Average Error: 1.0 → 0.0
Time: 26.1s
Precision: 64
Internal Precision: 128
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{\frac{4}{3}}{\pi}}{1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)}{\sqrt{-6 \cdot \left(v \cdot v\right) + 2}}\]

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  2. Initial simplification0.0

    \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}\]
  3. Using strategy rm

  4. Applied flip--0.0

    \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\pi}}{\color{blue}{\frac{1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)}{1 + v \cdot v}}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}\]

  5. Applied associate-/r/0.0

    \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{4}{3}}{\pi}}{1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}\]

  6. Final simplification0.0

    \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\pi}}{1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)}{\sqrt{-6 \cdot \left(v \cdot v\right) + 2}}\]

Reproduce

herbie shell --seed 2018362 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))

Details

Time bar (total: 20.6s)Debug log

start124.0ms

Algorithm
intervals

setup130.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize45.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ 4/3 PI)
0.1b
(* (* v v) -6)
0.0b
(/ (/ 4/3 PI) (- 1 (* v v)))
0.0b
(sqrt (+ 2 (* (* v v) -6)))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

14.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
5.0ms
(* (* v v) -6)
1.0ms
(sqrt (+ 2 (* (* v v) -6)))

series161.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

94.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
47.0ms
(sqrt (+ 2 (* (* v v) -6)))
20.0ms
(* (* v v) -6)
0.0ms
(/ 4/3 PI)

simplify4.2s

Counts
49 → 77
Calls

49 calls. Slowest were:

693.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
605.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
556.0ms
(sqrt (+ (pow 2 3) (pow (* (* v v) -6) 3)))

prune980.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize22.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ 4/3 PI)
0.1b
(* (* v v) -6)
0.1b
(* (* v v) (* v v))
0.0b
(/ (/ 4/3 PI) (- (* 1 1) (* (* v v) (* v v))))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

9.0ms
(/ (/ 4/3 PI) (- (* 1 1) (* (* v v) (* v v))))
7.0ms
(* (* v v) (* v v))
5.0ms
(* (* v v) -6)

series248.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

218.0ms
(/ (/ 4/3 PI) (- (* 1 1) (* (* v v) (* v v))))
16.0ms
(* (* v v) -6)
13.0ms
(* (* v v) (* v v))
0.0ms
(/ 4/3 PI)

simplify2.9s

Counts
60 → 86
Calls

60 calls. Slowest were:

474.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (+ (* 4/3 (/ 1 (* (pow v 8) PI))) (* 4/3 (/ 1 (* (pow v 12) PI))))))
447.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (+ (* 4/3 (/ 1 (* (pow v 8) PI))) (* 4/3 (/ 1 (* (pow v 12) PI))))))
440.0ms
(/ (* (* (/ 4/3 PI) (/ 4/3 PI)) (/ 4/3 PI)) (* (* (- (* 1 1) (* (* v v) (* v v))) (- (* 1 1) (* (* v v) (* v v)))) (- (* 1 1) (* (* v v) (* v v)))))

prune1.3s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize23.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v))))))
0.1b
(* 6 (* v v))
0.0b
(sqrt (- 2 (* 6 (* v v))))
0.0b
(* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 71
Calls

4 calls. Slowest were:

12.0ms
(* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))
4.0ms
(/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v))))))
1.0ms
(* 6 (* v v))

series424.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

225.0ms
(/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v))))))
145.0ms
(* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))
37.0ms
(sqrt (- 2 (* 6 (* v v))))
17.0ms
(* 6 (* v v))

simplify8.4s

Counts
52 → 83
Calls

52 calls. Slowest were:

894.0ms
(* (+ 1 (* v v)) (sqrt (+ (* 2 2) (+ (* (* 6 (* v v)) (* 6 (* v v))) (* 2 (* 6 (* v v)))))))
501.0ms
(* (+ 1 (* v v)) (sqrt (+ (* 2 2) (+ (* (* 6 (* v v)) (* 6 (* v v))) (* 2 (* 6 (* v v)))))))
465.0ms
(* (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))) (sqrt (+ (* 2 2) (+ (* (* 6 (* v v)) (* 6 (* v v))) (* 2 (* 6 (* v v)))))))

prune1.4s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes149.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms