Average Error: 0.0 → 0.0
Time: 53.5s
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 2018362 +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: 47.5s)Debug log

start176.0ms

Algorithm
intervals

setup257.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize34.0ms

Local error

Found 4 expressions with local error:

0.1b
(* 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)))
0.0b
(sqrt (- 1 (* 3 (* v v))))

rewrite38.0ms

Algorithm
rewrite-expression-head
Counts
4 → 96
Calls

4 calls. Slowest were:

17.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
16.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
2.0ms
(* 3 (* v v))

series275.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

135.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
83.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
45.0ms
(sqrt (- 1 (* 3 (* v v))))
12.0ms
(* 3 (* v v))

simplify15.5s

Counts
78 → 108
Calls

78 calls. Slowest were:

1.2s
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (sqrt (- 1 (* 3 (* v v)))) (sqrt (- 1 (* 3 (* v v))))) (sqrt (- 1 (* 3 (* v v))))))
740.0ms
(* (* 4 (sqrt (+ 1 (* 3 (* v v))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
639.0ms
(* (sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v)))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))

prune2.0s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize21.0ms

Local error

Found 4 expressions with local error:

1.0b
(* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v))))))
0.1b
(* 3 (* v v))
0.0b
(* (* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v)))))) (- 1 (* v v)))
0.0b
(sqrt (- 1 (* 3 (* v v))))

rewrite96.0ms

Algorithm
rewrite-expression-head
Counts
4 → 126
Calls

4 calls. Slowest were:

63.0ms
(* (* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v)))))) (- 1 (* v v)))
28.0ms
(* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v))))))
2.0ms
(* 3 (* v v))

series232.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

110.0ms
(* (* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v)))))) (- 1 (* v v)))
67.0ms
(* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v))))))
43.0ms
(sqrt (- 1 (* 3 (* v v))))
11.0ms
(* 3 (* v v))

simplify26.1s

Counts
123 → 138
Calls

123 calls. Slowest were:

750.0ms
(* (* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- (pow 1 3) (pow (* 3 (* v v)) 3))))) (- 1 (* v v)))
620.0ms
(* (* (sqrt 4) (* (sqrt 4) (sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v)))))))) (+ 1 (* v v)))
540.0ms
(* (* (sqrt 4) (sqrt (+ 1 (* 3 (* v v))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))

prune2.6s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes121.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms