Average Error: 0.0 → 0.0
Time: 36.7s
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot e^{\log \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right)}\]

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-exp-log0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019002 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))

Details

Time bar (total: 36.3s)Debug log

sample137.0ms

Algorithm
intervals

simplify216.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

216.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 3 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
3 → 46
Calls

3 calls. Slowest were:

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

series161.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

97.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
41.0ms
(- 1 (* 3 (* v v)))
23.0ms
(* 3 (* v v))

simplify3.2s

Counts
28 → 55
Calls

28 calls. Slowest were:

1.1s
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (sqrt (- 1 (* 3 (* v v)))) (sqrt (- 1 (* 3 (* v v))))) (sqrt (- 1 (* 3 (* v v))))))
342.0ms
(- (+ (* 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)))))
318.0ms
(- (* 1/4 (sqrt 2)) (+ (* 9/32 (* (sqrt 2) (pow v 4))) (* 3/8 (* (sqrt 2) (pow v 2)))))

prune541.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize19.0ms

Local error

Found 4 expressions with local error:

0.1b
(* 3 (* v v))
0.0b
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
0.0b
(- 1 (* 3 (* v v)))
0.0b
(exp (log (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

10.0ms
(exp (log (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))))
6.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
2.0ms
(* 3 (* v v))

series211.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

99.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
76.0ms
(exp (log (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))))
24.0ms
(- 1 (* 3 (* v v)))
12.0ms
(* 3 (* v v))

simplify6.0s

Counts
43 → 77
Calls

43 calls. Slowest were:

1.4s
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (sqrt (- 1 (* 3 (* v v)))) (sqrt (- 1 (* 3 (* v v))))) (sqrt (- 1 (* 3 (* v v))))))
364.0ms
(- (* 1/4 (sqrt 2)) (+ (* 9/32 (* (sqrt 2) (pow v 4))) (* 3/8 (* (sqrt 2) (pow v 2)))))
330.0ms
(- (* 1/4 (sqrt 2)) (+ (* 9/32 (* (sqrt 2) (pow v 4))) (* 3/8 (* (sqrt 2) (pow v 2)))))

prune974.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize18.0ms

Local error

Found 4 expressions with local error:

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

rewrite49.0ms

Algorithm
rewrite-expression-head
Counts
4 → 111
Calls

4 calls. Slowest were:

38.0ms
(* (sqrt (exp (log (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))))) (sqrt (exp (log (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))))))
5.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
1.0ms
(* 3 (* v v))

series214.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

98.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
75.0ms
(* (sqrt (exp (log (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))))) (sqrt (exp (log (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))))))
21.0ms
(* 3 (* v v))
20.0ms
(* 3 (* v v))

simplify19.3s

Counts
115 → 123
Calls

115 calls. Slowest were:

1.1s
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (sqrt (- 1 (* 3 (* v v)))) (sqrt (- 1 (* 3 (* v v))))) (sqrt (- 1 (* 3 (* v v))))))
423.0ms
(- (* 1/4 (sqrt 2)) (+ (* 9/32 (* (sqrt 2) (pow v 4))) (* 3/8 (* (sqrt 2) (pow v 2)))))
365.0ms
(* (sqrt (exp (log (* (/ (sqrt 2) 4) (sqrt (- (pow 1 3) (pow (* 3 (* v v)) 3))))))) (sqrt (exp (log (* (/ (sqrt 2) 4) (sqrt (- (* 1 1) (* (* 3 (* v v)) (* 3 (* v v))))))))))

prune2.1s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes56.0ms

Accuracy

0% (0.0b remaining)

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

bsearch5.0ms

end0.0ms

sample3.0s

Algorithm
intervals