Average Error: 16.6 → 6.5
Time: 22.5s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 226966812.55298555:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{{\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}}{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + \left(1.0 \cdot \left(\sqrt[3]{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}} \cdot \left(\sqrt[3]{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}}\right)\right) + 1.0 \cdot 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}{2.0}\\ \end{array}\]

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 < 226966812.55298555

    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 flip3--0.1

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

    8. Applied add-cube-cbrt0.1

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \color{blue}{\left(\left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}}\right) \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}}\right)} \cdot 1.0\right)}}{2.0}\]

    if 226966812.55298555 < alpha

    1. Initial program 49.1

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

    3. Applied div-sub49.1

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

    4. Applied associate-+l-47.5

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

    5. Taylor expanded around inf 19.2

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

    6. Simplified19.2

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 226966812.55298555:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{{\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}}{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + \left(1.0 \cdot \left(\sqrt[3]{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}} \cdot \left(\sqrt[3]{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}}\right)\right) + 1.0 \cdot 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019021 
(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: 21.7s)Debug log

sample61.0ms

Algorithm
intervals
Results
15.0ms35×body1280valid
14.0ms173×body80valid
9.0ms27×body640valid
7.0ms256×pre80true
3.0ms13×body320valid
1.0ms8×body160valid

simplify74.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
74.0ms
(/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 19.3b

localize34.0ms

Local error

Found 2 expressions with local error:

2.3b
(+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0)
0.0b
(/ (- beta alpha) (+ (+ alpha beta) 2.0))

rewrite39.0ms

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
28.0ms
(+ (/ (- 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
45.0ms
(+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0)
42.0ms
(/ (- beta alpha) (+ (+ alpha beta) 2.0))

simplify1.3s

Counts
37 → 49
Calls
37 calls:
Slowest
157.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 1) (* 2.0 (/ 1 beta)))
146.0ms
(/ (* (* (- beta alpha) (- beta alpha)) (- beta alpha)) (* (* (+ (+ alpha beta) 2.0) (+ (+ alpha beta) 2.0)) (+ (+ alpha beta) 2.0)))
136.0ms
(+ (* (+ alpha beta) (+ alpha beta)) (- (* 2.0 2.0) (* (+ alpha beta) 2.0)))
117.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 1) (* 2.0 (/ 1 beta)))
115.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))

prune423.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 18.6b

localize13.0ms

Local error

Found 3 expressions with local error:

2.3b
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
0.0b
(/ alpha (+ (+ alpha beta) 2.0))
0.0b
(/ beta (+ (+ alpha beta) 2.0))

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
6×associate-/r*
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×associate-/r/
3×add-log-exp
3×add-exp-log
3×add-cbrt-cube
3×pow1
2×div-inv
2×flip-+
2×flip3-+
2×frac-2neg
2×clear-num
1×flip--
1×flip3--
1×sub-neg
Counts
3 → 41
Calls
3 calls:
Slowest
9.0ms
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
3.0ms
(/ beta (+ (+ alpha beta) 2.0))
3.0ms
(/ alpha (+ (+ alpha beta) 2.0))

series122.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
59.0ms
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
33.0ms
(/ beta (+ (+ alpha beta) 2.0))
30.0ms
(/ alpha (+ (+ alpha beta) 2.0))

simplify1.7s

Counts
19 → 50
Calls
19 calls:
Slowest
376.0ms
(- (* 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)))))
253.0ms
(- (* 0.5 alpha) (+ (* 0.25 (pow alpha 2)) (* 0.25 (* beta alpha))))
239.0ms
(- (* 0.5 beta) (+ (* 0.25 (pow beta 2)) (* 0.25 (* beta alpha))))
182.0ms
(+ (* (+ alpha beta) (+ alpha beta)) (- (* 2.0 2.0) (* (+ alpha beta) 2.0)))

prune431.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 2.6b

localize47.0ms

Local error

Found 4 expressions with local error:

2.2b
(- (pow (/ alpha (+ (+ alpha beta) 2.0)) 3) (pow 1.0 3))
0.1b
(* (/ alpha (+ (+ alpha beta) 2.0)) (/ alpha (+ (+ alpha beta) 2.0)))
0.1b
(pow (/ alpha (+ (+ alpha beta) 2.0)) 3)
0.0b
(/ alpha (+ (+ alpha beta) 2.0))

rewrite29.0ms

Algorithm
rewrite-expression-head
Rules
15×add-sqr-sqrt
11×*-un-lft-identity
10×associate-/r/
10×pow1
9×add-cube-cbrt
7×add-exp-log
7×cube-prod
7×add-cbrt-cube
7×unpow-prod-down
6×add-log-exp
6×associate-*l*
6×associate-*r*
5×div-inv
5×flip-+
5×flip3-+
3×difference-of-squares
3×associate-/r*
1×rem-cube-cbrt
1×distribute-lft-out--
1×flip--
1×cbrt-unprod
1×pow-prod-up
1×*-commutative
1×associate-*r/
1×prod-exp
1×pow-exp
1×associate-*l/
1×pow-prod-down
1×unpow3
1×diff-log
1×flip3--
1×pow-plus
1×frac-2neg
1×pow-pow
1×sub-neg
1×clear-num
1×difference-cubes
1×pow-to-exp
1×frac-times
1×cube-mult
1×pow2
1×cube-div
Counts
4 → 87
Calls
4 calls:
Slowest
12.0ms
(- (pow (/ alpha (+ (+ alpha beta) 2.0)) 3) (pow 1.0 3))
9.0ms
(* (/ alpha (+ (+ alpha beta) 2.0)) (/ alpha (+ (+ alpha beta) 2.0)))
3.0ms
(/ alpha (+ (+ alpha beta) 2.0))
3.0ms
(pow (/ alpha (+ (+ alpha beta) 2.0)) 3)

series206.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
96.0ms
(- (pow (/ alpha (+ (+ alpha beta) 2.0)) 3) (pow 1.0 3))
52.0ms
(pow (/ alpha (+ (+ alpha beta) 2.0)) 3)
32.0ms
(* (/ alpha (+ (+ alpha beta) 2.0)) (/ alpha (+ (+ alpha beta) 2.0)))
25.0ms
(/ alpha (+ (+ alpha beta) 2.0))

simplify6.1s

Counts
63 → 99
Calls
63 calls:
Slowest
611.0ms
(pow (/ alpha (- (* (+ alpha beta) (+ alpha beta)) (* 2.0 2.0))) 3)
559.0ms
(pow (/ alpha (- (* (+ alpha beta) (+ alpha beta)) (* 2.0 2.0))) 3)
409.0ms
(- (* 24.0 (/ 1 (pow alpha 2))) (+ (* 6.0 (/ 1 alpha)) (* 80.0 (/ 1 (pow alpha 3)))))
401.0ms
(- (* 0.25 (pow alpha 2)) (+ (* 0.25 (pow alpha 3)) (* 0.25 (* beta (pow alpha 2)))))
373.0ms
(- (* 24.0 (/ 1 (pow alpha 2))) (+ (* 6.0 (/ 1 alpha)) (* 80.0 (/ 1 (pow alpha 3)))))

prune1.5s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 2.6b

localize25.0ms

Local error

Found 4 expressions with local error:

4.9b
(cbrt (/ alpha (+ (+ alpha beta) 2.0)))
4.9b
(cbrt (/ alpha (+ (+ alpha beta) 2.0)))
4.9b
(cbrt (/ alpha (+ (+ alpha beta) 2.0)))
2.2b
(- (pow (/ alpha (+ (+ alpha beta) 2.0)) 3) (pow 1.0 3))

rewrite25.0ms

Algorithm
rewrite-expression-head
Rules
18×cbrt-prod
13×add-sqr-sqrt
9×*-un-lft-identity
7×add-cube-cbrt
6×add-log-exp
6×associate-/r/
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×difference-of-squares
3×div-inv
3×flip-+
3×pow1/3
3×flip3-+
3×cbrt-div
1×distribute-lft-out--
1×flip--
1×cube-prod
1×diff-log
1×flip3--
1×unpow-prod-down
1×sub-neg
1×difference-cubes
Counts
4 → 62
Calls
4 calls:
Slowest
21.0ms
(- (pow (/ alpha (+ (+ alpha beta) 2.0)) 3) (pow 1.0 3))
1.0ms
(cbrt (/ alpha (+ (+ alpha beta) 2.0)))
1.0ms
(cbrt (/ alpha (+ (+ alpha beta) 2.0)))
1.0ms
(cbrt (/ alpha (+ (+ alpha beta) 2.0)))

series424.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
134.0ms
(cbrt (/ alpha (+ (+ alpha beta) 2.0)))
119.0ms
(cbrt (/ alpha (+ (+ alpha beta) 2.0)))
105.0ms
(cbrt (/ alpha (+ (+ alpha beta) 2.0)))
66.0ms
(- (pow (/ alpha (+ (+ alpha beta) 2.0)) 3) (pow 1.0 3))

simplify5.6s

Counts
43 → 74
Calls
43 calls:
Slowest
495.0ms
(+ (sqrt (pow (/ alpha (+ (+ alpha beta) 2.0)) 3)) (sqrt (pow 1.0 3)))
451.0ms
(cbrt (/ alpha (+ (pow (+ alpha beta) 3) (pow 2.0 3))))
412.0ms
(- (* 24.0 (/ 1 (pow alpha 2))) (+ (* 6.0 (/ 1 alpha)) (* 80.0 (/ 1 (pow alpha 3)))))
399.0ms
(- (exp (* 1/3 (+ (log alpha) (log 0.5)))) (+ (* 0.16666666666666666 (* (exp (* 1/3 (+ (log alpha) (log 0.5)))) alpha)) (* 0.16666666666666666 (* (exp (* 1/3 (+ (log alpha) (log 0.5)))) beta))))
376.0ms
(- (exp (* 1/3 (+ (log alpha) (log 0.5)))) (+ (* 0.16666666666666666 (* (exp (* 1/3 (+ (log alpha) (log 0.5)))) alpha)) (* 0.16666666666666666 (* (exp (* 1/3 (+ (log alpha) (log 0.5)))) beta))))

prune1.2s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 2.6b

regimes121.0ms

Accuracy

74.8% (3.2b remaining)

Error of 6.5b against oracle of 3.3b and baseline of 16.0b

bsearch93.0ms

end0.0ms

sample1.9s

Algorithm
intervals
Results
656.0ms5789×body80valid
462.0ms848×body1280valid
249.0ms721×body640valid
223.0ms8000×pre80true
145.0ms419×body320valid
69.0ms223×body160valid