Average Error: 45.3 → 0.0
Time: 7.3s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 227.50852513912608:\\ \;\;\;\;\frac{\frac{i}{4} \cdot i}{(\left(4 \cdot i\right) \cdot i + \left(-1.0\right))_*}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{0.015625}{i}\right) \cdot \left(\frac{1}{i}\right) + \frac{1}{16})_* + \frac{0.00390625}{{i}^{4}}\\ \end{array}\]

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 227.50852513912608

    1. Initial program 44.1

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

    2. Simplified0.0

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

    if 227.50852513912608 < i

    1. Initial program 46.4

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

    2. Simplified31.5

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

    4. Applied *-un-lft-identity31.5

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

    5. Applied times-frac31.7

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

    6. Simplified31.7

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

    7. Simplified31.7

      \[\leadsto i \cdot \color{blue}{\frac{i}{(16 \cdot \left(i \cdot i\right) + \left(-4 \cdot 1.0\right))_*}}\]
    8. Using strategy rm

    9. Applied div-inv31.7

      \[\leadsto i \cdot \color{blue}{\left(i \cdot \frac{1}{(16 \cdot \left(i \cdot i\right) + \left(-4 \cdot 1.0\right))_*}\right)}\]

    10. Applied associate-*r*31.6

      \[\leadsto \color{blue}{\left(i \cdot i\right) \cdot \frac{1}{(16 \cdot \left(i \cdot i\right) + \left(-4 \cdot 1.0\right))_*}}\]

    11. Taylor expanded around -inf 0.0

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

    12. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 227.50852513912608:\\ \;\;\;\;\frac{\frac{i}{4} \cdot i}{(\left(4 \cdot i\right) \cdot i + \left(-1.0\right))_*}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{0.015625}{i}\right) \cdot \left(\frac{1}{i}\right) + \frac{1}{16})_* + \frac{0.00390625}{{i}^{4}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(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: 6.9s)Debug log

sample35.0ms

Algorithm
intervals
Results
22.0ms256×body80valid
4.0ms256×pre80true

simplify146.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
146.0ms
(/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0))

prune14.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 12.2b

localize18.0ms

Local error

Found 1 expressions with local error:

12.2b
(/ (* i (/ i 4)) (fma (* 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
1×associate-/l/
1×add-log-exp
1×associate-/l*
1×div-inv
1×log1p-expm1-u
1×associate-*r/
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
1×pow1
1×expm1-log1p-u
Counts
1 → 22
Calls
1 calls:
Slowest
5.0ms
(/ (* i (/ i 4)) (fma (* i 4) i (- 1.0)))

series42.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
42.0ms
(/ (* i (/ i 4)) (fma (* i 4) i (- 1.0)))

simplify537.0ms

Counts
15 → 25
Calls
15 calls:
Slowest
155.0ms
(- (+ (* 0.25 (pow i 2)) (+ (* 1.0 (pow i 4)) (* 4.0 (pow i 6)))))
94.0ms
(- (log (* i (/ i 4))) (log (fma (* i 4) i (- 1.0))))
67.0ms
(+ (* 0.015625 (/ 1 (pow i 2))) (+ 1/16 (* 0.00390625 (/ 1 (pow i 4)))))
63.0ms
(/ (* (* (* i (/ i 4)) (* i (/ i 4))) (* i (/ i 4))) (* (* (fma (* i 4) i (- 1.0)) (fma (* i 4) i (- 1.0))) (fma (* i 4) i (- 1.0))))
40.0ms
(+ (* 0.015625 (/ 1 (pow i 2))) (+ 1/16 (* 0.00390625 (/ 1 (pow i 4)))))

prune165.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize27.0ms

Local error

Found 3 expressions with local error:

11.8b
(/ i (fma 16 (* i i) (* -4 1.0)))
0.1b
(* i (/ i (fma 16 (* i i) (* -4 1.0))))
0.0b
(fma 16 (* i i) (* -4 1.0))

rewrite4.0ms

Algorithm
rewrite-expression-head
Rules
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×associate-*r*
3×add-log-exp
3×log1p-expm1-u
3×add-exp-log
3×add-cbrt-cube
3×associate-/r*
3×pow1
3×expm1-log1p-u
2×div-inv
1×fma-udef
1×*-commutative
1×associate-*r/
1×frac-2neg
1×clear-num
Counts
3 → 40
Calls
3 calls:
Slowest
2.0ms
(* i (/ i (fma 16 (* i i) (* -4 1.0))))
1.0ms
(/ i (fma 16 (* i i) (* -4 1.0)))
0.0ms
(fma 16 (* i i) (* -4 1.0))

series104.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
47.0ms
(/ i (fma 16 (* i i) (* -4 1.0)))
32.0ms
(fma 16 (* i i) (* -4 1.0))
25.0ms
(* i (/ i (fma 16 (* i i) (* -4 1.0))))

simplify966.0ms

Counts
17 → 49
Calls
17 calls:
Slowest
262.0ms
(- (+ (* 0.25 i) (+ (* 1.0 (pow i 3)) (* 4.0 (pow i 5)))))
201.0ms
(- (+ (* 0.25 (pow i 2)) (+ (* 1.0 (pow i 4)) (* 4.0 (pow i 6)))))
144.0ms
(+ (* 0.00390625 (/ 1 (pow i 5))) (+ (* 0.015625 (/ 1 (pow i 3))) (* 1/16 (/ 1 i))))
112.0ms
(+ (* 0.00390625 (/ 1 (pow i 5))) (+ (* 0.015625 (/ 1 (pow i 3))) (* 1/16 (/ 1 i))))
39.0ms
(+ (* 0.015625 (/ 1 (pow i 2))) (+ 1/16 (* 0.00390625 (/ 1 (pow i 4)))))

prune433.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize12.0ms

Local error

Found 3 expressions with local error:

12.3b
(* (* i i) (/ 1 (fma 16 (* i i) (* -4 1.0))))
0.4b
(/ 1 (fma 16 (* i i) (* -4 1.0)))
0.0b
(fma 16 (* i i) (* -4 1.0))

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
8×add-exp-log
6×pow1
5×add-cube-cbrt
5×add-cbrt-cube
5×*-un-lft-identity
5×add-sqr-sqrt
4×associate-*r*
3×add-log-exp
3×log1p-expm1-u
3×associate-/r*
3×expm1-log1p-u
2×div-inv
2×prod-exp
2×rec-exp
1×inv-pow
1×pow-flip
1×fma-udef
1×cbrt-unprod
1×*-commutative
1×un-div-inv
1×associate-*r/
1×pow-prod-down
1×associate-*l*
1×frac-2neg
1×clear-num
Counts
3 → 49
Calls
3 calls:
Slowest
8.0ms
(* (* i i) (/ 1 (fma 16 (* i i) (* -4 1.0))))
1.0ms
(/ 1 (fma 16 (* i i) (* -4 1.0)))
0.0ms
(fma 16 (* i i) (* -4 1.0))

series109.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
45.0ms
(* (* i i) (/ 1 (fma 16 (* i i) (* -4 1.0))))
35.0ms
(fma 16 (* i i) (* -4 1.0))
29.0ms
(/ 1 (fma 16 (* i i) (* -4 1.0)))

simplify1.2s

Counts
23 → 58
Calls
23 calls:
Slowest
300.0ms
(* (* (* (* i i) (* i i)) (* i i)) (* (* (/ 1 (fma 16 (* i i) (* -4 1.0))) (/ 1 (fma 16 (* i i) (* -4 1.0)))) (/ 1 (fma 16 (* i i) (* -4 1.0)))))
188.0ms
(- (+ (* 0.25 (pow i 2)) (+ (* 1.0 (pow i 4)) (* 4.0 (pow i 6)))))
126.0ms
(- (+ (* 1.0 (pow i 2)) (+ (* 4.0 (pow i 4)) 0.25)))
77.0ms
(+ (* 0.015625 (/ 1 (pow i 2))) (+ 1/16 (* 0.00390625 (/ 1 (pow i 4)))))
74.0ms
(+ (* 0.015625 (/ 1 (pow i 2))) (+ 1/16 (* 0.00390625 (/ 1 (pow i 4)))))

prune525.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize25.0ms

Local error

Found 3 expressions with local error:

0.2b
(fma (/ 0.015625 i) (/ 1 i) 1/16)
0.2b
(/ 0.00390625 (pow i 4))
0.0b
(+ (fma (/ 0.015625 i) (/ 1 i) 1/16) (/ 0.00390625 (pow i 4)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
7×*-un-lft-identity
5×add-log-exp
5×add-cube-cbrt
5×add-sqr-sqrt
3×log1p-expm1-u
3×add-exp-log
3×add-cbrt-cube
3×fma-def
3×associate-/r*
3×pow1
3×expm1-log1p-u
2×fma-udef
1×distribute-lft-out
1×div-inv
1×flip-+
1×flip3-+
1×frac-2neg
1×sum-log
1×clear-num
1×associate-+l+
1×+-commutative
Counts
3 → 43
Calls
3 calls:
Slowest
11.0ms
(+ (fma (/ 0.015625 i) (/ 1 i) 1/16) (/ 0.00390625 (pow i 4)))
1.0ms
(/ 0.00390625 (pow i 4))
0.0ms
(fma (/ 0.015625 i) (/ 1 i) 1/16)

series79.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
30.0ms
(+ (fma (/ 0.015625 i) (/ 1 i) 1/16) (/ 0.00390625 (pow i 4)))
30.0ms
(fma (/ 0.015625 i) (/ 1 i) 1/16)
18.0ms
(/ 0.00390625 (pow i 4))

simplify502.0ms

Counts
17 → 52
Calls
17 calls:
Slowest
103.0ms
(* (exp (fma (/ 0.015625 i) (/ 1 i) 1/16)) (exp (/ 0.00390625 (pow i 4))))
57.0ms
(+ (* 0.015625 (/ 1 (pow i 2))) (+ 1/16 (* 0.00390625 (/ 1 (pow i 4)))))
53.0ms
(+ (* 0.015625 (/ 1 (pow i 2))) (+ 1/16 (* 0.00390625 (/ 1 (pow i 4)))))
52.0ms
(* (cbrt (fma (/ 0.015625 i) (/ 1 i) 1/16)) (cbrt (fma (/ 0.015625 i) (/ 1 i) 1/16)))
47.0ms
(+ (* 0.015625 (/ 1 (pow i 2))) (+ 1/16 (* 0.00390625 (/ 1 (pow i 4)))))

prune458.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes45.0ms

Accuracy

100% (0.0b remaining)

Error of 0.0b against oracle of 0.0b and baseline of 15.8b

bsearch70.0ms

end0.0ms

sample1.3s

Algorithm
intervals
Results
1.1s8000×body80valid
188.0ms8000×pre80true