Average Error: 29.1 → 0.0
Time: 4.7s
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. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019010 +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: 4.4s)Debug log

sample99.0ms

Algorithm
intervals

simplify3.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
3.0ms
(- (exp (* a x)) 1)

prune7.0ms

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))

rewrite0.0ms

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
0.0ms
(expm1 (* a x))

series36.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
36.0ms
(expm1 (* a x))

simplify122.0ms

Counts
4 → 14
Calls
4 calls:
Slowest
113.0ms
(+ (* 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)
1.0ms
(* a x)

prune92.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 3 expressions with local error:

7.9b
(log (expm1 (* a x)))
1.3b
(exp (log (expm1 (* a x))))
0.0b
(expm1 (* a x))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
6×add-cube-cbrt
6×*-un-lft-identity
6×log-prod
6×add-sqr-sqrt
5×pow1
4×log1p-expm1-u
4×add-exp-log
4×exp-prod
3×add-log-exp
3×exp-sum
3×add-cbrt-cube
3×expm1-log1p-u
2×log-pow
1×rem-exp-log
1×expm1-udef
1×rem-log-exp
1×expm1-log1p
Counts
3 → 42
Calls
3 calls:
Slowest
4.0ms
(exp (log (expm1 (* a x))))
2.0ms
(log (expm1 (* a x)))
1.0ms
(expm1 (* a x))

series112.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
54.0ms
(log (expm1 (* a x)))
32.0ms
(exp (log (expm1 (* a x))))
26.0ms
(expm1 (* a x))

simplify640.0ms

Counts
22 → 51
Calls
22 calls:
Slowest
230.0ms
(+ (* 1/2 (* a x)) (+ (log a) (+ (* 1/24 (* (pow a 2) (pow x 2))) (log x))))
170.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
153.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
10.0ms
(exp (log (* (cbrt (expm1 (* a x))) (cbrt (expm1 (* a x))))))
7.0ms
(log (- (exp (* a x)) 1))

prune437.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes76.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample2.7s

Algorithm
intervals