Average Error: 53.1 → 38.1
Time: 1.4m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.3658131024756376 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{(\left(i + \left(\alpha + \beta\right)\right) \cdot i + \left(\alpha \cdot \beta\right))_*}{(i \cdot 2 + \left(\alpha + \beta\right))_*}}{\sqrt{(\left((i \cdot 2 + \left(\alpha + \beta\right))_*\right) \cdot \left((i \cdot 2 + \left(\alpha + \beta\right))_*\right) + \left(-1.0\right))_*}} \cdot \frac{\frac{(\left(\alpha + \beta\right) \cdot i + \left(i \cdot i\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{\sqrt{(\left((2 \cdot i + \left(\alpha + \beta\right))_*\right) \cdot \left((2 \cdot i + \left(\alpha + \beta\right))_*\right) + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if beta < 1.3658131024756376e+144

    1. Initial program 51.1

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

    3. Applied add-sqr-sqrt51.1

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

    4. Applied times-frac35.6

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

    5. Applied times-frac35.7

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

    6. Simplified35.7

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

    7. Simplified35.7

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

    if 1.3658131024756376e+144 < beta

    1. Initial program 62.5

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

    2. Taylor expanded around -inf 49.4

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification38.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.3658131024756376 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{(\left(i + \left(\alpha + \beta\right)\right) \cdot i + \left(\alpha \cdot \beta\right))_*}{(i \cdot 2 + \left(\alpha + \beta\right))_*}}{\sqrt{(\left((i \cdot 2 + \left(\alpha + \beta\right))_*\right) \cdot \left((i \cdot 2 + \left(\alpha + \beta\right))_*\right) + \left(-1.0\right))_*}} \cdot \frac{\frac{(\left(\alpha + \beta\right) \cdot i + \left(i \cdot i\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{\sqrt{(\left((2 \cdot i + \left(\alpha + \beta\right))_*\right) \cdot \left((2 \cdot i + \left(\alpha + \beta\right))_*\right) + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))

Details

Time bar (total: 1.4m)Debug log

sample63.0ms

Algorithm
intervals

simplify258.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

258.0ms
(/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))

prune14.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 51.3b

localize62.0ms

Local error

Found 4 expressions with local error:

45.5b
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
32.4b
(/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
0.1b
(* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
0.0b
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))

rewrite119.0ms

Algorithm
rewrite-expression-head
Counts
4 → 178
Calls

4 calls. Slowest were:

57.0ms
(/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
24.0ms
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
23.0ms
(* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))

series1.0s

Counts
4 → 12
Calls

4 calls. Slowest were:

721.0ms
(/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0))
173.0ms
(/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))
103.0ms
(* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
37.0ms
(* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))

simplify44.2s

Counts
191 → 190
Calls

191 calls. Slowest were:

1.0s
(* (* (* (* i (+ (+ alpha beta) i)) (* i (+ (+ alpha beta) i))) (* i (+ (+ alpha beta) i))) (* (* (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))))
846.0ms
(/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (pow (+ alpha beta) 3) (pow (* 2 i) 3)))) (sqrt (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))
802.0ms
(* (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))) (- (* (+ alpha beta) (+ alpha beta)) (* (* 2 i) (* 2 i))))

prune3.0s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 32.3b

localize102.0ms

Local error

Found 4 expressions with local error:

33.9b
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))
33.9b
(sqrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))
30.8b
(/ (fma (+ i (+ alpha beta)) i (* beta alpha)) (fma i 2 (+ alpha beta)))
30.4b
(/ (fma (+ alpha beta) i (* i i)) (fma 2 i (+ alpha beta)))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 86
Calls

4 calls. Slowest were:

6.0ms
(/ (fma (+ alpha beta) i (* i i)) (fma 2 i (+ alpha beta)))
6.0ms
(/ (fma (+ i (+ alpha beta)) i (* beta alpha)) (fma i 2 (+ alpha beta)))
2.0ms
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))

series341.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

113.0ms
(sqrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))
92.0ms
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))
80.0ms
(/ (fma (+ alpha beta) i (* i i)) (fma 2 i (+ alpha beta)))
56.0ms
(/ (fma (+ i (+ alpha beta)) i (* beta alpha)) (fma i 2 (+ alpha beta)))

simplify8.7s

Counts
70 → 98
Calls

70 calls. Slowest were:

623.0ms
(sqrt (* (cbrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0))) (cbrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))))
619.0ms
(sqrt (sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0))))
602.0ms
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))

prune2.3s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 32.3b

localize84.0ms

Local error

Found 4 expressions with local error:

33.9b
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))
33.9b
(sqrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))
31.7b
(cbrt (* (+ alpha i) (+ i beta)))
31.7b
(cbrt (* (+ alpha i) (+ i beta)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

4.0ms
(cbrt (* (+ alpha i) (+ i beta)))
2.0ms
(cbrt (* (+ alpha i) (+ i beta)))
2.0ms
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))

series732.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

290.0ms
(cbrt (* (+ alpha i) (+ i beta)))
253.0ms
(cbrt (* (+ alpha i) (+ i beta)))
102.0ms
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))
87.0ms
(sqrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))

simplify9.2s

Counts
38 → 78
Calls

38 calls. Slowest were:

639.0ms
(sqrt (sqrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0))))
628.0ms
(sqrt (sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0))))
610.0ms
(sqrt (* (cbrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0))) (cbrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))))

prune2.1s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 32.3b

localize49.0ms

Local error

Found 4 expressions with local error:

33.9b
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))
33.9b
(sqrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))
31.2b
(cbrt (fma (+ alpha beta) i (* i i)))
31.2b
(cbrt (fma (+ alpha beta) i (* i i)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

2.0ms
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))
2.0ms
(sqrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))
1.0ms
(cbrt (fma (+ alpha beta) i (* i i)))

series568.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

192.0ms
(cbrt (fma (+ alpha beta) i (* i i)))
152.0ms
(cbrt (fma (+ alpha beta) i (* i i)))
117.0ms
(sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))
107.0ms
(sqrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0)))

simplify6.0s

Counts
26 → 66
Calls

26 calls. Slowest were:

703.0ms
(sqrt (sqrt (fma (fma i 2 (+ alpha beta)) (fma i 2 (+ alpha beta)) (- 1.0))))
668.0ms
(sqrt (sqrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0))))
596.0ms
(sqrt (* (cbrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0))) (cbrt (fma (fma 2 i (+ alpha beta)) (fma 2 i (+ alpha beta)) (- 1.0)))))

prune1.8s

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 32.3b

regimes202.0ms

Accuracy

41.6% (1.4b remaining)

Error of 38.1b against oracle of 36.6b and baseline of 39.1b

bsearch112.0ms

end0.0ms

sample2.5s

Algorithm
intervals