Average Error: 23.9 → 12.3
Time: 37.3s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\sqrt[3]{\left((\left(\left(\beta - \alpha\right) \cdot \frac{1}{\left(2.0 + \alpha\right) + (i \cdot 2 + \beta)_*}\right) \cdot \left(\frac{\alpha + \beta}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_* \cdot (\left(\left(\beta - \alpha\right) \cdot \frac{1}{\left(2.0 + \alpha\right) + (i \cdot 2 + \beta)_*}\right) \cdot \left(\frac{\alpha + \beta}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*\right) \cdot (\left(\left(\beta - \alpha\right) \cdot \frac{1}{\left(2.0 + \alpha\right) + (i \cdot 2 + \beta)_*}\right) \cdot \left(\frac{\alpha + \beta}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.9

    \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

  2. Initial simplification12.3

    \[\leadsto \frac{(\left(\frac{\beta - \alpha}{(i \cdot 2 + \beta)_* + \left(\alpha + 2.0\right)}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]
  3. Using strategy rm

  4. Applied div-inv12.2

    \[\leadsto \frac{(\color{blue}{\left(\left(\beta - \alpha\right) \cdot \frac{1}{(i \cdot 2 + \beta)_* + \left(\alpha + 2.0\right)}\right)} \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube12.3

    \[\leadsto \frac{\color{blue}{\sqrt[3]{\left((\left(\left(\beta - \alpha\right) \cdot \frac{1}{(i \cdot 2 + \beta)_* + \left(\alpha + 2.0\right)}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_* \cdot (\left(\left(\beta - \alpha\right) \cdot \frac{1}{(i \cdot 2 + \beta)_* + \left(\alpha + 2.0\right)}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*\right) \cdot (\left(\left(\beta - \alpha\right) \cdot \frac{1}{(i \cdot 2 + \beta)_* + \left(\alpha + 2.0\right)}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}}}{2.0}\]

  7. Final simplification12.3

    \[\leadsto \frac{\sqrt[3]{\left((\left(\left(\beta - \alpha\right) \cdot \frac{1}{\left(2.0 + \alpha\right) + (i \cdot 2 + \beta)_*}\right) \cdot \left(\frac{\alpha + \beta}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_* \cdot (\left(\left(\beta - \alpha\right) \cdot \frac{1}{\left(2.0 + \alpha\right) + (i \cdot 2 + \beta)_*}\right) \cdot \left(\frac{\alpha + \beta}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*\right) \cdot (\left(\left(\beta - \alpha\right) \cdot \frac{1}{\left(2.0 + \alpha\right) + (i \cdot 2 + \beta)_*}\right) \cdot \left(\frac{\alpha + \beta}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}}{2.0}\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))

Details

Time bar (total: 29.3s)Debug log

start127.0ms

Algorithm
intervals

setup745.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 12.0b

localize64.0ms

Local error

Found 4 expressions with local error:

0.9b
(fma (/ (- beta alpha) (+ (fma i 2 beta) (+ alpha 2.0))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
0.3b
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
0.3b
(/ (- beta alpha) (+ (fma i 2 beta) (+ alpha 2.0)))
0.0b
(+ (fma 2 i alpha) beta)

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 97
Calls

4 calls. Slowest were:

11.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
8.0ms
(/ (- beta alpha) (+ (fma i 2 beta) (+ alpha 2.0)))
3.0ms
(+ (fma 2 i alpha) beta)

series278.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

141.0ms
(fma (/ (- beta alpha) (+ (fma i 2 beta) (+ alpha 2.0))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
58.0ms
(/ (- beta alpha) (+ (fma i 2 beta) (+ alpha 2.0)))
56.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
22.0ms
(+ (fma 2 i alpha) beta)

simplify2.8s

Counts
81 → 109
Calls

81 calls. Slowest were:

188.0ms
(- (* 0.5 beta) (+ (* 0.25 (pow beta 2)) (* 0.5 alpha)))
175.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
163.0ms
(/ (* (* (+ beta alpha) (+ beta alpha)) (+ beta alpha)) (* (* (+ (fma 2 i alpha) beta) (+ (fma 2 i alpha) beta)) (+ (fma 2 i alpha) beta)))

prune2.2s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 11.8b

localize25.0ms

Local error

Found 4 expressions with local error:

2.8b
(fma (* (- beta alpha) (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
0.3b
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
0.2b
(/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))
0.2b
(* (- beta alpha) (/ 1 (+ (fma i 2 beta) (+ alpha 2.0))))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

11.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
10.0ms
(* (- beta alpha) (/ 1 (+ (fma i 2 beta) (+ alpha 2.0))))
5.0ms
(/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))

series335.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

148.0ms
(fma (* (- beta alpha) (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
65.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
62.0ms
(/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))
60.0ms
(* (- beta alpha) (/ 1 (+ (fma i 2 beta) (+ alpha 2.0))))

simplify3.9s

Counts
68 → 105
Calls

68 calls. Slowest were:

673.0ms
(* (- (* beta beta) (* alpha alpha)) (/ 1 (+ (fma i 2 beta) (+ alpha 2.0))))
392.0ms
(* (* (* (- beta alpha) (- beta alpha)) (- beta alpha)) (* (* (/ 1 (+ (fma i 2 beta) (+ alpha 2.0))) (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))) (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))))
321.0ms
(* (+ beta alpha) (+ (fma i 2 beta) (+ alpha 2.0)))

prune2.1s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 11.8b

localize14.0ms

Local error

Found 4 expressions with local error:

2.8b
(fma (* (- beta alpha) (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
2.8b
(fma (* (- beta alpha) (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
2.8b
(fma (* (- beta alpha) (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
0.3b
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

5.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
0.0ms
(fma (* (- beta alpha) (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
0.0ms
(fma (* (- beta alpha) (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)

series427.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

143.0ms
(fma (* (- beta alpha) (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
124.0ms
(fma (* (- beta alpha) (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
115.0ms
(fma (* (- beta alpha) (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
45.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))

simplify2.1s

Counts
41 → 75
Calls

41 calls. Slowest were:

178.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
173.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
162.0ms
(/ (* (* (+ beta alpha) (+ beta alpha)) (+ beta alpha)) (* (* (+ (fma 2 i alpha) beta) (+ (fma 2 i alpha) beta)) (+ (fma 2 i alpha) beta)))

prune1.8s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 11.8b

localize26.0ms

Local error

Found 4 expressions with local error:

6.9b
(fma (* (* (- beta alpha) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0))))) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0))))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
0.4b
(* (* (- beta alpha) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0))))) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))))
0.3b
(* (- beta alpha) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))))
0.3b
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))

rewrite49.0ms

Algorithm
rewrite-expression-head
Counts
4 → 110
Calls

4 calls. Slowest were:

25.0ms
(* (* (- beta alpha) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0))))) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))))
11.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
10.0ms
(* (- beta alpha) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))))

series304.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

96.0ms
(fma (* (* (- beta alpha) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0))))) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0))))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
89.0ms
(* (- beta alpha) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))))
69.0ms
(* (* (- beta alpha) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0))))) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))))
50.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))

simplify8.4s

Counts
94 → 122
Calls

94 calls. Slowest were:

658.0ms
(* (* (* (* (- beta alpha) (- beta alpha)) (- beta alpha)) (* (* (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0))))) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))))) (* (* (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0))))) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0))))))
507.0ms
(* (* (+ beta alpha) (sqrt (+ (fma i 2 beta) (+ alpha 2.0)))) (sqrt (+ (fma i 2 beta) (+ alpha 2.0))))
419.0ms
(* (* (- (pow beta 3) (pow alpha 3)) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0))))) (sqrt (/ 1 (+ (fma i 2 beta) (+ alpha 2.0)))))

prune3.0s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 11.8b

regimes449.0ms

Accuracy

0% (0.3b remaining)

Error of 12.3b against oracle of 12.0b and baseline of 12.3b

bsearch3.0ms