Average Error: 0.0 → 0.0
Time: 36.4s
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{(\left(-9 \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right) + 1)_*} \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2}\right)}{4 \cdot \sqrt{\left(v \cdot v\right) \cdot 3 + 1}}\]

Error

Bits error versus v

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 flip--0.0

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

  4. Applied sqrt-div0.0

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

  5. Applied frac-times0.0

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

  6. Applied associate-*l/0.0

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

  7. Simplified0.0

    \[\leadsto \frac{\color{blue}{\left(\left(1 - v \cdot v\right) \cdot \sqrt{2}\right) \cdot \sqrt{(\left(-9 \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right) + 1)_*}}}{4 \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}}\]

  8. Final simplification0.0

    \[\leadsto \frac{\sqrt{(\left(-9 \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right) + 1)_*} \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2}\right)}{4 \cdot \sqrt{\left(v \cdot v\right) \cdot 3 + 1}}\]

Reproduce

herbie shell --seed 2019005 +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: 35.9s)Debug log

sample162.0ms

Algorithm
intervals

simplify205.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

205.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize39.0ms

Local error

Found 4 expressions with local error:

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

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 81
Calls

4 calls. Slowest were:

13.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
1.0ms
(sqrt (- 1 (* 3 (* v v))))
1.0ms
(* 3 (* v v))

series203.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

132.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
29.0ms
(sqrt (- 1 (* 3 (* v v))))
22.0ms
(- 1 (* v v))
19.0ms
(* 3 (* v v))

simplify10.6s

Counts
59 → 93
Calls

59 calls. Slowest were:

663.0ms
(* (* 4 (sqrt (+ 1 (* 3 (* v v))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
499.0ms
(* (sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v)))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
450.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- (* 1 1) (* (* 3 (* v v)) (* 3 (* v v)))))) (- (pow 1 3) (pow (* v v) 3)))

prune1.0s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize48.0ms

Local error

Found 4 expressions with local error:

0.2b
(* -9 (* v v))
0.2b
(* 3 (* v v))
0.0b
(/ (* (* (- 1 (* v v)) (sqrt 2)) (sqrt (fma (* -9 (* v v)) (* v v) 1))) (* 4 (sqrt (+ 1 (* 3 (* v v))))))
0.0b
(sqrt (fma (* -9 (* v v)) (* v v) 1))

rewrite41.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

34.0ms
(/ (* (* (- 1 (* v v)) (sqrt 2)) (sqrt (fma (* -9 (* v v)) (* v v) 1))) (* 4 (sqrt (+ 1 (* 3 (* v v))))))
2.0ms
(* -9 (* v v))
2.0ms
(* 3 (* v v))

series296.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

167.0ms
(/ (* (* (- 1 (* v v)) (sqrt 2)) (sqrt (fma (* -9 (* v v)) (* v v) 1))) (* 4 (sqrt (+ 1 (* 3 (* v v))))))
89.0ms
(sqrt (fma (* -9 (* v v)) (* v v) 1))
21.0ms
(* -9 (* v v))
19.0ms
(* 3 (* v v))

simplify4.6s

Counts
30 → 73
Calls

30 calls. Slowest were:

758.0ms
(/ (* (* (* (* (- 1 (* v v)) (- 1 (* v v))) (- 1 (* v v))) (* (* (sqrt 2) (sqrt 2)) (sqrt 2))) (* (* (sqrt (fma (* -9 (* v v)) (* v v) 1)) (sqrt (fma (* -9 (* v v)) (* v v) 1))) (sqrt (fma (* -9 (* v v)) (* v v) 1)))) (* (* (* 4 (sqrt (+ 1 (* 3 (* v v))))) (* 4 (sqrt (+ 1 (* 3 (* v v)))))) (* 4 (sqrt (+ 1 (* 3 (* v v)))))))
470.0ms
(sqrt (+ (* 1 1) (- (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v))))))
454.0ms
(- (+ (* 1/4 (sqrt 2)) (* 3/32 (* (sqrt 2) (pow v 4)))) (* 5/8 (* (sqrt 2) (pow v 2))))

prune1.1s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize8.0ms

Local error

Found 4 expressions with local error:

0.2b
(* 3 (* v v))
0.0b
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
0.0b
(sqrt (- 1 (* 3 (* v v))))
0.0b
(log1p (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 79
Calls

4 calls. Slowest were:

17.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
1.0ms
(sqrt (- 1 (* 3 (* v v))))
1.0ms
(* 3 (* v v))

series354.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

170.0ms
(log1p (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))
129.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
40.0ms
(sqrt (- 1 (* 3 (* v v))))
14.0ms
(* 3 (* v v))

simplify13.1s

Counts
60 → 91
Calls

60 calls. Slowest were:

652.0ms
(- (+ (log (- (* 1/4 (* (sqrt 2) (sqrt -3))))) (+ (* 4 (/ 1 (* (sqrt 2) (* (pow v 3) (sqrt -3))))) (* 1/2 (/ 1 (* (pow (sqrt -3) 2) (pow v 2)))))) (+ (* 3 (log (/ -1 v))) (/ 1 (pow v 2))))
621.0ms
(* (* 4 (sqrt (+ 1 (* 3 (* v v))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
553.0ms
(+ (log (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))) (log (- 1 (* v v))))

prune963.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes62.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample2.9s

Algorithm
intervals