Original	42.5
Target	42.3
Herbie	18.4

Derivation

Split input into 3 regimes
if i < -1164558.298979503
1. Initial program 26.3
  \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
2. Taylor expanded around inf 62.9
  \[\leadsto \color{blue}{100 \cdot \frac{\left(e^{\left(\log \left(\frac{1}{n}\right) - \log \left(\frac{1}{i}\right)\right) \cdot n} - 1\right) \cdot n}{i}}\]
3. Simplified17.6
  \[\leadsto \color{blue}{\frac{100}{\frac{i}{n}} \cdot \left({\left(\frac{i}{n}\right)}^{n} + -1\right)}\]
if -1164558.298979503 < i < 5.683451278453723e-38
1. Initial program 50.0
  \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
2. Using strategy rm
3. Applied div-inv50.0
  \[\leadsto 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\color{blue}{i \cdot \frac{1}{n}}}\]
4. Applied *-un-lft-identity50.0
  \[\leadsto 100 \cdot \frac{\color{blue}{1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{i \cdot \frac{1}{n}}\]
5. Applied times-frac50.3
  \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\right)}\]
6. Applied associate-*r*50.4
  \[\leadsto \color{blue}{\left(100 \cdot \frac{1}{i}\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}}\]
7. Simplified50.4
  \[\leadsto \left(100 \cdot \frac{1}{i}\right) \cdot \color{blue}{\left(n \cdot {\left(\frac{i}{n} + 1\right)}^{n} - n\right)}\]
8. Taylor expanded around -inf 49.8
  \[\leadsto \left(100 \cdot \frac{1}{i}\right) \cdot \color{blue}{\left(-1 \cdot \left(\left(1 - e^{i}\right) \cdot n\right)\right)}\]
9. Simplified49.8
  \[\leadsto \left(100 \cdot \frac{1}{i}\right) \cdot \color{blue}{\left(e^{i} \cdot n - n\right)}\]
10. Taylor expanded around 0 16.7
  \[\leadsto \color{blue}{\frac{50}{3} \cdot \left({i}^{2} \cdot n\right) + \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)}\]
11. Simplified16.7
  \[\leadsto \color{blue}{\left(i \cdot n\right) \cdot \left(50 + \frac{50}{3} \cdot i\right) + 100 \cdot n}\]
if 5.683451278453723e-38 < i
1. Initial program 36.7
  \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
2. Using strategy rm
3. Applied div-inv36.7
  \[\leadsto 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\color{blue}{i \cdot \frac{1}{n}}}\]
4. Applied *-un-lft-identity36.7
  \[\leadsto 100 \cdot \frac{\color{blue}{1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{i \cdot \frac{1}{n}}\]
5. Applied times-frac36.7
  \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\right)}\]
6. Applied associate-*r*36.7
  \[\leadsto \color{blue}{\left(100 \cdot \frac{1}{i}\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}}\]
7. Simplified36.7
  \[\leadsto \left(100 \cdot \frac{1}{i}\right) \cdot \color{blue}{\left(n \cdot {\left(\frac{i}{n} + 1\right)}^{n} - n\right)}\]
8. Taylor expanded around 0 25.6
  \[\leadsto \left(100 \cdot \frac{1}{i}\right) \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot \left({n}^{4} \cdot \left(\log i \cdot {\left(\log n\right)}^{2}\right)\right) + \left(\frac{1}{2} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right) + \left(\frac{1}{2} \cdot \left({n}^{3} \cdot {\left(\log i\right)}^{2}\right) + \left(\frac{1}{3} \cdot \left({n}^{4} \cdot \left({\left(\log n\right)}^{2} \cdot \log i\right)\right) + \left(\frac{1}{6} \cdot \left({n}^{4} \cdot {\left(\log i\right)}^{3}\right) + {n}^{2} \cdot \log i\right)\right)\right)\right)\right) - \left(\frac{1}{2} \cdot \left({n}^{3} \cdot \left(\log n \cdot \log i\right)\right) + \left(\frac{1}{2} \cdot \left({n}^{3} \cdot \left(\log i \cdot \log n\right)\right) + \left({n}^{2} \cdot \log n + \left(\frac{1}{6} \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{3}\right) + \left(\frac{1}{6} \cdot \left({n}^{4} \cdot \left(\log n \cdot {\left(\log i\right)}^{2}\right)\right) + \frac{1}{3} \cdot \left({n}^{4} \cdot \left({\left(\log i\right)}^{2} \cdot \log n\right)\right)\right)\right)\right)\right)\right)\right)}\]
9. Simplified25.6
  \[\leadsto \left(100 \cdot \frac{1}{i}\right) \cdot \color{blue}{\left(\left(\left(\left(\frac{1}{6} \cdot \left({n}^{4} \cdot \log i\right)\right) \cdot \left(\log n \cdot \log n\right) + \left(\left(\frac{1}{2} \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \log n + \log i \cdot \log i\right)\right) + \left(\left(\left(n \cdot n\right) \cdot \log i + \left({n}^{4} \cdot \frac{1}{6}\right) \cdot {\left(\log i\right)}^{3}\right) + \left(\left(\log n \cdot \log n\right) \cdot \log i\right) \cdot \left(\frac{1}{3} \cdot {n}^{4}\right)\right)\right) - \left(\left(\left(\left(\frac{1}{2} \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \log n + \log i \cdot \log n\right) + \left(\left(n \cdot n\right) \cdot \log n + {\left(\log n\right)}^{3} \cdot \left({n}^{4} \cdot \frac{1}{6}\right)\right)\right) + \left(\left(\frac{1}{3} \cdot \left({n}^{4} \cdot \log n\right)\right) \cdot \left(\log i \cdot \log i\right) + \left(\left({n}^{4} \cdot \frac{1}{6}\right) \cdot \log n\right) \cdot \left(\log i \cdot \log i\right)\right)\right)\right)}\]
Recombined 3 regimes into one program.
Final simplification18.4
\[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1164558.298979503:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 5.683451278453723 \cdot 10^{-38}:\\ \;\;\;\;100 \cdot n + \left(\frac{50}{3} \cdot i + 50\right) \cdot \left(n \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\log i \cdot \log i + \log n \cdot \log n\right) \cdot \left(\left(n \cdot n\right) \cdot \left(\frac{1}{2} \cdot n\right)\right) + \left(\left(\log i \cdot {n}^{4}\right) \cdot \frac{1}{6}\right) \cdot \left(\log n \cdot \log n\right)\right) + \left(\left({\left(\log i\right)}^{3} \cdot \left(\frac{1}{6} \cdot {n}^{4}\right) + \log i \cdot \left(n \cdot n\right)\right) + \left(\frac{1}{3} \cdot {n}^{4}\right) \cdot \left(\left(\log n \cdot \log n\right) \cdot \log i\right)\right)\right) - \left(\left(\left(\log i \cdot \log i\right) \cdot \left(\left(\frac{1}{6} \cdot {n}^{4}\right) \cdot \log n\right) + \left(\left({n}^{4} \cdot \log n\right) \cdot \frac{1}{3}\right) \cdot \left(\log i \cdot \log i\right)\right) + \left(\left(\log n \cdot \left(n \cdot n\right) + {\left(\log n\right)}^{3} \cdot \left(\frac{1}{6} \cdot {n}^{4}\right)\right) + \left(\left(n \cdot n\right) \cdot \left(\frac{1}{2} \cdot n\right)\right) \cdot \left(\log n \cdot \log i + \log n \cdot \log i\right)\right)\right)\right) \cdot \left(100 \cdot \frac{1}{i}\right)\\ \end{array}\]

Details

sample275.0ms

Algorithm

intervals

Results

106.0ms	36×	body	2560	valid
89.0ms	73×	body	1280	valid
33.0ms	40×	body	640	valid
14.0ms	119×	body	80	valid
9.0ms	87×	body	80	nan
7.0ms	18×	body	320	valid
3.0ms	12×	body	160	valid

simplify182.0ms

Counts

1 → 1

Calls

1 calls:

Slowest

182.0ms

(* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 41.5b

localize25.0ms

Local error

Found 4 expressions with local error:

12.2b	(/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))
9.3b	(pow (+ 1 (/ i n)) n)
1.6b	(- (pow (+ 1 (/ i n)) n) 1)
0.1b	(* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)))

rewrite51.0ms

Algorithm

rewrite-expression-head

Rules

46×	`add-sqr-sqrt`
40×	`times-frac`
26×	`add-cube-cbrt`
26×	`*-un-lft-identity`
25×	`associate-r`
20×	`difference-of-sqr-1`
13×	`div-inv`
13×	`unpow-prod-down`
7×	`add-exp-log`
6×	`add-cbrt-cube`
5×	`associate-/l*`
5×	`pow1`
4×	`add-log-exp`
4×	`associate-/r*`
2×	`associate-/l/`
2×	`flip--`
2×	`associate-/r/`
2×	`flip3--`
1×	`*-commutative`
1×	`associate-*r/`
1×	`pow-exp`
1×	`div-exp`
1×	`div-sub`
1×	`frac-2neg`
1×	`pow-pow`
1×	`sub-neg`
1×	`clear-num`
1×	`cbrt-undiv`
1×	`pow-to-exp`

Counts

4 → 105

Calls

4 calls:

Slowest

24.0ms	(/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))
12.0ms	(* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)))
11.0ms	(- (pow (+ 1 (/ i n)) n) 1)
2.0ms	(pow (+ 1 (/ i n)) n)

series616.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

193.0ms	(/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))
179.0ms	(* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)))
126.0ms	(- (pow (+ 1 (/ i n)) n) 1)
117.0ms	(pow (+ 1 (/ i n)) n)

simplify2.8s

Counts

97 → 117

Calls

97 calls:

Slowest

257.0ms	(/ (* (* (- (pow (+ 1 (/ i n)) n) 1) (- (pow (+ 1 (/ i n)) n) 1)) (- (pow (+ 1 (/ i n)) n) 1)) (* (* (/ i n) (/ i n)) (/ i n)))
240.0ms	(* 100 (/ (* n (- (exp (* n (- (log (/ -1 n)) (log (/ -1 i))))) 1)) i))
229.0ms	(/ (* (- (exp (* (- (log (/ 1 n)) (log (/ 1 i))) n)) 1) n) i)
216.0ms	(* 100 (/ (* (- (exp (* (- (log (/ 1 n)) (log (/ 1 i))) n)) 1) n) i))
170.0ms	(- (exp (* (- (log (/ 1 n)) (log (/ 1 i))) n)) 1)

prune941.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 19.1b

localize14.0ms

Local error

Found 4 expressions with local error:

12.2b	(/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))
12.2b	(/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))
12.2b	(/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))
9.3b	(pow (+ 1 (/ i n)) n)

rewrite353.0ms

Algorithm

rewrite-expression-head

Rules

68×	`add-sqr-sqrt`
60×	`times-frac`
38×	`add-cube-cbrt`
38×	`*-un-lft-identity`
30×	`difference-of-sqr-1`
21×	`div-inv`
18×	`unpow-prod-down`
15×	`associate-/l*`
12×	`associate-/r*`
11×	`add-exp-log`
10×	`add-cbrt-cube`
6×	`associate-/l/`
5×	`pow1`
4×	`add-log-exp`
3×	`flip--`
3×	`associate-/r/`
3×	`div-exp`
3×	`div-sub`
3×	`flip3--`
3×	`frac-2neg`
3×	`clear-num`
3×	`cbrt-undiv`
1×	`pow-exp`
1×	`pow-pow`
1×	`pow-to-exp`

Counts

4 → 148

Calls

4 calls:

Slowest

302.0ms	(/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))
24.0ms	(/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))
23.0ms	(/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))
2.0ms	(pow (+ 1 (/ i n)) n)

series616.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

197.0ms	(/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))
172.0ms	(/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))
148.0ms	(/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))
98.0ms	(pow (+ 1 (/ i n)) n)

simplify4.1s

Counts

176 → 160

Calls

176 calls:

Slowest

284.0ms	(/ (* (* (- (pow (+ 1 (/ i n)) n) 1) (- (pow (+ 1 (/ i n)) n) 1)) (- (pow (+ 1 (/ i n)) n) 1)) (* (* (/ i n) (/ i n)) (/ i n)))
277.0ms	(/ (* (- (exp (* (- (log (/ 1 n)) (log (/ 1 i))) n)) 1) n) i)
276.0ms	(/ (* (- (exp (* (- (log (/ 1 n)) (log (/ 1 i))) n)) 1) n) i)
267.0ms	(/ (* (- (exp (* (- (log (/ 1 n)) (log (/ 1 i))) n)) 1) n) i)
255.0ms	(/ (* (* (- (pow (+ 1 (/ i n)) n) 1) (- (pow (+ 1 (/ i n)) n) 1)) (- (pow (+ 1 (/ i n)) n) 1)) (* (* (/ i n) (/ i n)) (/ i n)))

prune1.9s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 19.1b

localize30.0ms

Local error

Found 4 expressions with local error:

9.3b	(pow (+ (/ i n) 1) n)
1.6b	(- (* n (pow (+ (/ i n) 1) n)) n)
1.0b	(* (* 100 (/ 1 i)) (- (* n (pow (+ (/ i n) 1) n)) n))
0.2b	(* 100 (/ 1 i))

rewrite36.0ms

Algorithm

rewrite-expression-head

Rules

7×	`add-cube-cbrt`
7×	`add-exp-log`
7×	`associate-r`
7×	`*-un-lft-identity`
7×	`pow1`
7×	`add-sqr-sqrt`
6×	`associate-*r/`
6×	`add-cbrt-cube`
4×	`add-log-exp`
4×	`flip--`
4×	`un-div-inv`
4×	`flip3--`
4×	`frac-times`
3×	`unpow-prod-down`
3×	`sub-neg`
2×	`*-commutative`
2×	`associate-*l/`
2×	`distribute-lft-in`
2×	`distribute-rgt-in`
1×	`div-inv`
1×	`cbrt-unprod`
1×	`prod-exp`
1×	`pow-exp`
1×	`pow-prod-down`
1×	`associate-l`
1×	`pow-pow`
1×	`pow-to-exp`

Counts

4 → 65

Calls

4 calls:

Slowest

22.0ms	(* (* 100 (/ 1 i)) (- (* n (pow (+ (/ i n) 1) n)) n))
7.0ms	(- (* n (pow (+ (/ i n) 1) n)) n)
5.0ms	(pow (+ (/ i n) 1) n)
1.0ms	(* 100 (/ 1 i))

series438.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

191.0ms	(- (* n (pow (+ (/ i n) 1) n)) n)
123.0ms	(* (* 100 (/ 1 i)) (- (* n (pow (+ (/ i n) 1) n)) n))
116.0ms	(pow (+ (/ i n) 1) n)
8.0ms	(* 100 (/ 1 i))

simplify4.2s

Counts

43 → 77

Calls

43 calls:

Slowest

420.0ms	(* 100 (- (* (* n (pow (+ (/ i n) 1) n)) (* n (pow (+ (/ i n) 1) n))) (* n n)))
334.0ms	(* (* 100 1) (- (* (* n (pow (+ (/ i n) 1) n)) (* n (pow (+ (/ i n) 1) n))) (* n n)))
323.0ms	(* 100 (- (pow (* n (pow (+ (/ i n) 1) n)) 3) (pow n 3)))
321.0ms	(* i (+ (* (* n (pow (+ (/ i n) 1) n)) (* n (pow (+ (/ i n) 1) n))) (+ (* n n) (* (* n (pow (+ (/ i n) 1) n)) n))))
293.0ms	(* i (+ (* (* n (pow (+ (/ i n) 1) n)) (* n (pow (+ (/ i n) 1) n))) (+ (* n n) (* (* n (pow (+ (/ i n) 1) n)) n))))

prune635.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 15.4b

localize15.0ms

Local error

Found 4 expressions with local error:

2.3b	(- (* (exp i) n) n)
1.3b	(* (* 100 (/ 1 i)) (- (* (exp i) n) n))
0.2b	(* 100 (/ 1 i))
0.0b	(* (exp i) n)

rewrite29.0ms

Algorithm

rewrite-expression-head

Rules

7×	`add-cube-cbrt`
7×	`associate-r`
7×	`*-un-lft-identity`
7×	`add-sqr-sqrt`
6×	`add-exp-log`
6×	`associate-*r/`
6×	`add-cbrt-cube`
6×	`pow1`
4×	`add-log-exp`
4×	`flip--`
4×	`un-div-inv`
4×	`associate-l`
4×	`flip3--`
4×	`frac-times`
3×	`*-commutative`
3×	`sub-neg`
2×	`associate-*l/`
2×	`distribute-lft-in`
2×	`distribute-rgt-in`
1×	`div-inv`
1×	`cbrt-unprod`
1×	`prod-exp`
1×	`pow-prod-down`

Counts

4 → 63

Calls

4 calls:

Slowest

19.0ms	(* (* 100 (/ 1 i)) (- (* (exp i) n) n))
6.0ms	(- (* (exp i) n) n)
2.0ms	(* (exp i) n)
1.0ms	(* 100 (/ 1 i))

series119.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

39.0ms	(* (exp i) n)
37.0ms	(- (* (exp i) n) n)
35.0ms	(* (* 100 (/ 1 i)) (- (* (exp i) n) n))
8.0ms	(* 100 (/ 1 i))

simplify4.4s

Counts

40 → 75

Calls

40 calls:

Slowest

467.0ms	(+ (* 1/2 (* (pow i 2) n)) (+ (* 1/6 (* (pow i 3) n)) (* i n)))
407.0ms	(* (* 100 1) (- (* (* (exp i) n) (* (exp i) n)) (* n n)))
365.0ms	(* 100 (- (* (* (exp i) n) (* (exp i) n)) (* n n)))
357.0ms	(* i (+ (* (* (exp i) n) (* (exp i) n)) (+ (* n n) (* (* (exp i) n) n))))
296.0ms	(* i (+ (* (* (exp i) n) (* (exp i) n)) (+ (* n n) (* (* (exp i) n) n))))

prune573.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 3.7b

regimes170.0ms

Accuracy

50.3% (14.4b remaining)

Error of 18.4b against oracle of 3.9b and baseline of 32.9b

bsearch476.0ms

end0.0ms

sample7.3s

Algorithm

intervals

Results

2.8s	2459×	body	1280	valid
2.2s	987×	body	2560	valid
884.0ms	1636×	body	640	valid
644.0ms	3073×	body	80	valid
296.0ms	828×	body	320	valid
163.0ms	2182×	body	80	nan
86.0ms	453×	body	160	valid
7.0ms	2×	body	5120	valid

Error

Try it out

Target

Derivation

`if i < -1164558.298979503`

`if -1164558.298979503 < i < 5.683451278453723e-38`

`if 5.683451278453723e-38 < i`

Reproduce

Details

sample275.0ms

simplify182.0ms

prune5.0ms

localize25.0ms

rewrite51.0ms

series616.0ms

simplify2.8s

prune941.0ms

localize14.0ms

rewrite353.0ms

series616.0ms

simplify4.1s

prune1.9s

localize30.0ms

rewrite36.0ms

series438.0ms

simplify4.2s

prune635.0ms

localize15.0ms

rewrite29.0ms

series119.0ms

simplify4.4s

prune573.0ms

regimes170.0ms

bsearch476.0ms

end0.0ms

sample7.3s

In
Out