Average Error: 1.0 → 0.0
Time: 16.0s
Precision: 64
Internal Precision: 128
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{\frac{4}{3}}{\pi}}{\sqrt{1 - v \cdot v}} \cdot \frac{1}{\sqrt{1 - v \cdot v}}}{\sqrt{\left(v \cdot v\right) \cdot -6 + 2}}\]

Error

Bits error versus v

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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 add-sqr-sqrt0.0

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

  5. Applied *-un-lft-identity0.0

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

  6. Applied times-frac0.0

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

  7. Final simplification0.0

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

Runtime

Time bar (total: 16.0s)Debug logProfile

start287.0ms

Algorithm
intervals

setup222.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize41.0ms

Local error

Found 3 expressions with local error:

1.0b
(/ 4/3 PI)
0.2b
(* (* v v) -6)
0.0b
(/ (/ 4/3 PI) (- 1 (* v v)))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
3 → 53
Calls

3 calls. Slowest were:

7.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
4.0ms
(* (* v v) -6)
0.0ms
(/ 4/3 PI)

series110.0ms

Counts
3 → 7
Calls

3 calls. Slowest were:

90.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
20.0ms
(* (* v v) -6)
0.0ms
(/ 4/3 PI)

simplify2.4s

Counts
40 → 60
Calls

40 calls. Slowest were:

621.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
594.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
410.0ms
(/ (* (* (/ 4/3 PI) (/ 4/3 PI)) (/ 4/3 PI)) (* (* (- 1 (* v v)) (- 1 (* v v))) (- 1 (* v v))))

prune816.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize24.0ms

Local error

Found 3 expressions with local error:

1.0b
(/ 4/3 PI)
0.2b
(* (* v v) -6)
0.0b
(/ (/ 4/3 PI) (sqrt (- 1 (* v v))))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
3 → 68
Calls

3 calls. Slowest were:

7.0ms
(/ (/ 4/3 PI) (sqrt (- 1 (* v v))))
4.0ms
(* (* v v) -6)
0.0ms
(/ 4/3 PI)

series196.0ms

Counts
3 → 7
Calls

3 calls. Slowest were:

172.0ms
(/ (/ 4/3 PI) (sqrt (- 1 (* v v))))
24.0ms
(* (* v v) -6)
0.0ms
(/ 4/3 PI)

simplify1.7s

Counts
68 → 75
Calls

68 calls. Slowest were:

324.0ms
(/ (* (* (/ 4/3 PI) (/ 4/3 PI)) (/ 4/3 PI)) (* (* (sqrt (- 1 (* v v))) (sqrt (- 1 (* v v)))) (sqrt (- 1 (* v v)))))
289.0ms
(- (* 4/3 (/ (sqrt -1) (* v PI))) (+ (* 2/3 (/ 1 (* (pow v 5) (* (sqrt -1) PI)))) (+ (* 2/3 (/ 1 (* (pow v 3) (* (sqrt -1) PI)))) (* 1/6 (/ 1 (* (pow v 5) (* (pow (sqrt -1) 3) PI)))))))
211.0ms
(- (+ (* 2/3 (/ 1 (* (pow v 5) (* (sqrt -1) PI)))) (+ (* 2/3 (/ 1 (* (pow v 3) (* (sqrt -1) PI)))) (* 1/6 (/ 1 (* (pow v 5) (* (pow (sqrt -1) 3) PI)))))) (* 4/3 (/ (sqrt -1) (* v PI))))

prune1.2s

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

herbie shell --seed 2018357 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))