Average Error: 0.4 → 0.1
Time: 4.9m
Precision: 64
Internal Precision: 128
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{\pi}}{\sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_* \cdot 2}}}{\left(1 - v \cdot v\right) \cdot t}\]

Error

Bits error versus v

Bits error versus t

Derivation

  1. Initial program 0.4

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

  2. Initial simplification0.3

    \[\leadsto \frac{\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{\pi}}{\sqrt{2 \cdot (\left(v \cdot v\right) \cdot -3 + 1)_*} \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)}\]
  3. Using strategy rm

  4. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{\pi}}{\sqrt{2 \cdot (\left(v \cdot v\right) \cdot -3 + 1)_*}}}{t \cdot \left(1 - v \cdot v\right)}}\]

  5. Final simplification0.1

    \[\leadsto \frac{\frac{\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{\pi}}{\sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_* \cdot 2}}}{\left(1 - v \cdot v\right) \cdot t}\]

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))

Details

Time bar (total: 4.9m)Debug log

start254.0ms

Algorithm
intervals

setup128.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize46.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))
0.4b
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))))
0.0b
(sqrt (* 2 (fma (* v v) -3 1)))
0.0b
(/ (fma (* v v) -5 1) PI)

rewrite39.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

16.0ms
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))))
14.0ms
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))
5.0ms
(/ (fma (* v v) -5 1) PI)

series707.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

366.0ms
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))))
173.0ms
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))
114.0ms
(/ (fma (* v v) -5 1) PI)
54.0ms
(sqrt (* 2 (fma (* v v) -3 1)))

simplify6.3s

Counts
51 → 94
Calls

51 calls. Slowest were:

369.0ms
(- (log (/ (fma (* v v) -5 1) PI)) (+ (log (sqrt (* 2 (fma (* v v) -3 1)))) (log (* t (- 1 (* v v))))))
347.0ms
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))
335.0ms
(- (* t (sqrt 2)) (* 5/2 (* t (* (sqrt 2) (pow v 2)))))

prune1.8s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.1b
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))) (* t (- 1 (* v v))))
0.0b
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))
0.0b
(sqrt (* 2 (fma (* v v) -3 1)))
0.0b
(/ (fma (* v v) -5 1) PI)

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
4 → 125
Calls

4 calls. Slowest were:

18.0ms
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))) (* t (- 1 (* v v))))
11.0ms
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))
5.0ms
(/ (fma (* v v) -5 1) PI)

series671.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

335.0ms
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))) (* t (- 1 (* v v))))
171.0ms
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))
106.0ms
(/ (fma (* v v) -5 1) PI)
58.0ms
(sqrt (* 2 (fma (* v v) -3 1)))

simplify10.1s

Counts
124 → 137
Calls

124 calls. Slowest were:

457.0ms
(/ (/ (sqrt (/ (fma (* v v) -5 1) PI)) (sqrt (fma (* v v) -3 1))) (- 1 (* v v)))
360.0ms
(- (/ 1 (* t (* (sqrt 2) PI))) (+ (* 53/8 (/ (pow v 4) (* t (* (sqrt 2) PI)))) (* 5/2 (/ (pow v 2) (* t (* (sqrt 2) PI))))))
311.0ms
(- (+ (* (sqrt 2) (* v (sqrt -3))) (* 1/2 (/ (sqrt 2) (* v (sqrt -3))))) (* 1/8 (/ (sqrt 2) (* (pow v 3) (pow (sqrt -3) 3)))))

prune2.6s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize78.0ms

Local error

Found 4 expressions with local error:

0.4b
(/ (/ (sqrt (/ (fma v (* v -5) 1) PI)) (* (cbrt (sqrt (fma v (* v -6) 2))) (cbrt (sqrt (fma v (* v -6) 2))))) (/ (* t (- 1 (* v v))) (/ (sqrt (/ (fma (* v v) -5 1) PI)) (cbrt (sqrt (* 2 (fma (* v v) -3 1)))))))
0.1b
(/ (* t (- 1 (* v v))) (/ (sqrt (/ (fma (* v v) -5 1) PI)) (cbrt (sqrt (* 2 (fma (* v v) -3 1))))))
0.0b
(cbrt (sqrt (* 2 (fma (* v v) -3 1))))
0.0b
(cbrt (sqrt (fma v (* v -6) 2)))

rewrite79.0ms

Algorithm
rewrite-expression-head
Counts
4 → 895
Calls

4 calls. Slowest were:

25.0ms
(/ (/ (sqrt (/ (fma v (* v -5) 1) PI)) (* (cbrt (sqrt (fma v (* v -6) 2))) (cbrt (sqrt (fma v (* v -6) 2))))) (/ (* t (- 1 (* v v))) (/ (sqrt (/ (fma (* v v) -5 1) PI)) (cbrt (sqrt (* 2 (fma (* v v) -3 1)))))))
12.0ms
(/ (* t (- 1 (* v v))) (/ (sqrt (/ (fma (* v v) -5 1) PI)) (cbrt (sqrt (* 2 (fma (* v v) -3 1))))))
2.0ms
(cbrt (sqrt (* 2 (fma (* v v) -3 1))))

series1.8s

Counts
4 → 12
Calls

4 calls. Slowest were:

818.0ms
(/ (/ (sqrt (/ (fma v (* v -5) 1) PI)) (* (cbrt (sqrt (fma v (* v -6) 2))) (cbrt (sqrt (fma v (* v -6) 2))))) (/ (* t (- 1 (* v v))) (/ (sqrt (/ (fma (* v v) -5 1) PI)) (cbrt (sqrt (* 2 (fma (* v v) -3 1)))))))
517.0ms
(/ (* t (- 1 (* v v))) (/ (sqrt (/ (fma (* v v) -5 1) PI)) (cbrt (sqrt (* 2 (fma (* v v) -3 1))))))
294.0ms
(cbrt (sqrt (* 2 (fma (* v v) -3 1))))
134.0ms
(cbrt (sqrt (fma v (* v -6) 2)))

simplify3.8m

Counts
1552 → 907
Calls

1552 calls. Slowest were:

692.0ms
(/ (/ (* (cbrt (sqrt (/ (fma v (* v -5) 1) PI))) (cbrt (sqrt (/ (fma v (* v -5) 1) PI)))) (cbrt (sqrt (fma v (* v -6) 2)))) (/ t (/ (sqrt 1) 1)))
625.0ms
(/ (/ (* (* (sqrt (/ (fma v (* v -5) 1) PI)) (sqrt (/ (fma v (* v -5) 1) PI))) (sqrt (/ (fma v (* v -5) 1) PI))) (* (* (* (cbrt (sqrt (fma v (* v -6) 2))) (cbrt (sqrt (fma v (* v -6) 2)))) (* (cbrt (sqrt (fma v (* v -6) 2))) (cbrt (sqrt (fma v (* v -6) 2))))) (* (cbrt (sqrt (fma v (* v -6) 2))) (cbrt (sqrt (fma v (* v -6) 2)))))) (/ (* (* (* t (- 1 (* v v))) (* t (- 1 (* v v)))) (* t (- 1 (* v v)))) (* (* (/ (sqrt (/ (fma (* v v) -5 1) PI)) (cbrt (sqrt (* 2 (fma (* v v) -3 1))))) (/ (sqrt (/ (fma (* v v) -5 1) PI)) (cbrt (sqrt (* 2 (fma (* v v) -3 1)))))) (/ (sqrt (/ (fma (* v v) -5 1) PI)) (cbrt (sqrt (* 2 (fma (* v v) -3 1))))))))
622.0ms
(/ (sqrt (/ (sqrt (/ (fma v (* v -5) 1) PI)) (* (cbrt (sqrt (fma v (* v -6) 2))) (cbrt (sqrt (fma v (* v -6) 2)))))) (/ (- 1 (* v v)) (/ (sqrt (/ (fma (* v v) -5 1) PI)) (cbrt (sqrt (* 2 (fma (* v v) -3 1)))))))

prune19.6s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize40.0ms

Local error

Found 4 expressions with local error:

1.0b
(* (/ (sqrt (/ (fma (* v v) -5 1) PI)) (- 1 (* v v))) (sqrt (sqrt (/ (fma (* v v) -5 1) PI))))
0.2b
(* (/ (/ (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (cbrt (sqrt (fma v (* v -6) 2))))) t) (/ (* (/ (sqrt (/ (fma (* v v) -5 1) PI)) (- 1 (* v v))) (sqrt (sqrt (/ (fma (* v v) -5 1) PI)))) (* (* (cbrt (sqrt (fma (* v v) -6 2))) (cbrt (sqrt (fma (* v v) -6 2)))) (sqrt (cbrt (sqrt (fma (* v v) -6 2)))))))
0.1b
(/ (/ (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (cbrt (sqrt (fma v (* v -6) 2))))) t)
0.0b
(cbrt (sqrt (fma v (* v -6) 2)))

rewrite69.0ms

Algorithm
rewrite-expression-head
Counts
4 → 251
Calls

4 calls. Slowest were:

44.0ms
(* (/ (/ (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (cbrt (sqrt (fma v (* v -6) 2))))) t) (/ (* (/ (sqrt (/ (fma (* v v) -5 1) PI)) (- 1 (* v v))) (sqrt (sqrt (/ (fma (* v v) -5 1) PI)))) (* (* (cbrt (sqrt (fma (* v v) -6 2))) (cbrt (sqrt (fma (* v v) -6 2)))) (sqrt (cbrt (sqrt (fma (* v v) -6 2)))))))
10.0ms
(/ (/ (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (cbrt (sqrt (fma v (* v -6) 2))))) t)
9.0ms
(* (/ (sqrt (/ (fma (* v v) -5 1) PI)) (- 1 (* v v))) (sqrt (sqrt (/ (fma (* v v) -5 1) PI))))

series1.2s

Counts
4 → 12
Calls

4 calls. Slowest were:

541.0ms
(/ (/ (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (cbrt (sqrt (fma v (* v -6) 2))))) t)
290.0ms
(* (/ (/ (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (cbrt (sqrt (fma v (* v -6) 2))))) t) (/ (* (/ (sqrt (/ (fma (* v v) -5 1) PI)) (- 1 (* v v))) (sqrt (sqrt (/ (fma (* v v) -5 1) PI)))) (* (* (cbrt (sqrt (fma (* v v) -6 2))) (cbrt (sqrt (fma (* v v) -6 2)))) (sqrt (cbrt (sqrt (fma (* v v) -6 2)))))))
264.0ms
(* (/ (sqrt (/ (fma (* v v) -5 1) PI)) (- 1 (* v v))) (sqrt (sqrt (/ (fma (* v v) -5 1) PI))))
97.0ms
(cbrt (sqrt (fma v (* v -6) 2)))

simplify12.1s

Counts
220 → 263
Calls

220 calls. Slowest were:

828.0ms
(* (* (* (/ (/ (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (cbrt (sqrt (fma v (* v -6) 2))))) t) (/ (/ (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (cbrt (sqrt (fma v (* v -6) 2))))) t)) (/ (/ (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (cbrt (sqrt (fma v (* v -6) 2))))) t)) (/ (* (* (* (/ (sqrt (/ (fma (* v v) -5 1) PI)) (- 1 (* v v))) (/ (sqrt (/ (fma (* v v) -5 1) PI)) (- 1 (* v v)))) (/ (sqrt (/ (fma (* v v) -5 1) PI)) (- 1 (* v v)))) (* (* (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (sqrt (/ (fma (* v v) -5 1) PI)))) (sqrt (sqrt (/ (fma (* v v) -5 1) PI))))) (* (* (* (* (cbrt (sqrt (fma (* v v) -6 2))) (cbrt (sqrt (fma (* v v) -6 2)))) (sqrt (cbrt (sqrt (fma (* v v) -6 2))))) (* (* (cbrt (sqrt (fma (* v v) -6 2))) (cbrt (sqrt (fma (* v v) -6 2)))) (sqrt (cbrt (sqrt (fma (* v v) -6 2)))))) (* (* (cbrt (sqrt (fma (* v v) -6 2))) (cbrt (sqrt (fma (* v v) -6 2)))) (sqrt (cbrt (sqrt (fma (* v v) -6 2))))))))
814.0ms
(* (* (* (/ (/ (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (cbrt (sqrt (fma v (* v -6) 2))))) t) (/ (/ (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (cbrt (sqrt (fma v (* v -6) 2))))) t)) (/ (/ (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (cbrt (sqrt (fma v (* v -6) 2))))) t)) (/ (* (/ (* (* (sqrt (/ (fma (* v v) -5 1) PI)) (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (/ (fma (* v v) -5 1) PI))) (* (* (- 1 (* v v)) (- 1 (* v v))) (- 1 (* v v)))) (* (* (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (sqrt (/ (fma (* v v) -5 1) PI)))) (sqrt (sqrt (/ (fma (* v v) -5 1) PI))))) (* (* (* (* (cbrt (sqrt (fma (* v v) -6 2))) (cbrt (sqrt (fma (* v v) -6 2)))) (sqrt (cbrt (sqrt (fma (* v v) -6 2))))) (* (* (cbrt (sqrt (fma (* v v) -6 2))) (cbrt (sqrt (fma (* v v) -6 2)))) (sqrt (cbrt (sqrt (fma (* v v) -6 2)))))) (* (* (cbrt (sqrt (fma (* v v) -6 2))) (cbrt (sqrt (fma (* v v) -6 2)))) (sqrt (cbrt (sqrt (fma (* v v) -6 2))))))))
725.0ms
(* (* (* (/ (/ (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (cbrt (sqrt (fma v (* v -6) 2))))) t) (/ (/ (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (cbrt (sqrt (fma v (* v -6) 2))))) t)) (/ (/ (sqrt (sqrt (/ (fma (* v v) -5 1) PI))) (sqrt (cbrt (sqrt (fma v (* v -6) 2))))) t)) (/ (* (* (* (/ (sqrt (/ (fma (* v v) -5 1) PI)) (- 1 (* v v))) (sqrt (sqrt (/ (fma (* v v) -5 1) PI)))) (* (/ (sqrt (/ (fma (* v v) -5 1) PI)) (- 1 (* v v))) (sqrt (sqrt (/ (fma (* v v) -5 1) PI))))) (* (/ (sqrt (/ (fma (* v v) -5 1) PI)) (- 1 (* v v))) (sqrt (sqrt (/ (fma (* v v) -5 1) PI))))) (* (* (* (* (cbrt (sqrt (fma (* v v) -6 2))) (cbrt (sqrt (fma (* v v) -6 2)))) (sqrt (cbrt (sqrt (fma (* v v) -6 2))))) (* (* (cbrt (sqrt (fma (* v v) -6 2))) (cbrt (sqrt (fma (* v v) -6 2)))) (sqrt (cbrt (sqrt (fma (* v v) -6 2)))))) (* (* (cbrt (sqrt (fma (* v v) -6 2))) (cbrt (sqrt (fma (* v v) -6 2)))) (sqrt (cbrt (sqrt (fma (* v v) -6 2))))))))

prune5.7s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes110.0ms

Accuracy

0% (0.1b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 0.1b

bsearch2.0ms