Average Error: 15.2 → 6.0
Time: 18.3s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.141367796861846 \cdot 10^{+16}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)}{2.0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}{2.0}\\ \end{array}\]

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 < 4.141367796861846e+16

    1. Initial program 0.4

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

    3. Applied div-sub0.4

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

    4. Applied associate-+l-0.4

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

    6. Applied add-exp-log0.4

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

    if 4.141367796861846e+16 < alpha

    1. Initial program 49.4

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

    3. Applied div-sub49.4

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

    4. Applied associate-+l-47.8

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

    6. Applied clear-num47.8

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

    7. Taylor expanded around inf 18.8

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

    8. Simplified18.8

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.141367796861846 \cdot 10^{+16}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)}{2.0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 
(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: 17.2s)Debug log

sample43.0ms

Algorithm
intervals

simplify69.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
69.0ms
(/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)

prune4.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 18.0b

localize32.0ms

Local error

Found 2 expressions with local error:

1.9b
(+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0)
0.0b
(/ (- beta alpha) (+ (+ alpha beta) 2.0))

rewrite45.0ms

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
33.0ms
(+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0)
10.0ms
(/ (- beta alpha) (+ (+ alpha beta) 2.0))

series87.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
54.0ms
(+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0)
33.0ms
(/ (- beta alpha) (+ (+ alpha beta) 2.0))

simplify1.3s

Counts
37 → 49
Calls
37 calls:
Slowest
169.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
135.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
124.0ms
(+ (* (+ alpha beta) (+ alpha beta)) (- (* 2.0 2.0) (* (+ alpha beta) 2.0)))
116.0ms
(- (* 0.5 beta) (+ (* 0.25 (pow beta 2)) (* 0.5 alpha)))
102.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 1) (* 2.0 (/ 1 beta)))

prune507.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 17.1b

localize14.0ms

Local error

Found 4 expressions with local error:

1.9b
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
0.0b
(/ beta (+ (+ alpha beta) 2.0))
0.0b
(/ alpha (+ (+ alpha beta) 2.0))
0.0b
(/ (- (/ beta (+ (+ alpha beta) 2.0)) (- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)) 2.0)

rewrite58.0ms

Algorithm
rewrite-expression-head
Rules
9×*-un-lft-identity
9×add-sqr-sqrt
7×add-cube-cbrt
6×associate-/r*
5×associate-/l*
4×associate-/l/
4×add-log-exp
4×associate-/r/
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×flip--
3×div-inv
3×flip3--
3×frac-2neg
3×clear-num
2×flip-+
2×frac-sub
2×flip3-+
1×difference-of-squares
1×distribute-lft-out--
1×div-sub
1×sub-neg
Counts
4 → 61
Calls
4 calls:
Slowest
49.0ms
(/ (- (/ beta (+ (+ alpha beta) 2.0)) (- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)) 2.0)
5.0ms
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
2.0ms
(/ beta (+ (+ alpha beta) 2.0))
2.0ms
(/ alpha (+ (+ alpha beta) 2.0))

series182.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
60.0ms
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
49.0ms
(/ (- (/ beta (+ (+ alpha beta) 2.0)) (- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)) 2.0)
38.0ms
(/ beta (+ (+ alpha beta) 2.0))
35.0ms
(/ alpha (+ (+ alpha beta) 2.0))

simplify3.0s

Counts
30 → 73
Calls
30 calls:
Slowest
508.0ms
(- (* 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))))
363.0ms
(- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 2.0 (/ 1 alpha)) (* 8.0 (/ 1 (pow alpha 3)))))
341.0ms
(- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 2.0 (/ 1 alpha)) (* 8.0 (/ 1 (pow alpha 3)))))
271.0ms
(- (* 0.5 beta) (+ (* 0.25 (pow beta 2)) (* 0.25 (* beta alpha))))
251.0ms
(- (* 0.5 alpha) (+ (* 0.25 (pow alpha 2)) (* 0.25 (* beta alpha))))

prune841.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 3.5b

localize13.0ms

Local error

Found 4 expressions with local error:

1.9b
(- (/ 1 (/ (+ (+ alpha beta) 2.0) alpha)) 1.0)
0.3b
(/ 1 (/ (+ (+ alpha beta) 2.0) alpha))
0.0b
(/ beta (+ (+ alpha beta) 2.0))
0.0b
(/ (+ (+ alpha beta) 2.0) alpha)

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
7×add-cube-cbrt
7×*-un-lft-identity
7×associate-/r*
7×add-sqr-sqrt
5×add-exp-log
5×pow1
4×add-log-exp
4×div-inv
4×add-cbrt-cube
3×associate-/l*
3×associate-/r/
3×frac-2neg
3×clear-num
2×associate-/l/
2×flip-+
2×flip3-+
1×inv-pow
1×flip--
1×pow-flip
1×flip3--
1×sub-neg
1×rec-exp
Counts
4 → 59
Calls
4 calls:
Slowest
7.0ms
(/ (+ (+ alpha beta) 2.0) alpha)
3.0ms
(- (/ 1 (/ (+ (+ alpha beta) 2.0) alpha)) 1.0)
2.0ms
(/ 1 (/ (+ (+ alpha beta) 2.0) alpha))
2.0ms
(/ beta (+ (+ alpha beta) 2.0))

series99.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
37.0ms
(- (/ 1 (/ (+ (+ alpha beta) 2.0) alpha)) 1.0)
25.0ms
(/ 1 (/ (+ (+ alpha beta) 2.0) alpha))
23.0ms
(/ beta (+ (+ alpha beta) 2.0))
13.0ms
(/ (+ (+ alpha beta) 2.0) alpha)

simplify2.1s

Counts
26 → 71
Calls
26 calls:
Slowest
436.0ms
(+ (pow (+ alpha beta) 3) (pow 2.0 3))
432.0ms
(- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 2.0 (/ 1 alpha)) (* 8.0 (/ 1 (pow alpha 3)))))
361.0ms
(- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 2.0 (/ 1 alpha)) (* 8.0 (/ 1 (pow alpha 3)))))
215.0ms
(- (* 0.5 beta) (+ (* 0.25 (pow beta 2)) (* 0.25 (* beta alpha))))
209.0ms
(- (* 0.5 alpha) (+ (* 0.25 (pow alpha 2)) (* 0.25 (* beta alpha))))

prune724.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 3.5b

localize22.0ms

Local error

Found 4 expressions with local error:

6.5b
(cbrt (/ beta (+ (+ alpha beta) 2.0)))
6.5b
(cbrt (/ beta (+ (+ alpha beta) 2.0)))
6.5b
(cbrt (/ beta (+ (+ alpha beta) 2.0)))
1.9b
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
18×cbrt-prod
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×associate-/r/
4×add-log-exp
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×div-inv
3×flip-+
3×pow1/3
3×flip3-+
3×cbrt-div
1×flip--
1×flip3--
1×sub-neg
Counts
4 → 56
Calls
4 calls:
Slowest
5.0ms
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
1.0ms
(cbrt (/ beta (+ (+ alpha beta) 2.0)))
1.0ms
(cbrt (/ beta (+ (+ alpha beta) 2.0)))
1.0ms
(cbrt (/ beta (+ (+ alpha beta) 2.0)))

series454.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
160.0ms
(cbrt (/ beta (+ (+ alpha beta) 2.0)))
126.0ms
(cbrt (/ beta (+ (+ alpha beta) 2.0)))
102.0ms
(cbrt (/ beta (+ (+ alpha beta) 2.0)))
65.0ms
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)

simplify4.2s

Counts
33 → 68
Calls
33 calls:
Slowest
392.0ms
(- (exp (* 1/3 (+ (log beta) (log 0.5)))) (+ (* 0.16666666666666666 (* beta (exp (* 1/3 (+ (log beta) (log 0.5)))))) (* 0.16666666666666666 (* (exp (* 1/3 (+ (log beta) (log 0.5)))) alpha))))
391.0ms
(- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 2.0 (/ 1 alpha)) (* 8.0 (/ 1 (pow alpha 3)))))
351.0ms
(cbrt (/ beta (+ (pow (+ alpha beta) 3) (pow 2.0 3))))
348.0ms
(cbrt (/ beta (+ (pow (+ alpha beta) 3) (pow 2.0 3))))
343.0ms
(- (exp (* 1/3 (+ (log beta) (log 0.5)))) (+ (* 0.16666666666666666 (* beta (exp (* 1/3 (+ (log beta) (log 0.5)))))) (* 0.16666666666666666 (* (exp (* 1/3 (+ (log beta) (log 0.5)))) alpha))))

prune1.2s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 3.5b

regimes203.0ms

Accuracy

73.6% (3.2b remaining)

Error of 6.0b against oracle of 2.8b and baseline of 14.8b

bsearch97.0ms

end0.0ms

sample2.0s

Algorithm
intervals