Average Error: 45.2 → 0.0
Time: 6.4s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 222.3032347187608:\\ \;\;\;\;\frac{\frac{i}{4}}{\left(4 \cdot i\right) \cdot i - 1.0} \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{16} + \frac{0.00390625}{{i}^{4}}\right) + \frac{\frac{0.015625}{i}}{i}\\ \end{array}\]

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if i < 222.3032347187608

    1. Initial program 43.6

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{i \cdot \frac{i}{4}}{i \cdot \left(4 \cdot i\right) - 1.0}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity0.0

      \[\leadsto \frac{i \cdot \frac{i}{4}}{\color{blue}{1 \cdot \left(i \cdot \left(4 \cdot i\right) - 1.0\right)}}\]

    5. Applied times-frac0.0

      \[\leadsto \color{blue}{\frac{i}{1} \cdot \frac{\frac{i}{4}}{i \cdot \left(4 \cdot i\right) - 1.0}}\]

    6. Simplified0.0

      \[\leadsto \color{blue}{i} \cdot \frac{\frac{i}{4}}{i \cdot \left(4 \cdot i\right) - 1.0}\]

    if 222.3032347187608 < i

    1. Initial program 46.8

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

    2. Simplified32.4

      \[\leadsto \color{blue}{\frac{i \cdot \frac{i}{4}}{i \cdot \left(4 \cdot i\right) - 1.0}}\]

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.015625 \cdot \frac{1}{{i}^{2}} + \left(\frac{1}{16} + 0.00390625 \cdot \frac{1}{{i}^{4}}\right)}\]

    4. Simplified0.0

      \[\leadsto \color{blue}{\frac{\frac{0.015625}{i}}{i} + \left(\frac{1}{16} + \frac{0.00390625}{{i}^{4}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 222.3032347187608:\\ \;\;\;\;\frac{\frac{i}{4}}{\left(4 \cdot i\right) \cdot i - 1.0} \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{16} + \frac{0.00390625}{{i}^{4}}\right) + \frac{\frac{0.015625}{i}}{i}\\ \end{array}\]

Reproduce

herbie shell --seed 2019021 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0)))

Details

Time bar (total: 6.2s)Debug log

sample54.0ms

Algorithm
intervals
Results
37.0ms256×body80valid
6.0ms256×pre80true

simplify145.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
145.0ms
(/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0))

prune11.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 17.4b

localize15.0ms

Local error

Found 1 expressions with local error:

17.4b
(/ (* i (/ i 4)) (- (* i (* 4 i)) 1.0))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
3×add-cube-cbrt
3×add-exp-log
3×add-cbrt-cube
3×times-frac
3×*-un-lft-identity
3×associate-/r*
3×add-sqr-sqrt
2×associate-/r/
1×associate-/l/
1×add-log-exp
1×associate-/l*
1×flip--
1×div-inv
1×associate-*r/
1×div-exp
1×flip3--
1×frac-2neg
1×clear-num
1×cbrt-undiv
1×pow1
Counts
1 → 22
Calls
1 calls:
Slowest
5.0ms
(/ (* i (/ i 4)) (- (* i (* 4 i)) 1.0))

series37.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
37.0ms
(/ (* i (/ i 4)) (- (* i (* 4 i)) 1.0))

simplify692.0ms

Counts
17 → 25
Calls
17 calls:
Slowest
146.0ms
(- (log (* i (/ i 4))) (log (- (* i (* 4 i)) 1.0)))
119.0ms
(+ (* (* i (* 4 i)) (* i (* 4 i))) (+ (* 1.0 1.0) (* (* i (* 4 i)) 1.0)))
113.0ms
(- (+ (* 0.25 (pow i 2)) (+ (* 1.0 (pow i 4)) (* 4.0 (pow i 6)))))
107.0ms
(/ (* (* (* i (/ i 4)) (* i (/ i 4))) (* i (/ i 4))) (* (* (- (* i (* 4 i)) 1.0) (- (* i (* 4 i)) 1.0)) (- (* i (* 4 i)) 1.0)))
52.0ms
(+ (* 0.015625 (/ 1 (pow i 2))) (+ 1/16 (* 0.00390625 (/ 1 (pow i 4)))))

prune210.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize5.0ms

Local error

Found 2 expressions with local error:

16.8b
(/ (/ i 4) (- (* i (* 4 i)) 1.0))
0.1b
(* i (/ (/ i 4) (- (* i (* 4 i)) 1.0)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
24×times-frac
19×add-cube-cbrt
19×*-un-lft-identity
19×add-sqr-sqrt
18×associate-*r*
9×div-inv
4×associate-/l*
4×associate-/r/
4×add-exp-log
4×add-cbrt-cube
3×associate-/r*
2×add-log-exp
2×flip--
2×flip3--
2×pow1
1×associate-/l/
1×*-commutative
1×associate-*r/
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
2 → 61
Calls
2 calls:
Slowest
5.0ms
(/ (/ i 4) (- (* i (* 4 i)) 1.0))
4.0ms
(* i (/ (/ i 4) (- (* i (* 4 i)) 1.0)))

series93.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
50.0ms
(* i (/ (/ i 4) (- (* i (* 4 i)) 1.0)))
42.0ms
(/ (/ i 4) (- (* i (* 4 i)) 1.0))

simplify1.9s

Counts
58 → 67
Calls
58 calls:
Slowest
272.0ms
(/ (* (* (/ i 4) (/ i 4)) (/ i 4)) (* (* (- (* i (* 4 i)) 1.0) (- (* i (* 4 i)) 1.0)) (- (* i (* 4 i)) 1.0)))
158.0ms
(+ (* (* i (* 4 i)) (* i (* 4 i))) (+ (* 1.0 1.0) (* (* i (* 4 i)) 1.0)))
144.0ms
(- (+ (* 0.25 i) (+ (* 1.0 (pow i 3)) (* 4.0 (pow i 5)))))
117.0ms
(+ (* (* i (* 4 i)) (* i (* 4 i))) (+ (* 1.0 1.0) (* (* i (* 4 i)) 1.0)))
113.0ms
(- (+ (* 0.25 (pow i 2)) (+ (* 1.0 (pow i 4)) (* 4.0 (pow i 6)))))

prune431.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize25.0ms

Local error

Found 3 expressions with local error:

0.1b
(/ 0.00390625 (pow i 4))
0.1b
(/ (/ 0.015625 i) i)
0.0b
(+ (/ (/ 0.015625 i) i) (+ 1/16 (/ 0.00390625 (pow i 4))))

rewrite18.0ms

Algorithm
rewrite-expression-head
Rules
7×*-un-lft-identity
5×add-log-exp
5×add-cube-cbrt
5×add-sqr-sqrt
4×associate-/l*
3×div-inv
3×add-exp-log
3×add-cbrt-cube
3×associate-/r*
3×pow1
2×flip-+
2×frac-add
2×flip3-+
2×frac-2neg
2×clear-num
1×associate-/l/
1×distribute-lft-out
1×sum-log
1×associate-+r+
1×+-commutative
Counts
3 → 43
Calls
3 calls:
Slowest
11.0ms
(+ (/ (/ 0.015625 i) i) (+ 1/16 (/ 0.00390625 (pow i 4))))
5.0ms
(/ (/ 0.015625 i) i)
1.0ms
(/ 0.00390625 (pow i 4))

series44.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
19.0ms
(/ 0.00390625 (pow i 4))
15.0ms
(+ (/ (/ 0.015625 i) i) (+ 1/16 (/ 0.00390625 (pow i 4))))
10.0ms
(/ (/ 0.015625 i) i)

simplify741.0ms

Counts
21 → 52
Calls
21 calls:
Slowest
261.0ms
(* i (+ (* 1/16 1/16) (- (* (/ 0.00390625 (pow i 4)) (/ 0.00390625 (pow i 4))) (* 1/16 (/ 0.00390625 (pow i 4))))))
128.0ms
(+ (* (/ 0.015625 i) (- 1/16 (/ 0.00390625 (pow i 4)))) (* i (- (* 1/16 1/16) (* (/ 0.00390625 (pow i 4)) (/ 0.00390625 (pow i 4))))))
119.0ms
(+ (* (/ 0.015625 i) (+ (* 1/16 1/16) (- (* (/ 0.00390625 (pow i 4)) (/ 0.00390625 (pow i 4))) (* 1/16 (/ 0.00390625 (pow i 4)))))) (* i (+ (pow 1/16 3) (pow (/ 0.00390625 (pow i 4)) 3))))
48.0ms
(+ (* 0.015625 (/ 1 (pow i 2))) (+ 1/16 (* 0.00390625 (/ 1 (pow i 4)))))
44.0ms
(* (exp (/ (/ 0.015625 i) i)) (exp (+ 1/16 (/ 0.00390625 (pow i 4)))))

prune343.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes24.0ms

Accuracy

100% (0.0b remaining)

Error of 0.0b against oracle of 0.0b and baseline of 16.1b

bsearch62.0ms

end0.0ms

sample1.3s

Algorithm
intervals
Results
1.1s8000×body80valid
152.0ms8000×pre80true