Average Error: 16.0 → 5.8
Time: 51.8s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4048279473663.132:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{(\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) \cdot \left(\sqrt[3]{(e^{\log_* (1 + {\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3})} - 1)^*}\right) + \left(-{1.0}^{3}\right))_*}{\left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right) + \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - (\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(\frac{-2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 4048279473663.132

    1. Initial program 0.2

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

    3. Applied div-sub0.2

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

    4. Applied associate-+l-0.2

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

    6. Applied flip3--0.3

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

    8. Applied add-cube-cbrt0.3

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

    9. Applied fma-neg0.3

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

    10. Simplified0.3

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

    12. Applied expm1-log1p-u0.3

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

    if 4048279473663.132 < alpha

    1. Initial program 50.1

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

    3. Applied div-sub50.1

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

    4. Applied associate-+l-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. Taylor expanded around inf 17.9

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

    6. Simplified17.9

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

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4048279473663.132:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{(\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) \cdot \left(\sqrt[3]{(e^{\log_* (1 + {\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3})} - 1)^*}\right) + \left(-{1.0}^{3}\right))_*}{\left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right) + \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - (\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(\frac{-2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(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: 50.5s)Debug log

sample47.0ms

Algorithm
intervals
Results
11.0ms187×body80valid
6.0ms27×body640valid
6.0ms256×pre80true
5.0ms19×body1280valid
3.0ms15×body320valid
1.0ms8×body160valid

simplify63.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
63.0ms
(/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 16.3b

localize35.0ms

Local error

Found 2 expressions with local error:

2.7b
(+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0)
0.0b
(/ (- beta alpha) (+ (+ alpha beta) 2.0))

rewrite38.0ms

Algorithm
rewrite-expression-head
Rules
18×times-frac
17×add-cube-cbrt
17×*-un-lft-identity
17×add-sqr-sqrt
15×fma-def
4×associate-/r/
4×add-exp-log
4×add-cbrt-cube
3×associate-/l*
3×flip-+
3×flip3-+
3×associate-/r*
2×associate-/l/
2×add-log-exp
2×div-inv
2×log1p-expm1-u
2×div-sub
2×pow1
2×expm1-log1p-u
1×flip--
1×associate-+l-
1×div-exp
1×flip3--
1×frac-2neg
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
2 → 62
Calls
2 calls:
Slowest
32.0ms
(+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0)
5.0ms
(/ (- beta alpha) (+ (+ alpha beta) 2.0))

series74.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
41.0ms
(+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0)
33.0ms
(/ (- beta alpha) (+ (+ alpha beta) 2.0))

simplify2.5s

Counts
51 → 68
Calls
51 calls:
Slowest
488.0ms
(/ (- beta alpha) (+ (pow (+ alpha beta) 3) (pow 2.0 3)))
205.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
187.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 1) (* 2.0 (/ 1 beta)))
183.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
154.0ms
(/ (* (* (- beta alpha) (- beta alpha)) (- beta alpha)) (* (* (+ (+ alpha beta) 2.0) (+ (+ alpha beta) 2.0)) (+ (+ alpha beta) 2.0)))

prune696.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 15.4b

localize24.0ms

Local error

Found 3 expressions with local error:

2.7b
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
0.0b
(/ alpha (+ (+ alpha beta) 2.0))
0.0b
(/ beta (+ (+ alpha beta) 2.0))

rewrite20.0ms

Algorithm
rewrite-expression-head
Rules
6×add-cube-cbrt
6×associate-/r/
6×fma-neg
6×*-un-lft-identity
6×associate-/r*
6×add-sqr-sqrt
3×add-log-exp
3×div-inv
3×log1p-expm1-u
3×flip-+
3×add-exp-log
3×add-cbrt-cube
3×flip3-+
3×pow1
3×expm1-log1p-u
2×frac-2neg
2×clear-num
1×flip--
1×flip3--
1×sub-neg
Counts
3 → 53
Calls
3 calls:
Slowest
13.0ms
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
3.0ms
(/ alpha (+ (+ alpha beta) 2.0))
3.0ms
(/ beta (+ (+ alpha beta) 2.0))

series118.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
63.0ms
(- (/ alpha (+ (+ alpha beta) 2.0)) 1.0)
30.0ms
(/ beta (+ (+ alpha beta) 2.0))
24.0ms
(/ alpha (+ (+ alpha beta) 2.0))

simplify2.1s

Counts
23 → 62
Calls
23 calls:
Slowest
450.0ms
(/ alpha (+ (pow (+ alpha beta) 3) (pow 2.0 3)))
380.0ms
(- (* 0.5 beta) (+ (* 0.25 (pow beta 2)) (* 0.25 (* beta alpha))))
252.0ms
(- (* 0.5 alpha) (+ (* 0.25 (pow alpha 2)) (* 0.25 (* beta alpha))))
178.0ms
(+ (* (+ alpha beta) (+ alpha beta)) (- (* 2.0 2.0) (* (+ alpha beta) 2.0)))
175.0ms
(- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 2.0 (/ 1 alpha)) (* 8.0 (/ 1 (pow alpha 3)))))

prune600.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 4.0b

localize33.0ms

Local error

Found 4 expressions with local error:

2.7b
(- (pow (/ alpha (+ (+ alpha beta) 2.0)) 3) (pow 1.0 3))
0.1b
(pow (/ alpha (+ (+ alpha beta) 2.0)) 3)
0.1b
(* (/ alpha (+ (+ alpha beta) 2.0)) (/ alpha (+ (+ alpha beta) 2.0)))
0.0b
(/ alpha (+ (+ alpha beta) 2.0))

rewrite35.0ms

Algorithm
rewrite-expression-head
Rules
85×prod-diff
50×add-sqr-sqrt
46×*-un-lft-identity
44×add-cube-cbrt
43×cube-prod
43×unpow-prod-down
34×associate-/r/
24×unpow3
24×cube-mult
17×div-inv
17×flip-+
17×fma-neg
17×flip3-+
10×pow1
7×add-exp-log
7×add-cbrt-cube
6×add-log-exp
6×associate-*l*
6×associate-*r*
4×log1p-expm1-u
4×expm1-log1p-u
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×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×pow2
1×cube-div
Counts
4 → 197
Calls
4 calls:
Slowest
25.0ms
(- (pow (/ alpha (+ (+ alpha beta) 2.0)) 3) (pow 1.0 3))
4.0ms
(* (/ alpha (+ (+ alpha beta) 2.0)) (/ alpha (+ (+ alpha beta) 2.0)))
2.0ms
(/ alpha (+ (+ alpha beta) 2.0))
2.0ms
(pow (/ alpha (+ (+ alpha beta) 2.0)) 3)

series187.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
78.0ms
(- (pow (/ alpha (+ (+ alpha beta) 2.0)) 3) (pow 1.0 3))
46.0ms
(* (/ alpha (+ (+ alpha beta) 2.0)) (/ alpha (+ (+ alpha beta) 2.0)))
41.0ms
(pow (/ alpha (+ (+ alpha beta) 2.0)) 3)
22.0ms
(/ alpha (+ (+ alpha beta) 2.0))

simplify29.2s

Counts
249 → 209
Calls
249 calls:
Slowest
594.0ms
(fma (- (cbrt (pow 1.0 3))) (* (cbrt (pow 1.0 3)) (cbrt (pow 1.0 3))) (* (cbrt (pow 1.0 3)) (* (cbrt (pow 1.0 3)) (cbrt (pow 1.0 3)))))
521.0ms
(pow (/ alpha (- (* (+ alpha beta) (+ alpha beta)) (* 2.0 2.0))) 3)
501.0ms
(/ alpha (+ (pow (+ alpha beta) 3) (pow 2.0 3)))
454.0ms
(pow (/ alpha (- (* (+ alpha beta) (+ alpha beta)) (* 2.0 2.0))) 3)
451.0ms
(pow (/ alpha (- (* (+ alpha beta) (+ alpha beta)) (* 2.0 2.0))) 3)

prune3.1s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 4.0b

localize62.0ms

Local error

Found 4 expressions with local error:

24.2b
(cbrt (pow (/ alpha (+ (+ alpha beta) 2.0)) 3))
2.7b
(fma (* (/ alpha (+ 2.0 (+ beta alpha))) (/ alpha (+ 2.0 (+ beta alpha)))) (cbrt (pow (/ alpha (+ (+ alpha beta) 2.0)) 3)) (- (pow 1.0 3)))
0.1b
(pow (/ alpha (+ (+ alpha beta) 2.0)) 3)
0.1b
(* (/ alpha (+ (+ alpha beta) 2.0)) (/ alpha (+ (+ alpha beta) 2.0)))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
17×cbrt-prod
12×associate-/r/
12×cube-prod
12×unpow-prod-down
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
10×pow1
7×add-exp-log
7×add-cbrt-cube
6×div-inv
6×flip-+
6×associate-*l*
6×associate-*r*
6×flip3-+
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
2×unpow3
2×cube-mult
2×cube-div
1×rem-cube-cbrt
1×fma-udef
1×cbrt-unprod
1×rem-cbrt-cube
1×pow-prod-up
1×*-commutative
1×associate-*r/
1×prod-exp
1×pow-exp
1×associate-*l/
1×pow-prod-down
1×pow1/3
1×pow-plus
1×pow-pow
1×cbrt-div
1×pow-to-exp
1×frac-times
1×pow2
Counts
4 → 98
Calls
4 calls:
Slowest
9.0ms
(* (/ alpha (+ (+ alpha beta) 2.0)) (/ alpha (+ (+ alpha beta) 2.0)))
3.0ms
(pow (/ alpha (+ (+ alpha beta) 2.0)) 3)
3.0ms
(cbrt (pow (/ alpha (+ (+ alpha beta) 2.0)) 3))
0.0ms
(fma (* (/ alpha (+ 2.0 (+ beta alpha))) (/ alpha (+ 2.0 (+ beta alpha)))) (cbrt (pow (/ alpha (+ (+ alpha beta) 2.0)) 3)) (- (pow 1.0 3)))

series321.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
125.0ms
(fma (* (/ alpha (+ 2.0 (+ beta alpha))) (/ alpha (+ 2.0 (+ beta alpha)))) (cbrt (pow (/ alpha (+ (+ alpha beta) 2.0)) 3)) (- (pow 1.0 3)))
87.0ms
(cbrt (pow (/ alpha (+ (+ alpha beta) 2.0)) 3))
59.0ms
(pow (/ alpha (+ (+ alpha beta) 2.0)) 3)
50.0ms
(* (/ alpha (+ (+ alpha beta) 2.0)) (/ alpha (+ (+ alpha beta) 2.0)))

simplify6.5s

Counts
66 → 110
Calls
66 calls:
Slowest
568.0ms
(cbrt (pow (/ alpha (- (* (+ alpha beta) (+ alpha beta)) (* 2.0 2.0))) 3))
549.0ms
(/ alpha (+ (pow (+ alpha beta) 3) (pow 2.0 3)))
447.0ms
(pow (/ alpha (- (* (+ alpha beta) (+ alpha beta)) (* 2.0 2.0))) 3)
418.0ms
(pow (/ alpha (- (* (+ alpha beta) (+ alpha beta)) (* 2.0 2.0))) 3)
405.0ms
(cbrt (pow (/ alpha (- (* (+ alpha beta) (+ alpha beta)) (* 2.0 2.0))) 3))

prune2.1s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 4.0b

regimes186.0ms

Accuracy

76.6% (3.0b remaining)

Error of 5.8b against oracle of 2.9b and baseline of 15.5b

bsearch142.0ms

end0.0ms

sample2.4s

Algorithm
intervals
Results
1.1s5870×body80valid
412.0ms856×body1280valid
323.0ms8000×pre80true
282.0ms725×body640valid
93.0ms361×body320valid
37.0ms188×body160valid