Average Error: 0.0 → 0.0
Time: 34.2s
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \sqrt[3]{\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right)\right)}\]

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-cbrt-cube0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019007 +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: 33.9s)Debug log

sample138.0ms

Algorithm
intervals

simplify221.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

221.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize46.0ms

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)))))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

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

series183.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

89.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
40.0ms
(- 1 (* 3 (* v v)))
34.0ms
(sqrt (- 1 (* 3 (* v v))))
20.0ms
(* 3 (* v v))

simplify4.6s

Counts
37 → 80
Calls

37 calls. Slowest were:

968.0ms
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (sqrt (- 1 (* 3 (* v v)))) (sqrt (- 1 (* 3 (* v v))))) (sqrt (- 1 (* 3 (* v v))))))
436.0ms
(sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v))))))
346.0ms
(- (* 1/4 (sqrt 2)) (+ (* 9/32 (* (sqrt 2) (pow v 4))) (* 3/8 (* (sqrt 2) (pow v 2)))))

prune923.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize10.0ms

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)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

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

series157.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

75.0ms
(log (exp (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))))
38.0ms
(- 1 (* 3 (* v v)))
30.0ms
(sqrt (- 1 (* 3 (* v v))))
14.0ms
(* 3 (* v v))

simplify2.1s

Counts
24 → 68
Calls

24 calls. Slowest were:

326.0ms
(- (* 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))))))
325.0ms
(- (* 1/8 (/ 1 (* (pow (sqrt -3) 3) (pow v 3)))) (+ (* 1/2 (/ 1 (* (sqrt -3) v))) (* (sqrt -3) v)))
289.0ms
(- (+ (* 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)))))

prune874.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize12.0ms

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))

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
4 → 99
Calls

4 calls. Slowest were:

30.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))))
2.0ms
(* 3 (* v v))
2.0ms
(* 3 (* v v))

series296.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

242.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))))
19.0ms
(* 3 (* v v))
18.0ms
(* 3 (* v v))
17.0ms
(* 3 (* v v))

simplify19.3s

Counts
95 → 111
Calls

95 calls. Slowest were:

848.0ms
(* (sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v)))))) (sqrt (+ 1 (* 3 (* v v)))))
586.0ms
(* (* (* (* (/ (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)))))))
537.0ms
(* (* 4 (sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v))))))) (sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v)))))))

prune1.8s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes41.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample3.1s

Algorithm
intervals