Average Error: 29.0 → 0.8
Time: 18.2s
Precision: 64
Internal Precision: 128
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -194019.89548198052:\\ \;\;\;\;\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} + -1}{\sqrt[3]{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)} \cdot \left(\sqrt[3]{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)} \cdot \sqrt[3]{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(a \cdot \left(x \cdot \frac{1}{6}\right) + \frac{1}{2}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

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
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -194019.89548198052

    1. Initial program 0

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

    3. Applied flip--0

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]
    4. Using strategy rm

    5. Applied flip--0

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

    6. Applied associate-/l/0

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

    7. Simplified0

      \[\leadsto \frac{\color{blue}{e^{\left(x + x\right) \cdot \left(a + a\right)} + -1}}{\left(e^{a \cdot x} + 1\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1 \cdot 1\right)}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt0

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

    if -194019.89548198052 < (* a x)

    1. Initial program 43.0

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

    2. Taylor expanded around 0 13.9

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

    3. Simplified1.1

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -194019.89548198052:\\ \;\;\;\;\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} + -1}{\sqrt[3]{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)} \cdot \left(\sqrt[3]{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)} \cdot \sqrt[3]{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(a \cdot \left(x \cdot \frac{1}{6}\right) + \frac{1}{2}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019018 
(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: 17.7s)Debug log

sample72.0ms

Algorithm
intervals
Results
35.0ms91×body1280valid
11.0ms39×body640valid
8.0ms155×body80valid
4.0ms23×body320valid
2.0ms19×body160valid

simplify5.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
5.0ms
(- (exp (* a x)) 1)

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 30.8b

localize8.0ms

Local error

Found 2 expressions with local error:

4.9b
(- (exp (* a x)) 1)
0.0b
(exp (* a x))

rewrite5.0ms

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
4.0ms
(- (exp (* a x)) 1)
1.0ms
(exp (* a x))

series73.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
38.0ms
(- (exp (* a x)) 1)
34.0ms
(exp (* a x))

simplify477.0ms

Counts
9 → 27
Calls
9 calls:
Slowest
340.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
100.0ms
(+ (* a x) (+ (* 1/2 (* (pow a 2) (pow x 2))) 1))
10.0ms
(- (exp (* a x)) 1)
9.0ms
(- (exp (* a x)) 1)
6.0ms
(+ (sqrt (exp (* a x))) 1)

prune207.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.1b

localize19.0ms

Local error

Found 4 expressions with local error:

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

rewrite32.0ms

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
22.0ms
(* (* (* a x) (* a x)) (+ (* a (* x 1/6)) 1/2))
6.0ms
(* (* a x) (* a x))
2.0ms
(* a (* x 1/6))
0.0ms
(* x 1/6)

series105.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
53.0ms
(* (* (* a x) (* a x)) (+ (* a (* x 1/6)) 1/2))
18.0ms
(* a (* x 1/6))
17.0ms
(* (* a x) (* a x))
16.0ms
(* x 1/6)

simplify3.5s

Counts
32 → 67
Calls
32 calls:
Slowest
684.0ms
(* (* (* (* a x) (* a x)) (* a x)) (* (* (* a x) (* a x)) (* a x)))
586.0ms
(* (* (* a x) (* a x)) (+ (* a (* x 1/6)) 1/2))
323.0ms
(* (* (* (* (* 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)))
316.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (* 1/6 (* (pow a 3) (pow x 3))))
299.0ms
(+ (* 1/6 (* (pow a 3) (pow x 3))) (* 1/2 (* (pow a 2) (pow x 2))))

prune667.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.1b

localize15.0ms

Local error

Found 4 expressions with local error:

4.9b
(- (* (exp (* a x)) (exp (* a x))) (* 1 1))
0.0b
(* (exp (* a x)) (exp (* a x)))
0.0b
(/ (- (* (exp (* a x)) (exp (* a x))) (* 1 1)) (+ (exp (* a x)) 1))
0.0b
(exp (* a x))

rewrite51.0ms

Algorithm
rewrite-expression-head
Rules
15×add-cube-cbrt
15×*-un-lft-identity
15×add-sqr-sqrt
12×times-frac
9×pow1
8×add-cbrt-cube
7×add-log-exp
6×add-exp-log
5×difference-of-squares
5×exp-prod
4×associate-/l*
3×associate-*l*
3×associate-*r*
3×associate-/r*
2×associate-/l/
2×flip--
2×pow-prod-up
2×associate-/r/
2×pow-prod-down
2×flip3--
1×rem-exp-log
1×div-inv
1×flip-+
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×div-exp
1×div-sub
1×diff-log
1×pow-plus
1×flip3-+
1×frac-2neg
1×sub-neg
1×clear-num
1×cbrt-undiv
1×pow2
Counts
4 → 80
Calls
4 calls:
Slowest
28.0ms
(/ (- (* (exp (* a x)) (exp (* a x))) (* 1 1)) (+ (exp (* a x)) 1))
10.0ms
(* (exp (* a x)) (exp (* a x)))
10.0ms
(- (* (exp (* a x)) (exp (* a x))) (* 1 1))
1.0ms
(exp (* a x))

series149.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
58.0ms
(/ (- (* (exp (* a x)) (exp (* a x))) (* 1 1)) (+ (exp (* a x)) 1))
48.0ms
(- (* (exp (* a x)) (exp (* a x))) (* 1 1))
25.0ms
(* (exp (* a x)) (exp (* a x)))
17.0ms
(exp (* a x))

simplify3.2s

Counts
68 → 92
Calls
68 calls:
Slowest
356.0ms
(/ (* (* (- (* (exp (* a x)) (exp (* a x))) (* 1 1)) (- (* (exp (* a x)) (exp (* a x))) (* 1 1))) (- (* (exp (* a x)) (exp (* a x))) (* 1 1))) (* (* (+ (exp (* a x)) 1) (+ (exp (* a x)) 1)) (+ (exp (* a x)) 1)))
335.0ms
(+ (* 2 (* (pow a 2) (pow x 2))) (+ (* 2 (* a x)) (* 4/3 (* (pow a 3) (pow x 3)))))
331.0ms
(+ (* 1/2 (* (pow a 2) (pow x 2))) (+ (* a x) (* 1/6 (* (pow a 3) (pow x 3)))))
193.0ms
(* (* (* (exp (* a x)) (exp (* a x))) (exp (* a x))) (* (* (exp (* a x)) (exp (* a x))) (exp (* a x))))
145.0ms
(+ (* 2 (* a x)) (+ (* 2 (* (pow a 2) (pow x 2))) 1))

prune860.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.1b

localize32.0ms

Local error

Found 4 expressions with local error:

5.0b
(+ (exp (* (+ x x) (+ a a))) -1)
0.0b
(* (exp (* a x)) (exp (* a x)))
0.0b
(exp (* (+ x x) (+ a a)))
0.0b
(* (+ (exp (* a x)) 1) (+ (* (exp (* a x)) (exp (* a x))) (* 1 1)))

rewrite50.0ms

Algorithm
rewrite-expression-head
Rules
11×pow1
9×add-sqr-sqrt
8×add-cube-cbrt
8×add-cbrt-cube
8×*-un-lft-identity
7×flip-+
7×flip3-+
6×add-log-exp
6×add-exp-log
6×associate-*l*
6×associate-*r*
5×exp-prod
4×exp-sum
4×frac-times
3×pow-prod-down
3×distribute-lft-in
3×difference-of-sqr--1
3×distribute-rgt-in
2×cbrt-unprod
2×pow-prod-up
2×*-commutative
2×associate-*r/
2×prod-exp
2×associate-*l/
1×rem-exp-log
1×exp-to-pow
1×pow-plus
1×+-commutative
1×pow2
Counts
4 → 74
Calls
4 calls:
Slowest
31.0ms
(* (+ (exp (* a x)) 1) (+ (* (exp (* a x)) (exp (* a x))) (* 1 1)))
6.0ms
(exp (* (+ x x) (+ a a)))
6.0ms
(* (exp (* a x)) (exp (* a x)))
5.0ms
(+ (exp (* (+ x x) (+ a a))) -1)

series120.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
55.0ms
(* (+ (exp (* a x)) 1) (+ (* (exp (* a x)) (exp (* a x))) (* 1 1)))
36.0ms
(+ (exp (* (+ x x) (+ a a))) -1)
16.0ms
(* (exp (* a x)) (exp (* a x)))
13.0ms
(exp (* (+ x x) (+ a a)))

simplify4.7s

Counts
56 → 86
Calls
56 calls:
Slowest
480.0ms
(* (+ (* (exp (* a x)) (exp (* a x))) (- (* 1 1) (* (exp (* a x)) 1))) (- (* (exp (* a x)) (exp (* a x))) (* 1 1)))
348.0ms
(* (- (exp (* a x)) 1) (- (* (exp (* a x)) (exp (* a x))) (* 1 1)))
325.0ms
(+ (* 8 (* (pow a 2) (pow x 2))) (+ (* 4 (* a x)) (* 32/3 (* (pow a 3) (pow x 3)))))
275.0ms
(* (+ (pow (exp (* a x)) 3) (pow 1 3)) (- (* (* (exp (* a x)) (exp (* a x))) (* (exp (* a x)) (exp (* a x)))) (* (* 1 1) (* 1 1))))
273.0ms
(* (+ (pow (exp (* a x)) 3) (pow 1 3)) (+ (* (exp (* a x)) (exp (* a x))) (* 1 1)))

prune1.0s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.1b

regimes207.0ms

Accuracy

96.6% (0.7b remaining)

Error of 0.8b against oracle of 0.0b and baseline of 20.9b

bsearch9.0ms

end0.0ms

sample2.2s

Algorithm
intervals
Results
1.3s2789×body1280valid
340.0ms1245×body640valid
237.0ms5365×body80valid
115.0ms765×body320valid
38.0ms397×body160valid