Average Error: 42.0 → 19.0
Time: 26.7s
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -1.6699497128870854 \cdot 10^{-36}:\\ \;\;\;\;100 \cdot \frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -5.921036522656228 \cdot 10^{-174}:\\ \;\;\;\;\left(\frac{1}{i} \cdot 100\right) \cdot (\left({\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot n + \left(-n\right))_*\\ \mathbf{elif}\;n \le -1.259835294447841 \cdot 10^{-283}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{i}\\ \mathbf{elif}\;n \le 1.668841499988779 \cdot 10^{-260}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{\frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original42.0
Target41.8
Herbie19.0
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if n < -1.6699497128870854e-36 or 1.668841499988779e-260 < n

    1. Initial program 49.6

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
    2. Using strategy rm

    3. Applied pow-to-exp53.0

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{\frac{i}{n}}\]

    4. Applied expm1-def48.4

      \[\leadsto 100 \cdot \frac{\color{blue}{(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1)^*}}{\frac{i}{n}}\]

    5. Simplified20.7

      \[\leadsto 100 \cdot \frac{(e^{\color{blue}{n \cdot \log_* (1 + \frac{i}{n})}} - 1)^*}{\frac{i}{n}}\]
    6. Using strategy rm

    7. Applied *-commutative20.7

      \[\leadsto \color{blue}{\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}} \cdot 100}\]

    if -1.6699497128870854e-36 < n < -5.921036522656228e-174

    1. Initial program 17.7

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
    2. Using strategy rm

    3. Applied div-inv17.7

      \[\leadsto 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\color{blue}{i \cdot \frac{1}{n}}}\]

    4. Applied *-un-lft-identity17.7

      \[\leadsto 100 \cdot \frac{\color{blue}{1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{i \cdot \frac{1}{n}}\]

    5. Applied times-frac18.0

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\right)}\]

    6. Applied associate-*r*18.6

      \[\leadsto \color{blue}{\left(100 \cdot \frac{1}{i}\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}}\]

    7. Simplified18.6

      \[\leadsto \left(100 \cdot \frac{1}{i}\right) \cdot \color{blue}{(\left({\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot n + \left(-n\right))_*}\]

    if -5.921036522656228e-174 < n < -1.259835294447841e-283

    1. Initial program 12.1

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
    2. Using strategy rm

    3. Applied pow-to-exp12.1

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{\frac{i}{n}}\]

    4. Applied expm1-def2.4

      \[\leadsto 100 \cdot \frac{\color{blue}{(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1)^*}}{\frac{i}{n}}\]

    5. Simplified8.3

      \[\leadsto 100 \cdot \frac{(e^{\color{blue}{n \cdot \log_* (1 + \frac{i}{n})}} - 1)^*}{\frac{i}{n}}\]
    6. Using strategy rm

    7. Applied *-commutative8.3

      \[\leadsto \color{blue}{\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}} \cdot 100}\]
    8. Using strategy rm

    9. Applied associate-*l/8.2

      \[\leadsto \color{blue}{\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^* \cdot 100}{\frac{i}{n}}}\]
    10. Using strategy rm

    11. Applied div-inv8.2

      \[\leadsto \frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^* \cdot 100}{\color{blue}{i \cdot \frac{1}{n}}}\]

    12. Applied times-frac9.1

      \[\leadsto \color{blue}{\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{i} \cdot \frac{100}{\frac{1}{n}}}\]

    13. Simplified9.1

      \[\leadsto \frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{i} \cdot \color{blue}{\left(100 \cdot n\right)}\]

    if -1.259835294447841e-283 < n < 1.668841499988779e-260

    1. Initial program 23.8

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
    2. Using strategy rm

    3. Applied pow-to-exp23.8

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{\frac{i}{n}}\]

    4. Applied expm1-def18.7

      \[\leadsto 100 \cdot \frac{\color{blue}{(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1)^*}}{\frac{i}{n}}\]

    5. Simplified16.9

      \[\leadsto 100 \cdot \frac{(e^{\color{blue}{n \cdot \log_* (1 + \frac{i}{n})}} - 1)^*}{\frac{i}{n}}\]
    6. Using strategy rm

    7. Applied *-commutative16.9

      \[\leadsto \color{blue}{\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}} \cdot 100}\]
    8. Using strategy rm

    9. Applied associate-*l/16.8

      \[\leadsto \color{blue}{\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^* \cdot 100}{\frac{i}{n}}}\]

    10. Taylor expanded around 0 7.0

      \[\leadsto \color{blue}{0}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification19.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.6699497128870854 \cdot 10^{-36}:\\ \;\;\;\;100 \cdot \frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -5.921036522656228 \cdot 10^{-174}:\\ \;\;\;\;\left(\frac{1}{i} \cdot 100\right) \cdot (\left({\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot n + \left(-n\right))_*\\ \mathbf{elif}\;n \le -1.259835294447841 \cdot 10^{-283}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{i}\\ \mathbf{elif}\;n \le 1.668841499988779 \cdot 10^{-260}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100 (/ (- (exp (* n (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) 1) (/ i n)))

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

Details

Time bar (total: 25.7s)Debug log

sample343.0ms

Algorithm
intervals

simplify361.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
360.0ms
(* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 43.7b

localize17.0ms

Local error

Found 4 expressions with local error:

11.6b
(/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))
6.5b
(pow (+ 1 (/ i n)) n)
1.3b
(- (pow (+ 1 (/ i n)) n) 1)
0.1b
(* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)))

rewrite81.0ms

Algorithm
rewrite-expression-head
Rules
48×add-sqr-sqrt
40×times-frac
28×add-cube-cbrt
28×*-un-lft-identity
25×associate-*r*
20×difference-of-sqr-1
16×unpow-prod-down
13×div-inv
9×add-exp-log
6×fma-neg
6×add-cbrt-cube
5×associate-/l*
5×pow1
4×add-log-exp
4×log1p-expm1-u
4×associate-/r*
4×expm1-log1p-u
3×expm1-def
2×associate-/l/
2×flip--
2×associate-/r/
2×pow-exp
2×flip3--
2×pow-to-exp
1×*-commutative
1×associate-*r/
1×div-exp
1×div-sub
1×frac-2neg
1×pow-pow
1×sub-neg
1×clear-num
1×cbrt-undiv
Counts
4 → 122
Calls
4 calls:
Slowest
33.0ms
(/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))
25.0ms
(- (pow (+ 1 (/ i n)) n) 1)
18.0ms
(* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)))
3.0ms
(pow (+ 1 (/ i n)) n)

series712.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
229.0ms
(* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)))
216.0ms
(/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))
145.0ms
(- (pow (+ 1 (/ i n)) n) 1)
121.0ms
(pow (+ 1 (/ i n)) n)

simplify3.9s

Counts
105 → 134
Calls
105 calls:
Slowest
337.0ms
(/ (* (* (- (pow (+ 1 (/ i n)) n) 1) (- (pow (+ 1 (/ i n)) n) 1)) (- (pow (+ 1 (/ i n)) n) 1)) (* (* (/ i n) (/ i n)) (/ i n)))
268.0ms
(/ (* (- (exp (* (- (log (/ 1 n)) (log (/ 1 i))) n)) 1) n) i)
257.0ms
(* 100 (/ (* n (- (exp (* n (- (log (/ -1 n)) (log (/ -1 i))))) 1)) i))
240.0ms
(* 100 (/ (* (- (exp (* (- (log (/ 1 n)) (log (/ 1 i))) n)) 1) n) i))
191.0ms
(/ (* n (- (exp (* n (- (log (/ -1 n)) (log (/ -1 i))))) 1)) i)

prune1.3s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 11.0b

localize26.0ms

Local error

Found 4 expressions with local error:

11.5b
(* n (log1p (/ i n)))
11.5b
(/ (expm1 (* n (log1p (/ i n)))) (/ i n))
5.8b
(log1p (/ i n))
0.1b
(* 100 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
24×times-frac
22×add-cube-cbrt
22×*-un-lft-identity
22×add-sqr-sqrt
20×associate-*r*
9×div-inv
6×add-exp-log
6×add-cbrt-cube
5×expm1-log1p-u
4×add-log-exp
4×log1p-expm1-u
4×associate-/r*
4×pow1
3×associate-/l*
2×*-commutative
2×associate-/r/
1×expm1-udef
1×associate-*r/
1×log1p-expm1
1×div-exp
1×div-sub
1×frac-2neg
1×clear-num
1×log1p-udef
1×cbrt-undiv
Counts
4 → 87
Calls
4 calls:
Slowest
6.0ms
(/ (expm1 (* n (log1p (/ i n)))) (/ i n))
3.0ms
(* 100 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
2.0ms
(* n (log1p (/ i n)))
1.0ms
(log1p (/ i n))

series706.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
314.0ms
(/ (expm1 (* n (log1p (/ i n)))) (/ i n))
302.0ms
(* 100 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
46.0ms
(log1p (/ i n))
44.0ms
(* n (log1p (/ i n)))

simplify1.1s

Counts
66 → 99
Calls
66 calls:
Slowest
230.0ms
(/ (* (* (expm1 (* n (log1p (/ i n)))) (expm1 (* n (log1p (/ i n))))) (expm1 (* n (log1p (/ i n))))) (* (* (/ i n) (/ i n)) (/ i n)))
55.0ms
(- (log (/ -1 n)) (log (/ -1 i)))
48.0ms
(/ (* (- (* 100 (exp i)) 100) n) i)
47.0ms
(/ (* (- (* 100 (exp i)) 100) n) i)
46.0ms
(/ (* (- (exp i) 1) n) i)

prune991.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 10.8b

localize3.0ms

Local error

Found 4 expressions with local error:

11.5b
(* n (log1p (/ i n)))
11.5b
(/ (expm1 (* n (log1p (/ i n)))) (/ i n))
5.8b
(log1p (/ i n))
0.1b
(* (/ (expm1 (* n (log1p (/ i n)))) (/ i n)) 100)

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
24×times-frac
22×add-cube-cbrt
22×*-un-lft-identity
22×add-sqr-sqrt
17×associate-*l*
9×div-inv
6×add-exp-log
6×add-cbrt-cube
5×expm1-log1p-u
4×add-log-exp
4×log1p-expm1-u
4×associate-/r*
4×pow1
3×associate-/l*
3×associate-*r*
2×*-commutative
2×associate-/r/
1×expm1-udef
1×log1p-expm1
1×associate-*l/
1×div-exp
1×div-sub
1×frac-2neg
1×clear-num
1×log1p-udef
1×cbrt-undiv
Counts
4 → 87
Calls
4 calls:
Slowest
4.0ms
(* (/ (expm1 (* n (log1p (/ i n)))) (/ i n)) 100)
3.0ms
(/ (expm1 (* n (log1p (/ i n)))) (/ i n))
1.0ms
(* n (log1p (/ i n)))
0.0ms
(log1p (/ i n))

series689.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
324.0ms
(/ (expm1 (* n (log1p (/ i n)))) (/ i n))
271.0ms
(* (/ (expm1 (* n (log1p (/ i n)))) (/ i n)) 100)
47.0ms
(log1p (/ i n))
46.0ms
(* n (log1p (/ i n)))

simplify922.0ms

Counts
66 → 99
Calls
66 calls:
Slowest
194.0ms
(/ (* (* (expm1 (* n (log1p (/ i n)))) (expm1 (* n (log1p (/ i n))))) (expm1 (* n (log1p (/ i n))))) (* (* (/ i n) (/ i n)) (/ i n)))
65.0ms
(/ (* (- (* 100 (exp i)) 100) n) i)
45.0ms
(- (log (/ -1 n)) (log (/ -1 i)))
45.0ms
(/ (* (- (* 100 (exp i)) 100) n) i)
44.0ms
(/ (* (- (exp i) 1) n) i)

prune972.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 10.8b

localize12.0ms

Local error

Found 4 expressions with local error:

11.5b
(* n (log1p (/ i n)))
11.5b
(/ (* (expm1 (* n (log1p (/ i n)))) 100) (/ i n))
5.8b
(log1p (/ i n))
0.1b
(* (expm1 (* n (log1p (/ i n)))) 100)

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×add-exp-log
6×add-cbrt-cube
5×expm1-log1p-u
4×add-log-exp
4×log1p-expm1-u
4×times-frac
4×associate-/r*
4×pow1
3×div-inv
3×associate-*l*
3×associate-*r*
2×*-commutative
1×associate-/l*
1×associate-/r/
1×log1p-expm1
1×div-exp
1×frac-2neg
1×clear-num
1×log1p-udef
1×cbrt-undiv
Counts
4 → 61
Calls
4 calls:
Slowest
11.0ms
(/ (* (expm1 (* n (log1p (/ i n)))) 100) (/ i n))
3.0ms
(* (expm1 (* n (log1p (/ i n)))) 100)
2.0ms
(* n (log1p (/ i n)))
0.0ms
(log1p (/ i n))

series551.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
297.0ms
(/ (* (expm1 (* n (log1p (/ i n)))) 100) (/ i n))
152.0ms
(* (expm1 (* n (log1p (/ i n)))) 100)
57.0ms
(log1p (/ i n))
45.0ms
(* n (log1p (/ i n)))

simplify537.0ms

Counts
33 → 73
Calls
33 calls:
Slowest
88.0ms
(- (+ (* 50 (* (pow n 2) (pow (log n) 2))) (+ (* 50/3 (* (pow n 3) (pow (log i) 3))) (+ (* 100 (* n (log i))) (+ (* 100/3 (* (pow n 3) (* (pow (log n) 2) (log i)))) (+ (* 50/3 (* (pow n 3) (* (log i) (pow (log n) 2)))) (* 50 (* (pow n 2) (pow (log i) 2)))))))) (+ (* 100/3 (* (pow n 3) (* (pow (log i) 2) (log n)))) (+ (* 50 (* (pow n 2) (* (log n) (log i)))) (+ (* 50 (* (pow n 2) (* (log i) (log n)))) (+ (* 50/3 (* (pow n 3) (pow (log n) 3))) (+ (* 50/3 (* (pow n 3) (* (log n) (pow (log i) 2)))) (* 100 (* n (log n)))))))))
82.0ms
(/ (* (* (* (expm1 (* n (log1p (/ i n)))) 100) (* (expm1 (* n (log1p (/ i n)))) 100)) (* (expm1 (* n (log1p (/ i n)))) 100)) (* (* (/ i n) (/ i n)) (/ i n)))
75.0ms
(/ (* (- (* 100 (exp i)) 100) n) i)
69.0ms
(- (log (* (expm1 (* n (log1p (/ i n)))) 100)) (log (/ i n)))
68.0ms
(/ (* (- (* 100 (exp i)) 100) n) i)

prune754.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 10.8b

regimes235.0ms

Accuracy

12.7% (9.2b remaining)

Error of 19.0b against oracle of 9.7b and baseline of 20.3b

bsearch745.0ms

end0.0ms

sample10.8s

Algorithm
intervals