Average Error: 16.6 → 16.6
Time: 1.9m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\sqrt[3]{\left(1.0 + \frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \left(\beta - \alpha\right)\right) \cdot \left(\left(1.0 + \frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \left(\beta - \alpha\right)\right) \cdot \left(1.0 + \frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \left(\beta - \alpha\right)\right)\right)}}{2.0}\]

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. Initial program 16.6

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

  2. Initial simplification16.6

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

  4. Applied div-inv16.6

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

  6. Applied add-cbrt-cube16.6

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

  7. Final simplification16.6

    \[\leadsto \frac{\sqrt[3]{\left(1.0 + \frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \left(\beta - \alpha\right)\right) \cdot \left(\left(1.0 + \frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \left(\beta - \alpha\right)\right) \cdot \left(1.0 + \frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \left(\beta - \alpha\right)\right)\right)}}{2.0}\]

Reproduce

herbie shell --seed 2018362 
(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: 1.8m)Debug log

start123.0ms

Algorithm
intervals

setup110.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 15.1b

localize29.0ms

Local error

Found 2 expressions with local error:

1.4b
(+ 1.0 (/ (- beta alpha) (+ (+ alpha beta) 2.0)))
0.0b
(/ (- beta alpha) (+ (+ alpha beta) 2.0))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
2 → 43
Calls

2 calls. Slowest were:

12.0ms
(/ (- beta alpha) (+ (+ alpha beta) 2.0))
6.0ms
(+ 1.0 (/ (- beta alpha) (+ (+ alpha beta) 2.0)))

series78.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

47.0ms
(+ 1.0 (/ (- beta alpha) (+ (+ alpha beta) 2.0)))
31.0ms
(/ (- beta alpha) (+ (+ alpha beta) 2.0))

simplify1.4s

Counts
37 → 49
Calls

37 calls. Slowest were:

160.0ms
(+ (* (+ alpha beta) (+ alpha beta)) (- (* 2.0 2.0) (* (+ alpha beta) 2.0)))
154.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
154.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))

prune638.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 14.7b

localize24.0ms

Local error

Found 3 expressions with local error:

3.5b
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))
0.1b
(* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0)))
0.0b
(/ 1 (+ (+ alpha beta) 2.0))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
3 → 55
Calls

3 calls. Slowest were:

9.0ms
(* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0)))
3.0ms
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))
3.0ms
(/ 1 (+ (+ alpha beta) 2.0))

series110.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

44.0ms
(* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0)))
35.0ms
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))
31.0ms
(/ 1 (+ (+ alpha beta) 2.0))

simplify3.1s

Counts
35 → 64
Calls

35 calls. Slowest were:

530.0ms
(* (+ (* beta beta) (+ (* alpha alpha) (* beta alpha))) (+ (+ alpha beta) 2.0))
342.0ms
(* (* (* (- beta alpha) (- beta alpha)) (- beta alpha)) (* (* (/ 1 (+ (+ alpha beta) 2.0)) (/ 1 (+ (+ alpha beta) 2.0))) (/ 1 (+ (+ alpha beta) 2.0))))
294.0ms
(* (+ beta alpha) (+ (+ alpha beta) 2.0))

prune821.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 14.7b

localize21.0ms

Local error

Found 4 expressions with local error:

3.5b
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))
3.5b
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))
3.5b
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))
0.1b
(* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0)))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 57
Calls

4 calls. Slowest were:

8.0ms
(* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0)))
3.0ms
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))
3.0ms
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))

series180.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

53.0ms
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))
52.0ms
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))
45.0ms
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))
30.0ms
(* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0)))

simplify3.6s

Counts
32 → 69
Calls

32 calls. Slowest were:

561.0ms
(* (+ (* beta beta) (+ (* alpha alpha) (* beta alpha))) (+ (+ alpha beta) 2.0))
395.0ms
(* (* (* (- beta alpha) (- beta alpha)) (- beta alpha)) (* (* (/ 1 (+ (+ alpha beta) 2.0)) (/ 1 (+ (+ alpha beta) 2.0))) (/ 1 (+ (+ alpha beta) 2.0))))
259.0ms
(* (+ beta alpha) (+ (+ alpha beta) 2.0))

prune1.2s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 14.7b

localize21.0ms

Local error

Found 4 expressions with local error:

10.2b
(cbrt (* (* (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))) (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0)))) (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0)))))
3.5b
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))
3.5b
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))
3.5b
(+ 1.0 (cbrt (* (* (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))) (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0)))) (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))))

rewrite73.0ms

Algorithm
rewrite-expression-head
Counts
4 → 382
Calls

4 calls. Slowest were:

55.0ms
(cbrt (* (* (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))) (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0)))) (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0)))))
2.0ms
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))
2.0ms
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))

series152.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

53.0ms
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))
36.0ms
(+ 1.0 (cbrt (* (* (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))) (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0)))) (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))))
34.0ms
(+ 1.0 (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))))
28.0ms
(cbrt (* (* (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))) (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0)))) (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0)))))

simplify1.4m

Counts
356 → 394
Calls

356 calls. Slowest were:

1.0s
(cbrt (* (* (* (- beta alpha) 1) (* (- (* beta beta) (* alpha alpha)) (/ 1 (+ (+ alpha beta) 2.0)))) (* (- beta alpha) 1)))
934.0ms
(cbrt (* (* (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0))) (* (- beta alpha) (/ 1 (+ (+ alpha beta) 2.0)))) (* (- beta alpha) 1)))
932.0ms
(cbrt (* (* (* (- beta alpha) 1) (* (- (* beta beta) (* alpha alpha)) (/ 1 (+ (+ alpha beta) 2.0)))) (- beta alpha)))

prune10.1s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 14.7b

regimes241.0ms

Accuracy

0% (0.3b remaining)

Error of 16.6b against oracle of 16.3b and baseline of 16.6b

bsearch4.0ms