Average Error: 0.0 → 0.0
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \left(\left(\sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}} \cdot \sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}}\right) \cdot \frac{\sqrt{2}}{4}\right)\]

Error

Bits error versus v

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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-sqr-sqrt0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2018365 +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: 1.2m)Debug log

start221.0ms

Algorithm
intervals

setup213.0ms

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

rewrite43.0ms

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

series264.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

120.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
89.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
29.0ms
(sqrt (- 1 (* 3 (* v v))))
25.0ms
(* 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))))))
753.0ms
(* (* 4 (sqrt (+ 1 (* 3 (* v v))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
527.0ms
(* (sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v)))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))

prune1.8s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize10.0ms

Local error

Found 4 expressions with local error:

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

rewrite48.0ms

Algorithm
rewrite-expression-head
Counts
4 → 106
Calls

4 calls. Slowest were:

33.0ms
(* (/ (sqrt 2) 4) (* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v)))))))
11.0ms
(* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v))))))
1.0ms
(* 3 (* v v))

series138.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

81.0ms
(* (/ (sqrt 2) 4) (* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v)))))))
34.0ms
(* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v))))))
11.0ms
(* 3 (* v v))
11.0ms
(* 3 (* v v))

simplify13.1s

Counts
87 → 118
Calls

87 calls. Slowest were:

857.0ms
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v)))))) (* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v))))))) (* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v))))))))
433.0ms
(* (sqrt (sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v))))))) (sqrt (sqrt (+ 1 (* 3 (* v v))))))
429.0ms
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v)))))) (sqrt (sqrt (- 1 (* 3 (* v v)))))) (* (* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v)))))) (sqrt (sqrt (- 1 (* 3 (* v v))))))))

prune2.4s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize29.0ms

Local error

Found 4 expressions with local error:

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

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 87
Calls

4 calls. Slowest were:

5.0ms
(* (/ (sqrt 2) 4) (/ (sqrt 2) 4))
4.0ms
(* (/ (sqrt 2) 4) (fma (* v v) -3 1))
1.0ms
(pow (- 1 (* v v)) 3)

series135.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

77.0ms
(* (/ (sqrt 2) 4) (fma (* v v) -3 1))
39.0ms
(sqrt (fma (* v v) -3 1))
19.0ms
(pow (- 1 (* v v)) 3)
0.0ms
(* (/ (sqrt 2) 4) (/ (sqrt 2) 4))

simplify5.3s

Counts
53 → 97
Calls

53 calls. Slowest were:

1.2s
(pow (- (* 1 1) (* (* v v) (* v v))) 3)
684.0ms
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (fma (* v v) -3 1) (fma (* v v) -3 1)) (fma (* v v) -3 1)))
464.0ms
(* (- 1 (* v v)) (- 1 (* v v)))

prune2.0s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize10.0ms

Local error

Found 4 expressions with local error:

1.0b
(* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v))))))
0.2b
(* 3 (* v v))
0.0b
(sqrt (- 1 (* 3 (* v v))))
0.0b
(* (* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v)))))) (- 1 (* v v)))

rewrite66.0ms

Algorithm
rewrite-expression-head
Counts
4 → 126
Calls

4 calls. Slowest were:

37.0ms
(* (* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v)))))) (- 1 (* v v)))
22.0ms
(* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v))))))
2.0ms
(sqrt (- 1 (* 3 (* v v))))

series258.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

144.0ms
(* (* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v)))))) (- 1 (* v v)))
66.0ms
(* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v))))))
35.0ms
(sqrt (- 1 (* 3 (* v v))))
14.0ms
(* 3 (* v v))

simplify25.5s

Counts
123 → 138
Calls

123 calls. Slowest were:

593.0ms
(sqrt (- (* 1 1) (* (* 3 (* v v)) (* 3 (* v v)))))
592.0ms
(* (* (sqrt (sqrt 2)) (* (sqrt (sqrt 2)) (sqrt (- 1 (* 3 (* v v)))))) (- 1 (* v v)))
539.0ms
(* (sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v)))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))

prune2.4s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes55.0ms

Accuracy

0% (0.0b remaining)

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

bsearch4.0ms