Average Error: 0.0 → 0.0
Time: 30.5s
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \log \left(e^{\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}}\right)\]

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. Using strategy rm

  3. Applied add-log-exp0.0

    \[\leadsto \color{blue}{\log \left(e^{\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}\right)} \cdot \left(1 - v \cdot v\right)\]

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019007 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))

Details

Time bar (total: 30.0s)Debug log

sample163.0ms

Algorithm
intervals

simplify179.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

179.0ms
(* (* (/ (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

localize26.0ms

Local error

Found 4 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

5.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
1.0ms
(* 3 (* v v))
1.0ms
(- 1 (* 3 (* v v)))

series182.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

80.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
42.0ms
(sqrt (- 1 (* 3 (* v v))))
40.0ms
(- 1 (* 3 (* v v)))
19.0ms
(* 3 (* v v))

simplify4.2s

Counts
37 → 72
Calls

37 calls. Slowest were:

821.0ms
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (sqrt (- 1 (* 3 (* v v)))) (sqrt (- 1 (* 3 (* v v))))) (sqrt (- 1 (* 3 (* v v))))))
604.0ms
(- (+ (* 1/4 (* (sqrt 2) (* v (sqrt -3)))) (* 1/8 (/ (sqrt 2) (* v (sqrt -3))))) (* 1/32 (/ (sqrt 2) (* (pow v 3) (pow (sqrt -3) 3)))))
343.0ms
(- (* 1/4 (sqrt 2)) (+ (* 9/32 (* (sqrt 2) (pow v 4))) (* 3/8 (* (sqrt 2) (pow v 2)))))

prune694.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

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

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

3.0ms
(log (exp (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))))
1.0ms
(* 3 (* v v))
1.0ms
(- 1 (* 3 (* v v)))

series168.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

88.0ms
(log (exp (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))))
41.0ms
(- 1 (* 3 (* v v)))
28.0ms
(sqrt (- 1 (* 3 (* v v))))
11.0ms
(* 3 (* v v))

simplify1.9s

Counts
24 → 60
Calls

24 calls. Slowest were:

307.0ms
(- (* 1/4 (sqrt 2)) (+ (* 9/32 (* (sqrt 2) (pow v 4))) (* 3/8 (* (sqrt 2) (pow v 2)))))
264.0ms
(- (* 1/32 (/ (sqrt 2) (* (pow v 3) (pow (sqrt -3) 3)))) (+ (* 1/4 (* (sqrt 2) (* v (sqrt -3)))) (* 1/8 (/ (sqrt 2) (* v (sqrt -3))))))
263.0ms
(- (+ (* 1/4 (* (sqrt 2) (* v (sqrt -3)))) (* 1/8 (/ (sqrt 2) (* v (sqrt -3))))) (* 1/32 (/ (sqrt 2) (* (pow v 3) (pow (sqrt -3) 3)))))

prune683.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize24.0ms

Local error

Found 4 expressions with local error:

1.0b
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))))
0.2b
(* 3 (* v v))
0.2b
(* 3 (* v v))
0.2b
(* 3 (* v v))

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 91
Calls

4 calls. Slowest were:

23.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))))
1.0ms
(* 3 (* v v))
1.0ms
(* 3 (* v v))

series287.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

249.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))))
14.0ms
(* 3 (* v v))
13.0ms
(* 3 (* v v))
11.0ms
(* 3 (* v v))

simplify16.6s

Counts
95 → 103
Calls

95 calls. Slowest were:

757.0ms
(* (* (sqrt 2) (sqrt (- (* 1 1) (* (* 3 (* v v)) (* 3 (* v v)))))) (* (sqrt 2) (sqrt (- (* 1 1) (* (* 3 (* v v)) (* 3 (* v v)))))))
740.0ms
(* (* (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))) (* (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))))
423.0ms
(* (* (sqrt 2) (sqrt (- 1 (* 3 (* v v))))) (* (/ (sqrt 2) 4) (sqrt (- (* 1 1) (* (* 3 (* v v)) (* 3 (* v v)))))))

prune1.6s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes50.0ms

Accuracy

0% (0.0b remaining)

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

bsearch5.0ms

end0.0ms

sample3.2s

Algorithm
intervals