Average Error: 0.0 → 0.0
Time: 17.9s
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(1 - v \cdot v\right)\]

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019022 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))

Details

Time bar (total: 17.5s)Debug log

sample101.0ms

Algorithm
intervals
Results
43.0ms256×body80valid
41.0ms246×body80nan

simplify177.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
177.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize45.0ms

Local error

Found 3 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 2) 4) (sqrt (- 1 (* 3 (* v v)))))

rewrite31.0ms

Algorithm
rewrite-expression-head
Rules
18×frac-times
16×sqrt-div
14×flip--
14×flip3--
10×add-exp-log
10×associate-*r/
10×associate-*r*
10×add-cbrt-cube
10×pow1
9×associate-*l/
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
5×associate-*l*
4×cbrt-unprod
4×prod-exp
4×pow-prod-down
3×add-log-exp
3×sqrt-prod
3×*-commutative
2×distribute-lft-in
2×sub-neg
2×distribute-rgt-in
1×div-inv
Counts
3 → 74
Calls
3 calls:
Slowest
18.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
10.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
2.0ms
(* 3 (* v v))

series190.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
101.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
77.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
12.0ms
(* 3 (* v v))

simplify12.1s

Counts
69 → 83
Calls
69 calls:
Slowest
920.0ms
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (sqrt (- 1 (* 3 (* v v)))) (sqrt (- 1 (* 3 (* v v))))) (sqrt (- 1 (* 3 (* v v))))))
597.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- (* v v)))
468.0ms
(* (- (* v v)) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))))
420.0ms
(* (* 4 (sqrt (+ 1 (* 3 (* v v))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
391.0ms
(* (* (sqrt 2) (sqrt (- 1 (* 3 (* v v))))) (- (* 1 1) (* (* v v) (* v v))))

prune716.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize54.0ms

Local error

Found 4 expressions with local error:

0.2b
(* 3 (* v v))
0.2b
(* 3 (* v v))
0.2b
(* 3 (* v v))
0.2b
(* 3 (* v v))

rewrite4.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×add-cube-cbrt
4×*-commutative
4×add-exp-log
4×associate-*r*
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×add-sqr-sqrt
Counts
4 → 36
Calls
4 calls:
Slowest
1.0ms
(* 3 (* v v))
1.0ms
(* 3 (* v v))
1.0ms
(* 3 (* v v))
1.0ms
(* 3 (* v v))

series74.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
26.0ms
(* 3 (* v v))
19.0ms
(* 3 (* v v))
16.0ms
(* 3 (* v v))
13.0ms
(* 3 (* v v))

simplify61.0ms

Counts
12 → 48
Calls
12 calls:
Slowest
7.0ms
(* 3 (pow v 2))
7.0ms
(* 3 (pow v 2))
7.0ms
(* 3 (pow v 2))
7.0ms
(* 3 (pow v 2))
6.0ms
(* 3 (pow v 2))

prune762.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes85.0ms

Accuracy

0% (0.0b remaining)

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

bsearch4.0ms

end0.0ms

sample3.1s

Algorithm
intervals
Results
1.7s7999×body80nan
1.1s8000×body80valid