Average Error: 1.0 → 0.0
Time: 6.5m
Precision: 64
Internal Precision: 128
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}{\sqrt{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}}}{\sqrt{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}}\]

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. Simplified0.0

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

  4. Applied add-sqr-sqrt1.0

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

  5. Applied associate-/r*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019002 +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: 6.5m)Debug log

sample133.0ms

Algorithm
intervals

simplify126.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

126.0ms
(/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v))))))

prune18.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize33.0ms

Local error

Found 4 expressions with local error:

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

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 211
Calls

4 calls. Slowest were:

12.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (fma -6 (* v v) 2)))
5.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
1.0ms
(sqrt (fma -6 (* v v) 2))

series295.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

191.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (fma -6 (* v v) 2)))
78.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
26.0ms
(sqrt (fma -6 (* v v) 2))
0.0ms
(/ 4/3 PI)

simplify27.3s

Counts
291 → 221
Calls

291 calls. Slowest were:

886.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
875.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
510.0ms
(/ (/ (/ 4/3 PI) (- (pow 1 3) (pow (* v v) 3))) (sqrt (* (cbrt (fma -6 (* v v) 2)) (cbrt (fma -6 (* v v) 2)))))

prune2.3s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize8.0ms

Local error

Found 4 expressions with local error:

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

rewrite145.0ms

Algorithm
rewrite-expression-head
Counts
4 → 1942
Calls

4 calls. Slowest were:

16.0ms
(/ (/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (sqrt (fma -6 (* v v) 2)))) (sqrt (sqrt (fma -6 (* v v) 2))))
8.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (sqrt (fma -6 (* v v) 2))))
3.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))

series559.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

234.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (sqrt (fma -6 (* v v) 2))))
233.0ms
(/ (/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (sqrt (fma -6 (* v v) 2)))) (sqrt (sqrt (fma -6 (* v v) 2))))
91.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
0.0ms
(/ 4/3 PI)

simplify5.1m

Counts
3567 → 1952
Calls

3567 calls. Slowest were:

625.0ms
(/ (/ (/ (/ 4/3 PI) (- (pow 1 3) (pow (* v v) 3))) (sqrt 1)) (sqrt (sqrt (sqrt (fma -6 (* v v) 2)))))
561.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
557.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))

prune19.2s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize12.0ms

Local error

Found 4 expressions with local error:

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

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 218
Calls

4 calls. Slowest were:

10.0ms
(/ (/ (* 4/3 (/ (/ 1 PI) (- 1 (* v v)))) (sqrt (sqrt (fma -6 (* v v) 2)))) (sqrt (sqrt (fma -6 (* v v) 2))))
6.0ms
(/ (* 4/3 (/ (/ 1 PI) (- 1 (* v v)))) (sqrt (sqrt (fma -6 (* v v) 2))))
1.0ms
(sqrt (fma -6 (* v v) 2))

series377.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

174.0ms
(/ (* 4/3 (/ (/ 1 PI) (- 1 (* v v)))) (sqrt (sqrt (fma -6 (* v v) 2))))
154.0ms
(/ (/ (* 4/3 (/ (/ 1 PI) (- 1 (* v v)))) (sqrt (sqrt (fma -6 (* v v) 2)))) (sqrt (sqrt (fma -6 (* v v) 2))))
24.0ms
(sqrt (fma -6 (* v v) 2))
24.0ms
(sqrt (fma -6 (* v v) 2))

simplify28.7s

Counts
310 → 230
Calls

310 calls. Slowest were:

468.0ms
(/ (* 4/3 (/ (/ 1 PI) (- 1 (* v v)))) (sqrt (sqrt (sqrt (fma -6 (* v v) 2)))))
460.0ms
(/ (sqrt (/ (* 4/3 (/ (/ 1 PI) (- 1 (* v v)))) (sqrt (sqrt (fma -6 (* v v) 2))))) (* (cbrt (sqrt (sqrt (fma -6 (* v v) 2)))) (cbrt (sqrt (sqrt (fma -6 (* v v) 2))))))
350.0ms
(/ (/ (* (* (* 4/3 (/ (/ 1 PI) (- 1 (* v v)))) (* 4/3 (/ (/ 1 PI) (- 1 (* v v))))) (* 4/3 (/ (/ 1 PI) (- 1 (* v v))))) (* (* (sqrt (sqrt (fma -6 (* v v) 2))) (sqrt (sqrt (fma -6 (* v v) 2)))) (sqrt (sqrt (fma -6 (* v v) 2))))) (* (* (sqrt (sqrt (fma -6 (* v v) 2))) (sqrt (sqrt (fma -6 (* v v) 2)))) (sqrt (sqrt (fma -6 (* v v) 2)))))

prune1.8s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes36.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample2.1s

Algorithm
intervals