Average Error: 29.4 → 0.0
Time: 7.6s
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

Original29.4
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 29.4

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

  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 2019005 +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: 7.1s)Debug log

sample69.0ms

Algorithm
intervals

simplify7.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune30.0ms

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))

series43.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

43.0ms
(expm1 (* a x))

simplify213.0ms

Counts
4 → 14
Calls

4 calls. Slowest were:

197.0ms
(+ (* 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)

prune122.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 4 expressions with local error:

18.0b
(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))

rewrite36.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

20.0ms
(* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x)))
7.0ms
(cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))
6.0ms
(* (expm1 (* a x)) (expm1 (* a x)))

series171.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

57.0ms
(* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x)))
45.0ms
(* (expm1 (* a x)) (expm1 (* a x)))
42.0ms
(cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))
26.0ms
(expm1 (* a x))

simplify1.1s

Counts
46 → 84
Calls

46 calls. Slowest were:

264.0ms
(+ (* 7/12 (* (pow a 4) (pow x 4))) (+ (* (pow a 3) (pow x 3)) (* (pow a 2) (pow x 2))))
198.0ms
(+ (* 3/2 (* (pow a 4) (pow x 4))) (+ (* 5/4 (* (pow a 5) (pow x 5))) (* (pow a 3) (pow x 3))))
180.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))

prune743.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize24.0ms

Local error

Found 4 expressions with local error:

18.0b
(cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))
18.0b
(cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))
18.0b
(cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))
9.0b
(cbrt (cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x)))))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 50
Calls

4 calls. Slowest were:

11.0ms
(cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))
7.0ms
(cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))
7.0ms
(cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))

series226.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

122.0ms
(cbrt (cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x)))))
39.0ms
(cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))
35.0ms
(cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))
30.0ms
(cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))

simplify933.0ms

Counts
22 → 62
Calls

22 calls. Slowest were:

289.0ms
(+ (* 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)))))))))
213.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
177.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))

prune732.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes101.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample2.5s

Algorithm
intervals