Average Error: 0.0 → 0.0
Time: 36.0s
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 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))

Details

Time bar (total: 35.7s)Debug log

sample148.0ms

Algorithm
intervals

simplify206.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

206.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))

prune20.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize36.0ms

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)))

rewrite41.0ms

Algorithm
rewrite-expression-head
Counts
4 → 96
Calls

4 calls. Slowest were:

26.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
10.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
1.0ms
(sqrt (- 1 (* 3 (* v v))))

series229.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

106.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
70.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
32.0ms
(sqrt (- 1 (* 3 (* v v))))
20.0ms
(* 3 (* v v))

simplify13.8s

Counts
78 → 108
Calls

78 calls. Slowest were:

1.1s
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (sqrt (- 1 (* 3 (* v v)))) (sqrt (- 1 (* 3 (* v v))))) (sqrt (- 1 (* 3 (* v v))))))
564.0ms
(* (sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v)))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
556.0ms
(* (* 4 (sqrt (+ 1 (* 3 (* v v))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))

prune1.4s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize15.0ms

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)))

rewrite45.0ms

Algorithm
rewrite-expression-head
Counts
4 → 96
Calls

4 calls. Slowest were:

28.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
11.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
2.0ms
(sqrt (- 1 (* 3 (* v v))))

series236.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

103.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
69.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
40.0ms
(sqrt (- 1 (* 3 (* v v))))
24.0ms
(* 3 (* v v))

simplify14.7s

Counts
78 → 108
Calls

78 calls. Slowest were:

1.0s
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (sqrt (- 1 (* 3 (* v v)))) (sqrt (- 1 (* 3 (* v v))))) (sqrt (- 1 (* 3 (* v v))))))
601.0ms
(* (* 4 (sqrt (+ 1 (* 3 (* v v))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
469.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- (* 1 1) (* (* 3 (* v v)) (* 3 (* v v)))))) (- (pow 1 3) (pow (* v v) 3)))

prune1.4s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes108.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample3.4s

Algorithm
intervals