Average Error: 16.3 → 5.7
Time: 18.1s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3166370434.8112526:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2.0 + \left(\beta + \alpha\right)} \cdot \beta - \frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 3166370434.8112526

    1. Initial program 0.1

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub0.1

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    4. Applied associate-+l-0.1

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

    6. Applied add-exp-log0.1

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}}}{2.0}\]
    7. Using strategy rm

    8. Applied rem-exp-log0.1

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]

    if 3166370434.8112526 < alpha

    1. Initial program 50.0

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub50.0

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    4. Applied associate-+l-48.5

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

    6. Applied div-inv48.5

      \[\leadsto \frac{\color{blue}{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

    7. Taylor expanded around -inf 17.3

      \[\leadsto \frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2.0}\]

    8. Simplified17.3

      \[\leadsto \frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3166370434.8112526:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2.0 + \left(\beta + \alpha\right)} \cdot \beta - \frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

Details

Time bar (total: 17.4s)Debug log

sample56.0ms

Algorithm
intervals
Results
256×(pre true 80)
195×(body real 80)
23×(body real 1280)
20×(body real 640)
9×(body real 320)
9×(body real 160)

simplify67.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
66.0ms
(/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 13.9b

localize17.0ms

Local error

Found 2 expressions with local error:

2.1b
(+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0)
0.0b
(/ (- beta alpha) (+ (+ alpha beta) 2.0))

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
10×add-cube-cbrt
10×*-un-lft-identity
10×add-sqr-sqrt
9×times-frac
4×add-exp-log
4×add-cbrt-cube
3×associate-/l*
3×associate-/r*
2×associate-/l/
2×add-log-exp
2×flip-+
2×associate-/r/
2×div-sub
2×flip3-+
2×pow1
1×flip--
1×div-inv
1×associate-+l-
1×div-exp
1×flip3--
1×frac-2neg
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
2 → 43
Calls
2 calls:
Slowest
14.0ms
(+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0)
5.0ms
(/ (- beta alpha) (+ (+ alpha beta) 2.0))

series85.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
60.0ms
(+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0)
25.0ms
(/ (- beta alpha) (+ (+ alpha beta) 2.0))

simplify1.7s

Counts
37 → 49
Calls
37 calls:
Slowest
535.0ms
(+ (* (+ alpha beta) (+ alpha beta)) (- (* 2.0 2.0) (* (+ alpha beta) 2.0)))
171.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
129.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
119.0ms
(- (* 0.5 beta) (+ (* 0.25 (pow beta 2)) (* 0.5 alpha)))
119.0ms
(/ (* (* (- beta alpha) (- beta alpha)) (- beta alpha)) (* (* (+ (+ alpha beta) 2.0) (+ (+ alpha beta) 2.0)) (+ (+ alpha beta) 2.0)))

prune438.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 13.5b

localize23.0ms

Local error

Found 4 expressions with local error:

2.1b
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
0.0b
(/ alpha (+ (+ alpha beta) 2.0))
0.0b
(/ beta (+ (+ alpha beta) 2.0))
0.0b
(- (/ beta (+ (+ alpha beta) 2.0)) (- (/ alpha (+ (+ alpha beta) 2.0)) 1.0))

rewrite42.0ms

Algorithm
rewrite-expression-head
Rules
8×*-un-lft-identity
8×add-sqr-sqrt
6×add-log-exp
6×add-cube-cbrt
6×associate-/r*
4×associate-/r/
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×flip--
3×flip3--
3×sub-neg
2×associate--r+
2×div-inv
2×flip-+
2×frac-sub
2×flip3-+
2×frac-2neg
2×clear-num
1×difference-of-squares
1×associate--r-
1×distribute-lft-out--
1×diff-log
Counts
4 → 60
Calls
4 calls:
Slowest
25.0ms
(- (/ beta (+ (+ alpha beta) 2.0)) (- (/ alpha (+ (+ alpha beta) 2.0)) 1.0))
10.0ms
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
3.0ms
(/ beta (+ (+ alpha beta) 2.0))
3.0ms
(/ alpha (+ (+ alpha beta) 2.0))

series176.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
62.0ms
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
39.0ms
(/ alpha (+ (+ alpha beta) 2.0))
38.0ms
(/ beta (+ (+ alpha beta) 2.0))
37.0ms
(- (/ beta (+ (+ alpha beta) 2.0)) (- (/ alpha (+ (+ alpha beta) 2.0)) 1.0))

simplify3.2s

Counts
32 → 72
Calls
32 calls:
Slowest
588.0ms
(- (* beta (+ (/ alpha (+ (+ alpha beta) 2.0)) 1.0)) (* (+ (+ alpha beta) 2.0) (- (* (/ alpha (+ (+ alpha beta) 2.0)) (/ alpha (+ (+ alpha beta) 2.0))) (* 1.0 1.0))))
384.0ms
(- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 2.0 (/ 1 alpha)) (* 8.0 (/ 1 (pow alpha 3)))))
371.0ms
(- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 2.0 (/ 1 alpha)) (* 8.0 (/ 1 (pow alpha 3)))))
276.0ms
(- (* 0.5 alpha) (+ (* 0.25 (pow alpha 2)) (* 0.25 (* beta alpha))))
230.0ms
(- (* 0.5 beta) (+ (* 0.25 (pow beta 2)) (* 0.25 (* beta alpha))))

prune724.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 1.3b

localize11.0ms

Local error

Found 4 expressions with local error:

2.1b
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
0.1b
(* beta (/ 1 (+ (+ alpha beta) 2.0)))
0.0b
(/ 1 (+ (+ alpha beta) 2.0))
0.0b
(/ alpha (+ (+ alpha beta) 2.0))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×associate-/r/
6×associate-*r*
6×associate-/r*
5×add-exp-log
5×pow1
4×add-log-exp
4×add-cbrt-cube
3×div-inv
3×flip-+
3×flip3-+
2×frac-2neg
2×clear-num
1×inv-pow
1×flip--
1×pow-flip
1×*-commutative
1×un-div-inv
1×associate-*r/
1×flip3--
1×sub-neg
1×rec-exp
Counts
4 → 60
Calls
4 calls:
Slowest
5.0ms
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
2.0ms
(/ 1 (+ (+ alpha beta) 2.0))
2.0ms
(/ alpha (+ (+ alpha beta) 2.0))
1.0ms
(* beta (/ 1 (+ (+ alpha beta) 2.0)))

series146.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
66.0ms
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
32.0ms
(/ alpha (+ (+ alpha beta) 2.0))
26.0ms
(* beta (/ 1 (+ (+ alpha beta) 2.0)))
22.0ms
(/ 1 (+ (+ alpha beta) 2.0))

simplify2.1s

Counts
31 → 72
Calls
31 calls:
Slowest
389.0ms
(- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 2.0 (/ 1 alpha)) (* 8.0 (/ 1 (pow alpha 3)))))
375.0ms
(- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 2.0 (/ 1 alpha)) (* 8.0 (/ 1 (pow alpha 3)))))
265.0ms
(- (* 0.5 beta) (+ (* 0.25 (pow beta 2)) (* 0.25 (* beta alpha))))
191.0ms
(+ (* (+ alpha beta) (+ alpha beta)) (- (* 2.0 2.0) (* (+ alpha beta) 2.0)))
171.0ms
(- (* 0.5 alpha) (+ (* 0.25 (pow alpha 2)) (* 0.25 (* beta alpha))))

prune713.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 1.3b

localize9.0ms

Local error

Found 4 expressions with local error:

11.9b
(log (- (/ beta (+ (+ alpha beta) 2.0)) (- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)))
2.1b
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
0.5b
(exp (log (- (/ beta (+ (+ alpha beta) 2.0)) (- (/ alpha (+ (+ alpha beta) 2.0)) 1.0))))
0.0b
(/ alpha (+ (+ alpha beta) 2.0))

rewrite39.0ms

Algorithm
rewrite-expression-head
Rules
12×*-un-lft-identity
12×add-sqr-sqrt
10×log-prod
8×add-cube-cbrt
8×log-div
6×pow1
5×flip--
5×add-exp-log
5×exp-sum
5×flip3--
4×add-log-exp
4×frac-sub
4×exp-diff
4×add-cbrt-cube
4×exp-prod
3×associate-/r*
2×difference-of-squares
2×distribute-lft-out--
2×log-pow
2×associate-/r/
1×rem-exp-log
1×div-inv
1×flip-+
1×rem-log-exp
1×flip3-+
1×frac-2neg
1×sub-neg
1×clear-num
Counts
4 → 65
Calls
4 calls:
Slowest
17.0ms
(log (- (/ beta (+ (+ alpha beta) 2.0)) (- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)))
11.0ms
(exp (log (- (/ beta (+ (+ alpha beta) 2.0)) (- (/ alpha (+ (+ alpha beta) 2.0)) 1.0))))
8.0ms
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
2.0ms
(/ alpha (+ (+ alpha beta) 2.0))

series209.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
80.0ms
(log (- (/ beta (+ (+ alpha beta) 2.0)) (- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)))
58.0ms
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
39.0ms
(exp (log (- (/ beta (+ (+ alpha beta) 2.0)) (- (/ alpha (+ (+ alpha beta) 2.0)) 1.0))))
31.0ms
(/ alpha (+ (+ alpha beta) 2.0))

simplify4.2s

Counts
41 → 77
Calls
41 calls:
Slowest
551.0ms
(exp (log (- (* beta (+ (/ alpha (+ (+ alpha beta) 2.0)) 1.0)) (* (+ (+ alpha beta) 2.0) (- (* (/ alpha (+ (+ alpha beta) 2.0)) (/ alpha (+ (+ alpha beta) 2.0))) (* 1.0 1.0))))))
471.0ms
(log (- (* beta (+ (/ alpha (+ (+ alpha beta) 2.0)) 1.0)) (* (+ (+ alpha beta) 2.0) (- (* (/ alpha (+ (+ alpha beta) 2.0)) (/ alpha (+ (+ alpha beta) 2.0))) (* 1.0 1.0)))))
402.0ms
(- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 2.0 (/ 1 alpha)) (* 8.0 (/ 1 (pow alpha 3)))))
363.0ms
(- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 2.0 (/ 1 alpha)) (* 8.0 (/ 1 (pow alpha 3)))))
296.0ms
(- (* 0.5 alpha) (+ (* 0.25 (pow alpha 2)) (* 0.25 (* beta alpha))))

prune859.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 1.3b

regimes138.0ms

Accuracy

77.6% (2.9b remaining)

Error of 5.7b against oracle of 2.8b and baseline of 15.8b

bsearch65.0ms

end0.0ms

sample2.3s

Algorithm
intervals
Results
8000×(pre true 80)
5831×(body real 80)
830×(body real 1280)
727×(body real 640)
419×(body real 320)
193×(body real 160)