Average Error: 23.3 → 12.6
Time: 33.7s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\sqrt[3]{(\left(\left(\beta - \alpha\right) \cdot \frac{1}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}\right) \cdot \left(\frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_* \cdot \left((\left(\left(\beta - \alpha\right) \cdot \frac{1}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}\right) \cdot \left(\frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_* \cdot (\left(\left(\beta - \alpha\right) \cdot \frac{1}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}\right) \cdot \left(\frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_*\right)}}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.3

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

  2. Simplified12.6

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

  4. Applied div-inv12.6

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

  6. Applied add-cbrt-cube12.6

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

  7. Final simplification12.6

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

Reproduce

herbie shell --seed 2019002 +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: 32.0s)Debug log

sample80.0ms

Algorithm
intervals

simplify643.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

643.0ms
(/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0)

prune25.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 12.3b

localize67.0ms

Local error

Found 3 expressions with local error:

1.1b
(fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
0.0b
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))
0.0b
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
3 → 81
Calls

3 calls. Slowest were:

13.0ms
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))
11.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
0.0ms
(fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)

series245.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

121.0ms
(fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
79.0ms
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))
45.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))

simplify3.4s

Counts
75 → 90
Calls

75 calls. Slowest were:

409.0ms
(+ (* (+ beta 2.0) (+ beta 2.0)) (- (* (fma 2 i alpha) (fma 2 i alpha)) (* (+ beta 2.0) (fma 2 i alpha))))
239.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
210.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))

prune1.3s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 12.1b

localize14.0ms

Local error

Found 4 expressions with local error:

2.3b
(fma (* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
0.2b
(* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))
0.0b
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
0.0b
(/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

6.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
5.0ms
(* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))
3.0ms
(/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))

series289.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

102.0ms
(fma (* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
80.0ms
(* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))
54.0ms
(/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))
53.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))

simplify4.8s

Counts
68 → 105
Calls

68 calls. Slowest were:

789.0ms
(+ (* (+ beta 2.0) (+ beta 2.0)) (- (* (fma 2 i alpha) (fma 2 i alpha)) (* (+ beta 2.0) (fma 2 i alpha))))
419.0ms
(* (* (* (- beta alpha) (- beta alpha)) (- beta alpha)) (* (* (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))))
418.0ms
(+ (* (+ beta 2.0) (+ beta 2.0)) (- (* (fma 2 i alpha) (fma 2 i alpha)) (* (+ beta 2.0) (fma 2 i alpha))))

prune1.6s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 12.1b

localize18.0ms

Local error

Found 4 expressions with local error:

2.3b
(fma (* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
2.3b
(fma (* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
2.3b
(fma (* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
0.2b
(* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

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

series417.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

123.0ms
(fma (* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
114.0ms
(fma (* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
111.0ms
(fma (* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
68.0ms
(* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))

simplify4.7s

Counts
32 → 71
Calls

32 calls. Slowest were:

542.0ms
(+ (* (+ beta 2.0) (+ beta 2.0)) (- (* (fma 2 i alpha) (fma 2 i alpha)) (* (+ beta 2.0) (fma 2 i alpha))))
452.0ms
(* (* (* (- beta alpha) (- beta alpha)) (- beta alpha)) (* (* (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))))
356.0ms
(* (- (pow beta 3) (pow alpha 3)) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))

prune1.2s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 12.1b

localize29.0ms

Local error

Found 4 expressions with local error:

7.9b
(fma (* (* (- beta alpha) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
2.3b
(fma (* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
2.3b
(fma (* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
0.3b
(* (* (- beta alpha) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

19.0ms
(* (* (- beta alpha) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))))
0.0ms
(fma (* (* (- beta alpha) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
0.0ms
(fma (* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)

series342.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

100.0ms
(fma (* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
99.0ms
(fma (* (- beta alpha) (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
94.0ms
(fma (* (* (- beta alpha) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
49.0ms
(* (* (- beta alpha) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))))

simplify7.7s

Counts
44 → 79
Calls

44 calls. Slowest were:

753.0ms
(* (* (* (* (- beta alpha) (- beta alpha)) (- beta alpha)) (* (* (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))))) (* (* (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))))
581.0ms
(* (* (- (* beta beta) (* alpha alpha)) (sqrt 1)) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha)))))
524.0ms
(* (* (- (* beta beta) (* alpha alpha)) (sqrt (/ 1 (+ (+ beta 2.0) (fma 2 i alpha))))) (sqrt 1))

prune1.6s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 12.1b

regimes707.0ms

Accuracy

0% (0.3b remaining)

Error of 12.6b against oracle of 12.4b and baseline of 12.6b

bsearch3.0ms

end0.0ms

sample2.9s

Algorithm
intervals