Average Error: 29.0 → 0.3
Time: 20.2s
Precision: 64
Internal Precision: 128
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.000729121522466105:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot x\right) \cdot a + \frac{1}{2}\right) + a \cdot x\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.0
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.000729121522466105

    1. Initial program 0.0

      \[e^{a \cdot x} - 1\]
    2. Using strategy rm

    3. Applied add-log-exp0.0

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

    if -0.000729121522466105 < (* a x)

    1. Initial program 43.7

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

    2. Taylor expanded around 0 13.7

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

    3. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.000729121522466105:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot x\right) \cdot a + \frac{1}{2}\right) + a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 1042149663 
(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: 19.0s)Debug log

sample126.0ms

Algorithm
intervals
Results
342×(pre true 80)
185×(body real 80)
94×(body real 1280)
37×(body real 640)
15×(body real 320)
11×(body real 160)

simplify12.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
11.0ms
(- (exp (* a x)) 1)

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 25.2b

localize23.0ms

Local error

Found 2 expressions with local error:

2.7b
(- (exp (* a x)) 1)
0.0b
(exp (* a x))

rewrite12.0ms

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
10.0ms
(- (exp (* a x)) 1)
2.0ms
(exp (* a x))

series71.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
49.0ms
(- (exp (* a x)) 1)
21.0ms
(exp (* a x))

simplify734.0ms

Counts
9 → 27
Calls
9 calls:
Slowest
531.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
158.0ms
(+ (* a x) (+ (* 1/2 (* (pow a 2) (pow x 2))) 1))
12.0ms
(- (exp (* a x)) 1)
11.0ms
(- (exp (* a x)) 1)
7.0ms
(+ (sqrt (exp (* a x))) 1)

prune383.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.1b

localize44.0ms

Local error

Found 4 expressions with local error:

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

rewrite37.0ms

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
26.0ms
(* (* (* a x) (* a x)) (+ (* a (* x 1/6)) 1/2))
8.0ms
(* (* a x) (* a x))
2.0ms
(* a (* x 1/6))
0.0ms
(* x 1/6)

series172.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
98.0ms
(* (* (* a x) (* a x)) (+ (* a (* x 1/6)) 1/2))
37.0ms
(* (* a x) (* a x))
22.0ms
(* a (* x 1/6))
15.0ms
(* x 1/6)

simplify4.4s

Counts
32 → 67
Calls
32 calls:
Slowest
878.0ms
(* (* (* (* a x) (* a x)) (* a x)) (* (* (* a x) (* a x)) (* a x)))
415.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (* 1/6 (* (pow a 3) (pow x 3))))
405.0ms
(+ (* 1/6 (* (pow a 3) (pow x 3))) (* 1/2 (* (pow a 2) (pow x 2))))
402.0ms
(+ (* 1/6 (* (pow a 3) (pow x 3))) (* 1/2 (* (pow a 2) (pow x 2))))
361.0ms
(* (* (* (* (* 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)))

prune918.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.1b

localize24.0ms

Local error

Found 4 expressions with local error:

2.7b
(- (exp (* a x)) 1)
2.7b
(- (exp (* a x)) 1)
2.7b
(- (exp (* a x)) 1)
0.1b
(cbrt (- (exp (* a x)) 1))

rewrite37.0ms

Algorithm
rewrite-expression-head
Rules
9×add-sqr-sqrt
5×add-cube-cbrt
5×*-un-lft-identity
4×add-log-exp
4×flip--
4×add-exp-log
4×difference-of-sqr-1
4×add-cbrt-cube
4×flip3--
4×pow1
4×cbrt-prod
3×sub-neg
2×cbrt-div
1×pow1/3
Counts
4 → 50
Calls
4 calls:
Slowest
10.0ms
(- (exp (* a x)) 1)
10.0ms
(- (exp (* a x)) 1)
10.0ms
(- (exp (* a x)) 1)
6.0ms
(cbrt (- (exp (* a x)) 1))

series319.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
170.0ms
(cbrt (- (exp (* a x)) 1))
52.0ms
(- (exp (* a x)) 1)
48.0ms
(- (exp (* a x)) 1)
48.0ms
(- (exp (* a x)) 1)

simplify2.2s

Counts
21 → 62
Calls
21 calls:
Slowest
525.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
523.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
522.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
298.0ms
(+ (* 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)))))))))
77.0ms
(cbrt (- (* (exp (* a x)) (exp (* a x))) (* 1 1)))

prune1.0s

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.1b

localize51.0ms

Local error

Found 4 expressions with local error:

2.8b
(- (sqrt (exp (* a x))) 1)
2.7b
(- (exp (* a x)) 1)
2.7b
(- (exp (* a x)) 1)
0.1b
(cbrt (- (exp (* a x)) 1))

rewrite45.0ms

Algorithm
rewrite-expression-head
Rules
10×add-sqr-sqrt
5×add-cube-cbrt
5×difference-of-sqr-1
5×*-un-lft-identity
4×add-log-exp
4×flip--
4×add-exp-log
4×add-cbrt-cube
4×flip3--
4×pow1
4×cbrt-prod
3×sub-neg
2×cbrt-div
1×sqrt-prod
1×pow1/3
Counts
4 → 51
Calls
4 calls:
Slowest
14.0ms
(- (sqrt (exp (* a x))) 1)
10.0ms
(- (exp (* a x)) 1)
10.0ms
(cbrt (- (exp (* a x)) 1))
10.0ms
(- (exp (* a x)) 1)

series339.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
173.0ms
(cbrt (- (exp (* a x)) 1))
67.0ms
(- (sqrt (exp (* a x))) 1)
49.0ms
(- (exp (* a x)) 1)
49.0ms
(- (exp (* a x)) 1)

simplify2.1s

Counts
22 → 63
Calls
22 calls:
Slowest
530.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
519.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
483.0ms
(+ (* 1/8 (* (pow a 2) (pow x 2))) (+ (* 1/2 (* a x)) (* 1/48 (* (pow a 3) (pow x 3)))))
295.0ms
(+ (* 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)))))))))
77.0ms
(cbrt (- (* (exp (* a x)) (exp (* a x))) (* 1 1)))

prune1.3s

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0.1b

regimes650.0ms

Accuracy

98.7% (0.3b remaining)

Error of 0.3b against oracle of 0.1b and baseline of 21.0b

bsearch7.0ms

end0.0ms

sample4.0s

Algorithm
intervals
Results
10627×(pre true 80)
5470×(body real 80)
2705×(body real 1280)
1363×(body real 640)
708×(body real 320)
381×(body real 160)