Average Error: 0.4 → 0.1
Time: 3.2m
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 2018360 +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: 3.1m)Debug log

start187.0ms

Algorithm
intervals

setup130.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize37.0ms

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

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

8.0ms
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))))
7.0ms
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))
3.0ms
(/ (fma (* v v) -5 1) PI)

series702.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

338.0ms
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))))
200.0ms
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))
116.0ms
(/ (fma (* v v) -5 1) PI)
49.0ms
(sqrt (* 2 (fma (* v v) -3 1)))

simplify6.8s

Counts
51 → 94
Calls

51 calls. Slowest were:

747.0ms
(- (sqrt 2) (+ (* 9/8 (* (sqrt 2) (pow v 4))) (* 3/2 (* (sqrt 2) (pow v 2)))))
364.0ms
(- (log (/ (fma (* v v) -5 1) PI)) (log (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))))
363.0ms
(- (* t (sqrt 2)) (* 5/2 (* t (* (sqrt 2) (pow v 2)))))

prune1.7s

Pruning

5 alts after pruning (5 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
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))
0.0b
(/ (fma (* v v) -5 1) PI)
0.0b
(sqrt (* 2 (fma (* v v) -3 1)))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 125
Calls

4 calls. Slowest were:

14.0ms
(/ (/ (/ (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))))
2.0ms
(/ (fma (* v v) -5 1) PI)

series746.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

378.0ms
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))) (* t (- 1 (* v v))))
211.0ms
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))
111.0ms
(/ (fma (* v v) -5 1) PI)
46.0ms
(sqrt (* 2 (fma (* v v) -3 1)))

simplify9.9s

Counts
124 → 137
Calls

124 calls. Slowest were:

717.0ms
(/ (* (* (/ (fma (* v v) -5 1) PI) (/ (fma (* v v) -5 1) PI)) (/ (fma (* v v) -5 1) PI)) (* (* (sqrt (* 2 (fma (* v v) -3 1))) (sqrt (* 2 (fma (* v v) -3 1)))) (sqrt (* 2 (fma (* v v) -3 1)))))
410.0ms
(- (sqrt 2) (+ (* 9/8 (* (sqrt 2) (pow v 4))) (* 3/2 (* (sqrt 2) (pow v 2)))))
325.0ms
(- (/ 1 (* t (* (sqrt 2) PI))) (+ (* 53/8 (/ (pow v 4) (* t (* (sqrt 2) PI)))) (* 5/2 (/ (pow v 2) (* t (* (sqrt 2) PI))))))

prune2.6s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize17.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ 1 (sqrt (fma -6 (* v v) 2)))
0.5b
(* (/ 1 (sqrt (fma -6 (* v v) 2))) (/ (/ (fma (* v v) -5 1) PI) (* t (- 1 (* v v)))))
0.4b
(/ (/ (fma (* v v) -5 1) PI) (* t (- 1 (* v v))))
0.0b
(/ (fma (* v v) -5 1) PI)

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 97
Calls

4 calls. Slowest were:

15.0ms
(* (/ 1 (sqrt (fma -6 (* v v) 2))) (/ (/ (fma (* v v) -5 1) PI) (* t (- 1 (* v v)))))
7.0ms
(/ (/ (fma (* v v) -5 1) PI) (* t (- 1 (* v v))))
2.0ms
(/ (fma (* v v) -5 1) PI)

series489.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

232.0ms
(* (/ 1 (sqrt (fma -6 (* v v) 2))) (/ (/ (fma (* v v) -5 1) PI) (* t (- 1 (* v v)))))
127.0ms
(/ (/ (fma (* v v) -5 1) PI) (* t (- 1 (* v v))))
85.0ms
(/ (fma (* v v) -5 1) PI)
44.0ms
(/ 1 (sqrt (fma -6 (* v v) 2)))

simplify6.5s

Counts
62 → 109
Calls

62 calls. Slowest were:

449.0ms
(* 1 (/ (/ (fma (* v v) -5 1) PI) (* t (- 1 (* v v)))))
398.0ms
(- (/ (sqrt -1/6) v) (+ (* 1/108 (/ 1 (* (pow v 5) (sqrt -1/6)))) (+ (* 1/2592 (/ 1 (* (pow v 5) (pow (sqrt -1/6) 3)))) (* 1/36 (/ 1 (* (pow v 3) (sqrt -1/6)))))))
333.0ms
(+ (- (log (sqrt (fma -6 (* v v) 2)))) (- (log (/ (fma (* v v) -5 1) PI)) (log (* t (- 1 (* v v))))))

prune1.9s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize43.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ (sqrt (/ (/ (fma v (* v -5) 1) PI) (sqrt (fma -6 (* v v) 2)))) (/ (* t (- 1 (* v v))) (sqrt (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))))))
0.3b
(/ (* t (- 1 (* v v))) (sqrt (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))))
0.0b
(/ (/ (fma v (* v -5) 1) PI) (sqrt (fma -6 (* v v) 2)))
0.0b
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))

rewrite98.0ms

Algorithm
rewrite-expression-head
Counts
4 → 1097
Calls

4 calls. Slowest were:

21.0ms
(/ (sqrt (/ (/ (fma v (* v -5) 1) PI) (sqrt (fma -6 (* v v) 2)))) (/ (* t (- 1 (* v v))) (sqrt (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))))))
10.0ms
(/ (* t (- 1 (* v v))) (sqrt (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))))
5.0ms
(/ (/ (fma v (* v -5) 1) PI) (sqrt (fma -6 (* v v) 2)))

series1.3s

Counts
4 → 12
Calls

4 calls. Slowest were:

535.0ms
(/ (sqrt (/ (/ (fma v (* v -5) 1) PI) (sqrt (fma -6 (* v v) 2)))) (/ (* t (- 1 (* v v))) (sqrt (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))))))
474.0ms
(/ (* t (- 1 (* v v))) (sqrt (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))))
173.0ms
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))
152.0ms
(/ (/ (fma v (* v -5) 1) PI) (sqrt (fma -6 (* v v) 2)))

simplify2.2m

Counts
1988 → 1109
Calls

1988 calls. Slowest were:

586.0ms
(/ (sqrt (sqrt (/ (/ (fma v (* v -5) 1) PI) (sqrt (fma -6 (* v v) 2))))) (sqrt (/ (* t (- 1 (* v v))) (sqrt (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))))))
423.0ms
(/ (* (* (/ (fma (* v v) -5 1) PI) (/ (fma (* v v) -5 1) PI)) (/ (fma (* v v) -5 1) PI)) (* (* (sqrt (* 2 (fma (* v v) -3 1))) (sqrt (* 2 (fma (* v v) -3 1)))) (sqrt (* 2 (fma (* v v) -3 1)))))
348.0ms
(/ (sqrt (/ (sqrt (/ (fma v (* v -5) 1) PI)) (sqrt (sqrt (fma -6 (* v v) 2))))) (/ (- 1 (* v v)) (sqrt (/ 1 (sqrt (* 2 (fma (* v v) -3 1)))))))

prune21.1s

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

regimes129.0ms

Accuracy

0% (0.1b remaining)

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

bsearch2.0ms