Average Error: 29.7 → 0.0
Time: 9.4s
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

Original29.7
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 29.7

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

  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 2019016 +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: 8.6s)Debug log

sample102.0ms

Algorithm
intervals
Results
359×(pre true 80)
189×(body real 80)
85×(body real 1280)
48×(body real 640)
23×(body real 320)
14×(body real 160)

simplify7.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
7.0ms
(- (exp (* a x)) 1)

prune21.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize7.0ms

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
0.0ms
(expm1 (* a x))

series32.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
32.0ms
(expm1 (* a x))

simplify172.0ms

Counts
4 → 14
Calls
4 calls:
Slowest
162.0ms
(+ (* 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)

prune93.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 4 expressions with local error:

18.6b
(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
Rules
23×pow1
11×add-exp-log
11×add-cbrt-cube
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×pow-prod-up
6×associate-*r*
6×pow-plus
5×log1p-expm1-u
4×add-log-exp
4×cbrt-unprod
4×prod-exp
4×pow-prod-down
4×associate-*l*
4×expm1-log1p-u
3×pow2
2×*-commutative
2×pow3
1×expm1-udef
1×rem-cbrt-cube
1×pow1/3
1×expm1-log1p
1×cbrt-prod
Counts
4 → 72
Calls
4 calls:
Slowest
17.0ms
(* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x)))
10.0ms
(cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))
7.0ms
(* (expm1 (* a x)) (expm1 (* a x)))
0.0ms
(expm1 (* a x))

series131.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
47.0ms
(* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x)))
30.0ms
(expm1 (* a x))
28.0ms
(* (expm1 (* a x)) (expm1 (* a x)))
26.0ms
(cbrt (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))

simplify971.0ms

Counts
46 → 84
Calls
46 calls:
Slowest
204.0ms
(+ (* 7/12 (* (pow a 4) (pow x 4))) (+ (* (pow a 3) (pow x 3)) (* (pow a 2) (pow x 2))))
188.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
174.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
132.0ms
(+ (* 3/2 (* (pow a 4) (pow x 4))) (+ (* 5/4 (* (pow a 5) (pow x 5))) (* (pow a 3) (pow x 3))))
66.0ms
(* (* (* (* (expm1 (* a x)) (expm1 (* a x))) (* (expm1 (* a x)) (expm1 (* a x)))) (* (expm1 (* a x)) (expm1 (* a x)))) (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))

prune717.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize17.0ms

Local error

Found 4 expressions with local error:

18.6b
(cbrt (expm1 (log1p (* (* (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
(log1p (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))

rewrite31.0ms

Algorithm
rewrite-expression-head
Rules
23×pow1
11×add-exp-log
11×add-cbrt-cube
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×pow-prod-up
6×associate-*r*
6×pow-plus
5×expm1-log1p-u
4×add-log-exp
4×log1p-expm1-u
4×cbrt-unprod
4×prod-exp
4×pow-prod-down
4×associate-*l*
3×pow2
3×cbrt-prod
2×*-commutative
1×log1p-expm1
1×pow3
1×pow1/3
1×log1p-udef
Counts
4 → 73
Calls
4 calls:
Slowest
20.0ms
(* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x)))
8.0ms
(* (expm1 (* a x)) (expm1 (* a x)))
1.0ms
(cbrt (expm1 (log1p (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))))
1.0ms
(log1p (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))

series508.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
355.0ms
(cbrt (expm1 (log1p (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))))
65.0ms
(log1p (* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x))))
46.0ms
(* (* (expm1 (* a x)) (expm1 (* a x))) (expm1 (* a x)))
42.0ms
(* (expm1 (* a x)) (expm1 (* a x)))

simplify2.4s

Counts
47 → 85
Calls
47 calls:
Slowest
470.0ms
(+ (* 1/6 (* (pow a 2) (* (pow x 2) (exp (* 1/3 (+ (* 3 (log a)) (* 3 (log x)))))))) (+ (* 1/2 (* a (* x (exp (* 1/3 (+ (* 3 (log a)) (* 3 (log x)))))))) (exp (* 1/3 (+ (* 3 (log a)) (* 3 (log x)))))))
328.0ms
(log (- (+ (pow (exp (* a x)) 3) (* 3 (exp (* a x)))) (* 3 (pow (exp (* a x)) 2))))
290.0ms
(log (- (+ (pow (exp (* a x)) 3) (* 3 (exp (* a x)))) (* 3 (pow (exp (* a x)) 2))))
256.0ms
(pow (- (+ (pow (exp (* a x)) 3) (* 3 (exp (* a x)))) (+ (* 3 (pow (exp (* a x)) 2)) 1)) 1/3)
255.0ms
(+ (* 7/12 (* (pow a 4) (pow x 4))) (+ (* (pow a 3) (pow x 3)) (* (pow a 2) (pow x 2))))

prune890.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes131.0ms

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
Results
10737×(pre true 80)
5528×(body real 80)
2772×(body real 1280)
1284×(body real 640)
748×(body real 320)
405×(body real 160)