Average Error: 1.0 → 0.0
Time: 17.5s
Precision: 64
Internal Precision: 128
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{\frac{4}{3}}{\pi}}{\sqrt{1 - v \cdot v}} \cdot \frac{1}{\sqrt{1 - v \cdot v}}}{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}\]

Error

Bits error versus v

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

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

  4. Applied add-sqr-sqrt0.0

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

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

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

  6. Applied times-frac0.0

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

  7. Final simplification0.0

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

Runtime

Time bar (total: 17.5s)Debug logProfile

start234.0ms

Algorithm
intervals

setup192.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize20.0ms

Local error

Found 2 expressions with local error:

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

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 48
Calls

2 calls. Slowest were:

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

series89.0ms

Counts
2 → 4
Calls

2 calls. Slowest were:

88.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
0.0ms
(/ 4/3 PI)

simplify3.3s

Counts
37 → 52
Calls

37 calls. Slowest were:

830.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
828.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
391.0ms
(/ (* (* (/ 4/3 PI) (/ 4/3 PI)) (/ 4/3 PI)) (* (* (- 1 (* v v)) (- 1 (* v v))) (- 1 (* v v))))

prune694.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize27.0ms

Local error

Found 2 expressions with local error:

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

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
2 → 63
Calls

2 calls. Slowest were:

8.0ms
(/ (/ 4/3 PI) (sqrt (- 1 (* v v))))
1.0ms
(/ 4/3 PI)

series151.0ms

Counts
2 → 4
Calls

2 calls. Slowest were:

151.0ms
(/ (/ 4/3 PI) (sqrt (- 1 (* v v))))
0.0ms
(/ 4/3 PI)

simplify2.6s

Counts
65 → 67
Calls

65 calls. Slowest were:

622.0ms
(- (* 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)))))))
440.0ms
(- (+ (* 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))))
355.0ms
(/ (* (* (/ 4/3 PI) (/ 4/3 PI)) (/ 4/3 PI)) (* (* (sqrt (- 1 (* v v))) (sqrt (- 1 (* v v)))) (sqrt (- 1 (* v v)))))

prune1.2s

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

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