Average Error: 29.7 → 0.0

Time: 2.8s

Precision: 64

Internal Precision: 128

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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `a`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	29.7
Target	0.2
Herbie	0.0

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

Derivation

Initial program 29.7
\[e^{a \cdot x} - 1\]
Simplified0.0
\[\leadsto \color{blue}{(e^{a \cdot x} - 1)^*}\]
Final simplification0.0
\[\leadsto (e^{a \cdot x} - 1)^*\]

Reproduce

herbie shell --seed 2019022 +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

sample87.0ms

Algorithm

intervals

Results

39.0ms	84×	body	1280	valid
15.0ms	175×	body	80	valid
13.0ms	43×	body	640	valid
5.0ms	24×	body	320	valid
2.0ms	17×	body	160	valid

simplify6.0ms

Counts

1 → 1

Calls

1 calls:

Slowest

6.0ms

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

prune14.0ms

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

Rules

2×	`log1p-expm1-u`
1×	`add-log-exp`
1×	`expm1-udef`
1×	`add-cube-cbrt`
1×	`add-exp-log`
1×	`add-cbrt-cube`
1×	`*-un-lft-identity`
1×	`expm1-log1p`
1×	`pow1`
1×	`expm1-log1p-u`
1×	`add-sqr-sqrt`

Counts

1 → 11

Calls

1 calls:

Slowest

1.0ms

(expm1 (* a x))

series40.0ms

Counts

1 → 3

Calls

1 calls:

Slowest

40.0ms

(expm1 (* a x))

simplify178.0ms

Counts

4 → 14

Calls

4 calls:

Slowest

168.0ms	(+ (* 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)
2.0ms	(* a x)

prune114.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0.0b

end0.0ms

sample2.1s

Algorithm

intervals

Results

987.0ms	2682×	body	1280	valid
411.0ms	1330×	body	640	valid
276.0ms	5460×	body	80	valid
191.0ms	760×	body	320	valid
96.0ms	431×	body	160	valid