Average Error: 0.0 → 0.0
Time: 23.7s
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(1 - v \cdot v\right)\]

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]

  2. Final simplification0.0

    \[\leadsto \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(1 - v \cdot v\right)\]

Reproduce

herbie shell --seed 2019004 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))

Details

Time bar (total: 23.3s)Debug log

sample171.0ms

Algorithm
intervals

simplify222.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

222.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 4 expressions with local error:

0.1b
(* 3 (* v v))
0.0b
(sqrt (- 1 (* 3 (* v v))))
0.0b
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
0.0b
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 88
Calls

4 calls. Slowest were:

13.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
7.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
1.0ms
(sqrt (- 1 (* 3 (* v v))))

series246.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

113.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
72.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
44.0ms
(sqrt (- 1 (* 3 (* v v))))
18.0ms
(* 3 (* v v))

simplify12.8s

Counts
78 → 100
Calls

78 calls. Slowest were:

883.0ms
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (sqrt (- 1 (* 3 (* v v)))) (sqrt (- 1 (* 3 (* v v))))) (sqrt (- 1 (* 3 (* v v))))))
430.0ms
(* (- (* v v)) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))))
414.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- (pow 1 3) (pow (* 3 (* v v)) 3)))) (- 1 (* v v)))

prune1.2s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize11.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (- (* v v)) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))))
0.1b
(* 3 (* v v))
0.1b
(* 3 (* v v))
0.0b
(sqrt (- 1 (* 3 (* v v))))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

13.0ms
(* (- (* v v)) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))))
1.0ms
(sqrt (- 1 (* 3 (* v v))))
1.0ms
(* 3 (* v v))

series167.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

107.0ms
(* (- (* v v)) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))))
26.0ms
(sqrt (- 1 (* 3 (* v v))))
18.0ms
(* 3 (* v v))
16.0ms
(* 3 (* v v))

simplify4.3s

Counts
33 → 71
Calls

33 calls. Slowest were:

539.0ms
(* (- (* v v)) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))))
439.0ms
(* (- (* v v)) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))))
334.0ms
(+ (log (- (* v v))) (log (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))))

prune920.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes61.0ms

Accuracy

0% (0.0b remaining)

Error of 0.0b against oracle of 0.0b and baseline of 0.0b

bsearch4.0ms

end0.0ms

sample3.2s

Algorithm
intervals