Average Error: 0.4 → 0.1
Time: 1.3m
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. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{\pi}}{\sqrt{2 \cdot (\left(v \cdot v\right) \cdot -3 + 1)_*} \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)}}\]
  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 2019008 +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: 1.3m)Debug log

sample232.0ms

Algorithm
intervals

simplify105.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

105.0ms
(/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v))))

prune27.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize65.0ms

Local error

Found 4 expressions with local error:

0.4b
(* (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
(* t (- 1 (* v v)))
0.0b
(fma (* v v) -3 1)

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
4 → 85
Calls

4 calls. Slowest were:

14.0ms
(/ (/ (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))))
1.0ms
(* t (- 1 (* v v)))

series587.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

345.0ms
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))))
168.0ms
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))
50.0ms
(* t (- 1 (* v v)))
25.0ms
(fma (* v v) -3 1)

simplify5.7s

Counts
57 → 97
Calls

57 calls. Slowest were:

380.0ms
(- (/ 1 (* t (* (sqrt 2) PI))) (+ (* 53/8 (/ (pow v 4) (* t (* (sqrt 2) PI)))) (* 5/2 (/ (pow v 2) (* t (* (sqrt 2) PI))))))
342.0ms
(- (log (/ (fma (* v v) -5 1) PI)) (+ (log (sqrt (* 2 (fma (* v v) -3 1)))) (log (* t (- 1 (* v v))))))
335.0ms
(- (* t (sqrt 2)) (* 5/2 (* t (* (sqrt 2) (pow v 2)))))

prune1.4s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

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

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 128
Calls

4 calls. Slowest were:

11.0ms
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))) (* t (- 1 (* v v))))
5.0ms
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))
1.0ms
(* t (- 1 (* v v)))

series583.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

309.0ms
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))) (* t (- 1 (* v v))))
207.0ms
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))
38.0ms
(* t (- 1 (* v v)))
29.0ms
(fma (* v v) -3 1)

simplify8.9s

Counts
130 → 140
Calls

130 calls. Slowest were:

292.0ms
(- (/ 1 (* t (* (sqrt 2) PI))) (+ (* 53/8 (/ (pow v 4) (* t (* (sqrt 2) PI)))) (* 5/2 (/ (pow v 2) (* t (* (sqrt 2) PI))))))
278.0ms
(- (log (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))) (log (* t (- 1 (* v v)))))
257.0ms
(/ (/ (/ (fma (* v v) -5 1) PI) (cbrt (sqrt (* 2 (fma (* v v) -3 1))))) (- 1 (* v v)))

prune1.9s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize26.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (/ (fma v (* v -5) 1) t) (/ (/ (/ 1 PI) (- 1 (* v v))) (sqrt (fma (* v v) -6 2))))
0.0b
(/ (/ (/ 1 PI) (- 1 (* v v))) (sqrt (fma (* v v) -6 2)))
0.0b
(/ (/ 1 PI) (- 1 (* v v)))
0.0b
(fma (* v v) -6 2)

rewrite49.0ms

Algorithm
rewrite-expression-head
Counts
4 → 331
Calls

4 calls. Slowest were:

15.0ms
(* (/ (fma v (* v -5) 1) t) (/ (/ (/ 1 PI) (- 1 (* v v))) (sqrt (fma (* v v) -6 2))))
14.0ms
(/ (/ (/ 1 PI) (- 1 (* v v))) (sqrt (fma (* v v) -6 2)))
6.0ms
(/ (/ 1 PI) (- 1 (* v v)))

series452.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

231.0ms
(* (/ (fma v (* v -5) 1) t) (/ (/ (/ 1 PI) (- 1 (* v v))) (sqrt (fma (* v v) -6 2))))
144.0ms
(/ (/ (/ 1 PI) (- 1 (* v v))) (sqrt (fma (* v v) -6 2)))
53.0ms
(/ (/ 1 PI) (- 1 (* v v)))
23.0ms
(fma (* v v) -6 2)

simplify35.8s

Counts
415 → 343
Calls

415 calls. Slowest were:

597.0ms
(/ (/ (/ 1 PI) (- 1 (* v v))) (sqrt (cbrt (fma (* v v) -6 2))))
529.0ms
(/ (* (* (/ (/ 1 PI) (- 1 (* v v))) (/ (/ 1 PI) (- 1 (* v v)))) (/ (/ 1 PI) (- 1 (* v v)))) (* (* (sqrt (fma (* v v) -6 2)) (sqrt (fma (* v v) -6 2))) (sqrt (fma (* v v) -6 2))))
429.0ms
(/ (/ (/ 1 PI) (- (pow 1 3) (pow (* v v) 3))) (sqrt 1))

prune3.9s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize17.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ (fma (* v v) -5 1) (sqrt 2))
0.4b
(* (/ (/ (fma (* v v) -5 1) (sqrt 2)) t) (/ (/ (/ 1 PI) (sqrt (fma (* v v) -3 1))) (- 1 (* v v))))
0.4b
(/ (/ (fma (* v v) -5 1) (sqrt 2)) t)
0.0b
(/ (/ 1 PI) (sqrt (fma (* v v) -3 1)))

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
4 → 218
Calls

4 calls. Slowest were:

12.0ms
(* (/ (/ (fma (* v v) -5 1) (sqrt 2)) t) (/ (/ (/ 1 PI) (sqrt (fma (* v v) -3 1))) (- 1 (* v v))))
8.0ms
(/ (/ (fma (* v v) -5 1) (sqrt 2)) t)
4.0ms
(/ (/ 1 PI) (sqrt (fma (* v v) -3 1)))

series451.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

253.0ms
(* (/ (/ (fma (* v v) -5 1) (sqrt 2)) t) (/ (/ (/ 1 PI) (sqrt (fma (* v v) -3 1))) (- 1 (* v v))))
93.0ms
(/ (/ 1 PI) (sqrt (fma (* v v) -3 1)))
53.0ms
(/ (fma (* v v) -5 1) (sqrt 2))
52.0ms
(/ (/ (fma (* v v) -5 1) (sqrt 2)) t)

simplify10.7s

Counts
212 → 230
Calls

212 calls. Slowest were:

509.0ms
(* (* (* (/ (/ (fma (* v v) -5 1) (sqrt 2)) t) (/ (/ (fma (* v v) -5 1) (sqrt 2)) t)) (/ (/ (fma (* v v) -5 1) (sqrt 2)) t)) (/ (* (* (/ (/ 1 PI) (sqrt (fma (* v v) -3 1))) (/ (/ 1 PI) (sqrt (fma (* v v) -3 1)))) (/ (/ 1 PI) (sqrt (fma (* v v) -3 1)))) (* (* (- 1 (* v v)) (- 1 (* v v))) (- 1 (* v v)))))
290.0ms
(/ (* (* (/ 1 PI) (/ 1 PI)) (/ 1 PI)) (* (* (sqrt (fma (* v v) -3 1)) (sqrt (fma (* v v) -3 1))) (sqrt (fma (* v v) -3 1))))
278.0ms
(* (/ (fma (* v v) -5 1) (sqrt 2)) (/ (/ 1 PI) (sqrt (fma (* v v) -3 1))))

prune2.3s

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

regimes96.0ms

Accuracy

0% (0.1b remaining)

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

bsearch2.0ms

end0.0ms

sample2.8s

Algorithm
intervals