+ (log a) (log x)))) (* 1/36 (* (pow a 2) (* (pow x 2) (exp (* 1/3 (+ (log a) (log x)))))))))

prune761.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize27.0ms

Local error

Found 4 expressions with local error:

10.1b
(cbrt (expm1 (* a x)))
10.1b
(cbrt (expm1 (* a x)))
10.1b
(cbrt (expm1 (* a x)))
10.1b
(cbrt (expm1 (* a x)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

4.0ms
(cbrt (expm1 (* a x)))
1.0ms
(cbrt (expm1 (* a x)))
1.0ms
(cbrt (expm1 (* a x)))

series412.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

121.0ms
(cbrt (expm1 (* a x)))
116.0ms
(cbrt (expm1 (* a x)))
88.0ms
(cbrt (expm1 (* a x)))
87.0ms
(cbrt (expm1 (* a x)))

simplify1.0s

Counts
24 → 64
Calls

24 calls. Slowest were:

290.0ms
(+ (* 1/6 (* a (* x (exp (* 1/3 (+ (log a) (log x))))))) (+ (exp (* 1/3 (+ (log a) (log x)))) (* 1/36 (* (pow a 2) (* (pow x 2) (exp (* 1/3 (+ (log a) (log x)))))))))
244.0ms
(+ (* 1/6 (* a (* x (exp (* 1/3 (+ (log a) (log x))))))) (+ (exp (* 1/3 (+ (log a) (log x)))) (* 1/36 (* (pow a 2) (* (pow x 2) (exp (* 1/3 (+ (log a) (log x)))))))))
208.0ms
(+ (* 1/6 (* a (* x (exp (* 1/3 (+ (log a) (log x))))))) (+ (exp (* 1/3 (+ (log a) (log x)))) (* 1/36 (* (pow a 2) (* (pow x 2) (exp (* 1/3 (+ (log a) (log x)))))))))

prune604.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize30.0ms

Local error

Found 4 expressions with local error:

10.1b
(cbrt (expm1 (* a x)))
10.1b
(cbrt (expm1 (* a x)))
10.1b
(cbrt (expm1 (* a x)))
10.1b
(cbrt (expm1 (* a x)))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

1.0ms
(cbrt (expm1 (* a x)))
1.0ms
(cbrt (expm1 (* a x)))
1.0ms
(cbrt (expm1 (* a x)))

series353.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

98.0ms
(cbrt (expm1 (* a x)))
86.0ms
(cbrt (expm1 (* a x)))
85.0ms
(cbrt (expm1 (* a x)))
84.0ms
(cbrt (expm1 (* a x)))

simplify1.2s

Counts
24 → 64
Calls

24 calls. Slowest were:

298.0ms
(+ (* 1/6 (* a (* x (exp (* 1/3 (+ (log a) (log x))))))) (+ (exp (* 1/3 (+ (log a) (log x)))) (* 1/36 (* (pow a 2) (* (pow x 2) (exp (* 1/3 (+ (log a) (log x)))))))))
290.0ms
(+ (* 1/6 (* a (* x (exp (* 1/3 (+ (log a) (log x))))))) (+ (exp (* 1/3 (+ (log a) (log x)))) (* 1/36 (* (pow a 2) (* (pow x 2) (exp (* 1/3 (+ (log a) (log x)))))))))
278.0ms
(+ (* 1/6 (* a (* x (exp (* 1/3 (+ (log a) (log x))))))) (+ (exp (* 1/3 (+ (log a) (log x)))) (* 1/36 (* (pow a 2) (* (pow x 2) (exp (* 1/3 (+ (log a) (log x)))))))))

prune803.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

regimes117.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample2.8s

Algorithm
intervals