Average Error: 29.7 → 0.5
Time: 9.7s
Precision: 64
Internal Precision: 128
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -8.353482356037643:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + a \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + a \cdot x\\ \end{array}\]

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
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -8.353482356037643

    1. Initial program 0

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

    if -8.353482356037643 < (* a x)

    1. Initial program 44.2

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

    2. Taylor expanded around 0 13.5

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(a \cdot x + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)}\]

    3. Simplified0.7

      \[\leadsto \color{blue}{\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \left(x \cdot \frac{1}{6}\right) + \frac{1}{2}\right) + a \cdot x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -8.353482356037643:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + a \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 
(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: 9.3s)Debug log

sample92.0ms

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)

simplify21.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
21.0ms
(- (exp (* a x)) 1)

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 29.5b

localize15.0ms

Local error

Found 2 expressions with local error:

3.6b
(- (exp (* a x)) 1)
0.0b
(exp (* a x))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
3×add-log-exp
3×add-sqr-sqrt
2×add-cube-cbrt
2×add-exp-log
2×add-cbrt-cube
2×*-un-lft-identity
2×pow1
1×rem-exp-log
1×flip--
1×difference-of-sqr-1
1×flip3--
1×sub-neg
1×exp-prod
Counts
2 → 21
Calls
2 calls:
Slowest
7.0ms
(- (exp (* a x)) 1)
1.0ms
(exp (* a x))

series39.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
28.0ms
(- (exp (* a x)) 1)
10.0ms
(exp (* a x))

simplify462.0ms

Counts
9 → 27
Calls
9 calls:
Slowest
315.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
117.0ms
(+ (* a x) (+ (* 1/2 (* (pow a 2) (pow x 2))) 1))
9.0ms
(- (exp (* a x)) 1)
8.0ms
(- (exp (* a x)) 1)
3.0ms
(exp (* a x))

prune204.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 4 expressions with local error:

0.4b
(* x 1/6)
0.3b
(* (* a x) (* a x))
0.2b
(* a (* x 1/6))
0.1b
(* (* (* a x) (* a x)) (+ (* a (* x 1/6)) 1/2))

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
14×pow1
11×add-exp-log
11×add-cbrt-cube
5×add-cube-cbrt
5×associate-*r*
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-log-exp
4×cbrt-unprod
4×*-commutative
4×prod-exp
4×pow-prod-down
2×associate-*r/
2×associate-*l*
1×flip-+
1×pow-prod-up
1×pow-plus
1×flip3-+
1×distribute-lft-in
1×distribute-rgt-in
1×pow2
Counts
4 → 55
Calls
4 calls:
Slowest
11.0ms
(* (* (* a x) (* a x)) (+ (* a (* x 1/6)) 1/2))
3.0ms
(* (* a x) (* a x))
1.0ms
(* a (* x 1/6))
0.0ms
(* x 1/6)

series107.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
64.0ms
(* (* (* a x) (* a x)) (+ (* a (* x 1/6)) 1/2))
19.0ms
(* (* a x) (* a x))
17.0ms
(* a (* x 1/6))
7.0ms
(* x 1/6)

simplify2.8s

Counts
32 → 67
Calls
32 calls:
Slowest
471.0ms
(* (* (* (* a x) (* a x)) (* a x)) (* (* (* a x) (* a x)) (* a x)))
288.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (* 1/6 (* (pow a 3) (pow x 3))))
284.0ms
(+ (* 1/6 (* (pow a 3) (pow x 3))) (* 1/2 (* (pow a 2) (pow x 2))))
264.0ms
(+ (* 1/6 (* (pow a 3) (pow x 3))) (* 1/2 (* (pow a 2) (pow x 2))))
217.0ms
(* (* (* (* (* a x) (* a x)) (* a x)) (* (* (* a x) (* a x)) (* a x))) (* (* (+ (* a (* x 1/6)) 1/2) (+ (* a (* x 1/6)) 1/2)) (+ (* a (* x 1/6)) 1/2)))

prune592.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 4 expressions with local error:

3.6b
(- (exp (* a x)) 1)
3.6b
(- (exp (* a x)) 1)
0.0b
(exp (* a x))
0.0b
(exp (* a x))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
6×add-log-exp
6×add-sqr-sqrt
4×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
2×rem-exp-log
2×flip--
2×difference-of-sqr-1
2×flip3--
2×sub-neg
2×exp-prod
Counts
4 → 42
Calls
4 calls:
Slowest
6.0ms
(- (exp (* a x)) 1)
4.0ms
(- (exp (* a x)) 1)
2.0ms
(exp (* a x))
2.0ms
(exp (* a x))

series96.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
35.0ms
(- (exp (* a x)) 1)
25.0ms
(- (exp (* a x)) 1)
18.0ms
(exp (* a x))
18.0ms
(exp (* a x))

simplify1.1s

Counts
18 → 54
Calls
18 calls:
Slowest
418.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
347.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
132.0ms
(+ (* a x) (+ (* 1/2 (* (pow a 2) (pow x 2))) 1))
122.0ms
(+ (* a x) (+ (* 1/2 (* (pow a 2) (pow x 2))) 1))
10.0ms
(- (exp (* a x)) 1)

prune587.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0.0b

regimes319.0ms

Accuracy

98% (0.4b remaining)

Error of 0.5b against oracle of 0.1b and baseline of 20.7b

bsearch4.0ms

end0.0ms

sample2.8s

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)