Average Error: 1.0 → 0.0
Time: 56.3s
Precision: 64
Internal Precision: 128
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}\]

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. Final simplification0.0

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

Reproduce

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

Details

Time bar (total: 50.6s)Debug log

start104.0ms

Algorithm
intervals

setup159.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

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

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 211
Calls

4 calls. Slowest were:

7.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (fma -6 (* v v) 2)))
3.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
1.0ms
(sqrt (fma -6 (* v v) 2))

series304.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

194.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (fma -6 (* v v) 2)))
75.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
34.0ms
(sqrt (fma -6 (* v v) 2))
0.0ms
(/ 4/3 PI)

simplify26.3s

Counts
291 → 221
Calls

291 calls. Slowest were:

850.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
835.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
601.0ms
(- (* 1/2 (/ 1 (* (pow v 3) (pow (sqrt -6) 3)))) (+ (/ 1 (* v (sqrt -6))) (* v (sqrt -6))))

prune3.5s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize21.0ms

Local error

Found 4 expressions with local error:

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

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 175
Calls

4 calls. Slowest were:

5.0ms
(/ (/ 1 (- 1 (* v v))) (sqrt (sqrt (fma -6 (* v v) 2))))
4.0ms
(/ (/ 4/3 PI) (sqrt (sqrt (fma -6 (* v v) 2))))
1.0ms
(sqrt (fma -6 (* v v) 2))

series414.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

221.0ms
(/ (/ 4/3 PI) (sqrt (sqrt (fma -6 (* v v) 2))))
162.0ms
(/ (/ 1 (- 1 (* v v))) (sqrt (sqrt (fma -6 (* v v) 2))))
31.0ms
(sqrt (fma -6 (* v v) 2))
0.0ms
(/ 4/3 PI)

simplify16.4s

Counts
224 → 185
Calls

224 calls. Slowest were:

536.0ms
(/ (/ 1 (- (pow 1 3) (pow (* v v) 3))) (* (cbrt (sqrt (sqrt (fma -6 (* v v) 2)))) (cbrt (sqrt (sqrt (fma -6 (* v v) 2))))))
420.0ms
(+ (* (/ (pow v 2) PI) (pow 1/2 1/4)) (+ (* 15/8 (* (/ (pow v 4) PI) (pow 1/2 1/4))) (* 4/3 (* (pow 1/2 1/4) (/ 1 PI)))))
392.0ms
(/ (* (* (/ 1 (- 1 (* v v))) (/ 1 (- 1 (* v v)))) (/ 1 (- 1 (* v v)))) (* (* (sqrt (sqrt (fma -6 (* v v) 2))) (sqrt (sqrt (fma -6 (* v v) 2)))) (sqrt (sqrt (fma -6 (* v v) 2)))))

prune3.2s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes89.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms