Average Error: 29.6 → 0.0
Time: 10.4s
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

Original29.6
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 29.6

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

  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 2019002 +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: 9.9s)Debug log

sample78.0ms

Algorithm
intervals

simplify3.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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
Counts
1 → 11
Calls

1 calls. Slowest were:

0.0ms
(expm1 (* a x))

series25.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

25.0ms
(expm1 (* a x))

simplify129.0ms

Counts
4 → 14
Calls

4 calls. Slowest were:

121.0ms
(+ (* 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)

prune110.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 4 expressions with local error:

9.7b
(cbrt (expm1 (* a x)))
9.7b
(cbrt (expm1 (* a x)))
9.7b
(cbrt (expm1 (* a x)))
0.3b
(* (cbrt (expm1 (* a x))) (cbrt (expm1 (* a x))))

rewrite7.0ms

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)))

series465.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

133.0ms
(cbrt (expm1 (* a x)))
113.0ms
(cbrt (expm1 (* a x)))
110.0ms
(cbrt (expm1 (* a x)))
109.0ms
(* (cbrt (expm1 (* a x))) (cbrt (expm1 (* a x))))

simplify1.1s

Counts
41 → 81
Calls

41 calls. Slowest were:

286.0ms
(+ (* 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)))))))))
250.0ms
(+ (* 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)))))))))
225.0ms
(+ (* 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)))))))))

prune472.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 4 expressions with local error:

9.7b
(cbrt (expm1 (* a x)))
9.7b
(cbrt (expm1 (* a x)))
9.7b
(cbrt (expm1 (* a x)))
9.7b
(cbrt (expm1 (* a x)))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

1.0ms
(cbrt (expm1 (* a x)))
1.0ms
(cbrt (expm1 (* a x)))
0.0ms
(cbrt (expm1 (* a x)))

series376.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

102.0ms
(cbrt (expm1 (* a x)))
102.0ms
(cbrt (expm1 (* a x)))
90.0ms
(cbrt (expm1 (* a x)))
81.0ms
(cbrt (expm1 (* a x)))

simplify1.1s

Counts
24 → 64
Calls

24 calls. Slowest were:

300.0ms
(+ (* 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)))))))))
276.0ms
(+ (* 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)))))))))
272.0ms
(+ (* 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)))))))))

prune679.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

10.7b
(pow (expm1 (* a x)) 1/3)
9.7b
(cbrt (expm1 (* a x)))
9.7b
(cbrt (expm1 (* a x)))
9.7b
(cbrt (expm1 (* a x)))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

1.0ms
(pow (expm1 (* a x)) 1/3)
1.0ms
(cbrt (expm1 (* a x)))
1.0ms
(cbrt (expm1 (* a x)))

series424.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

123.0ms
(cbrt (expm1 (* a x)))
121.0ms
(pow (expm1 (* a x)) 1/3)
94.0ms
(cbrt (expm1 (* a x)))
86.0ms
(cbrt (expm1 (* a x)))

simplify1.6s

Counts
26 → 67
Calls

26 calls. Slowest were:

599.0ms
(+ (* 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)))))))))
327.0ms
(+ (* 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)))))))))
319.0ms
(+ (* 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)))))))))

prune885.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

regimes116.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample2.3s

Algorithm
intervals