Average Error: 0.0 → 0.0
Time: 33.1s
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \frac{1 - v \cdot v}{\frac{4}{\sqrt{2}}}\]

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. Initial simplification0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2018362 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))

Details

Time bar (total: 27.3s)Debug log

start113.0ms

Algorithm
intervals

setup208.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize37.0ms

Local error

Found 4 expressions with local error:

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

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 98
Calls

4 calls. Slowest were:

10.0ms
(* (/ (- 1 (* v v)) (/ 4 (sqrt 2))) (sqrt (+ (* -3 (* v v)) 1)))
3.0ms
(/ (- 1 (* v v)) (/ 4 (sqrt 2)))
1.0ms
(* -3 (* v v))

series223.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

137.0ms
(* (/ (- 1 (* v v)) (/ 4 (sqrt 2))) (sqrt (+ (* -3 (* v v)) 1)))
65.0ms
(/ (- 1 (* v v)) (/ 4 (sqrt 2)))
20.0ms
(* -3 (* v v))
0.0ms
(/ 4 (sqrt 2))

simplify5.8s

Counts
83 → 108
Calls

83 calls. Slowest were:

561.0ms
(/ (* (* (- 1 (* v v)) (- 1 (* v v))) (- 1 (* v v))) (* (* (/ 4 (sqrt 2)) (/ 4 (sqrt 2))) (/ 4 (sqrt 2))))
346.0ms
(* (- 1 (* v v)) (sqrt (+ (pow (* -3 (* v v)) 3) (pow 1 3))))
329.0ms
(* (/ (- 1 (* v v)) (/ 4 (sqrt 2))) (sqrt (+ (* -3 (* v v)) 1)))

prune1.7s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

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

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 205
Calls

4 calls. Slowest were:

6.0ms
(/ (- 1 (* v v)) (/ (/ 4 (sqrt (sqrt 2))) (sqrt (sqrt 2))))
4.0ms
(/ (/ 4 (sqrt (sqrt 2))) (sqrt (sqrt 2)))
2.0ms
(* -3 (* v v))

series110.0ms

Counts
4 → 8
Calls

4 calls. Slowest were:

98.0ms
(/ (- 1 (* v v)) (/ (/ 4 (sqrt (sqrt 2))) (sqrt (sqrt 2))))
12.0ms
(* -3 (* v v))
0.0ms
(/ (/ 4 (sqrt (sqrt 2))) (sqrt (sqrt 2)))
0.0ms
(/ 4 (sqrt (sqrt 2)))

simplify15.8s

Counts
278 → 213
Calls

278 calls. Slowest were:

771.0ms
(/ (* (* (- 1 (* v v)) (- 1 (* v v))) (- 1 (* v v))) (* (* (/ (/ 4 (sqrt (sqrt 2))) (sqrt (sqrt 2))) (/ (/ 4 (sqrt (sqrt 2))) (sqrt (sqrt 2)))) (/ (/ 4 (sqrt (sqrt 2))) (sqrt (sqrt 2)))))
494.0ms
(/ (- 1 (* v v)) (/ (/ 1 (sqrt (sqrt 2))) (sqrt (sqrt 2))))
465.0ms
(/ (* (cbrt (- 1 (* v v))) (cbrt (- 1 (* v v)))) (/ (* (cbrt (/ 4 (sqrt (sqrt 2)))) (cbrt (/ 4 (sqrt (sqrt 2))))) (* (cbrt (sqrt (sqrt 2))) (cbrt (sqrt (sqrt 2))))))

prune3.2s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes70.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms