Average Error: 0.5 → 0.1
Time: 1.4m
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 0.5

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

  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 2019002 +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

sample182.0ms

Algorithm
intervals

simplify134.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

134.0ms
(/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v))))

prune23.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize33.0ms

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

rewrite20.0ms

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

series659.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

328.0ms
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))))
166.0ms
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))
112.0ms
(/ (fma (* v v) -5 1) PI)
52.0ms
(sqrt (* 2 (fma (* v v) -3 1)))

simplify6.7s

Counts
51 → 94
Calls

51 calls. Slowest were:

415.0ms
(- (log (/ (fma (* v v) -5 1) PI)) (+ (log (sqrt (* 2 (fma (* v v) -3 1)))) (log (* t (- 1 (* v v))))))
407.0ms
(- (/ 1 (* t (* (sqrt 2) PI))) (+ (* 53/8 (/ (pow v 4) (* t (* (sqrt 2) PI)))) (* 5/2 (/ (pow v 2) (* t (* (sqrt 2) PI))))))
369.0ms
(- (* 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: 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)
0.0b
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))
0.0b
(sqrt (* 2 (fma (* v v) -3 1)))

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
4 → 125
Calls

4 calls. Slowest were:

19.0ms
(/ (/ (/ (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)

series698.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

350.0ms
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))) (* t (- 1 (* v v))))
189.0ms
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))
111.0ms
(/ (fma (* v v) -5 1) PI)
48.0ms
(sqrt (* 2 (fma (* v v) -3 1)))

simplify10.0s

Counts
124 → 137
Calls

124 calls. Slowest were:

625.0ms
(- (+ (/ (sqrt -1/3) (* PI (* (sqrt 2) v))) (+ (* 5/648 (/ 1 (* (pow v 3) (* (sqrt 2) (* (pow (sqrt -1/3) 3) PI))))) (+ (* 1/27 (/ 1 (* (pow v 3) (* (sqrt 2) (* (sqrt -1/3) PI))))) (* 5/18 (/ 1 (* (sqrt 2) (* v (* PI (sqrt -1/3))))))))) (* 5 (/ (* v (sqrt -1/3)) (* (sqrt 2) PI))))
402.0ms
(- (/ 1 (* t (* (sqrt 2) PI))) (+ (* 53/8 (/ (pow v 4) (* t (* (sqrt 2) PI)))) (* 5/2 (/ (pow v 2) (* t (* (sqrt 2) PI))))))
305.0ms
(- (+ (* (sqrt 2) (* v (sqrt -3))) (* 1/2 (/ (sqrt 2) (* v (sqrt -3))))) (* 1/8 (/ (sqrt 2) (* (pow v 3) (pow (sqrt -3) 3)))))

prune1.9s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize16.0ms

Local error

Found 4 expressions with local error:

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

rewrite43.0ms

Algorithm
rewrite-expression-head
Counts
4 → 100
Calls

4 calls. Slowest were:

24.0ms
(/ (/ (* (cbrt (fma (* v v) -5 1)) (cbrt (fma (* v v) -5 1))) (/ PI (cbrt (fma (* v v) -5 1)))) (* (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
(cbrt (fma (* v v) -5 1))

series730.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

315.0ms
(/ (/ (* (cbrt (fma (* v v) -5 1)) (cbrt (fma (* v v) -5 1))) (/ PI (cbrt (fma (* v v) -5 1)))) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))))
160.0ms
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))
142.0ms
(cbrt (fma (* v v) -5 1))
112.0ms
(cbrt (fma (* v v) -5 1))

simplify11.2s

Counts
77 → 112
Calls

77 calls. Slowest were:

538.0ms
(/ (* (* (/ (* (cbrt (fma (* v v) -5 1)) (cbrt (fma (* v v) -5 1))) (/ PI (cbrt (fma (* v v) -5 1)))) (/ (* (cbrt (fma (* v v) -5 1)) (cbrt (fma (* v v) -5 1))) (/ PI (cbrt (fma (* v v) -5 1))))) (/ (* (cbrt (fma (* v v) -5 1)) (cbrt (fma (* v v) -5 1))) (/ PI (cbrt (fma (* v v) -5 1))))) (* (* (* (sqrt (* 2 (fma (* v v) -3 1))) (sqrt (* 2 (fma (* v v) -3 1)))) (sqrt (* 2 (fma (* v v) -3 1)))) (* (* (* t (- 1 (* v v))) (* t (- 1 (* v v)))) (* t (- 1 (* v v))))))
496.0ms
(/ (/ (* (* (* (cbrt (fma (* v v) -5 1)) (cbrt (fma (* v v) -5 1))) (* (cbrt (fma (* v v) -5 1)) (cbrt (fma (* v v) -5 1)))) (* (cbrt (fma (* v v) -5 1)) (cbrt (fma (* v v) -5 1)))) (* (* (/ PI (cbrt (fma (* v v) -5 1))) (/ PI (cbrt (fma (* v v) -5 1)))) (/ PI (cbrt (fma (* v v) -5 1))))) (* (* (* (sqrt (* 2 (fma (* v v) -3 1))) (sqrt (* 2 (fma (* v v) -3 1)))) (sqrt (* 2 (fma (* v v) -3 1)))) (* (* (* t (- 1 (* v v))) (* t (- 1 (* v v)))) (* t (- 1 (* v v))))))
436.0ms
(/ (* (* (/ (* (cbrt (fma (* v v) -5 1)) (cbrt (fma (* v v) -5 1))) (/ PI (cbrt (fma (* v v) -5 1)))) (/ (* (cbrt (fma (* v v) -5 1)) (cbrt (fma (* v v) -5 1))) (/ PI (cbrt (fma (* v v) -5 1))))) (/ (* (cbrt (fma (* v v) -5 1)) (cbrt (fma (* v v) -5 1))) (/ PI (cbrt (fma (* v v) -5 1))))) (* (* (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))))

prune1.9s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize69.0ms

Local error

Found 4 expressions with local error:

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

rewrite35.0ms

Algorithm
rewrite-expression-head
Counts
4 → 464
Calls

4 calls. Slowest were:

17.0ms
(* (/ (/ (cbrt (fma v (* v -5) 1)) (sqrt (fma -6 (* v v) 2))) (sqrt (/ PI (cbrt (fma v (* v -5) 1))))) (/ (/ (cbrt (fma (* v v) -5 1)) (sqrt (/ PI (cbrt (fma (* v v) -5 1))))) (* t (- 1 (* v v)))))
4.0ms
(/ (cbrt (fma v (* v -5) 1)) (sqrt (fma -6 (* v v) 2)))
1.0ms
(sqrt (/ PI (cbrt (fma (* v v) -5 1))))

series805.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

263.0ms
(* (/ (/ (cbrt (fma v (* v -5) 1)) (sqrt (fma -6 (* v v) 2))) (sqrt (/ PI (cbrt (fma v (* v -5) 1))))) (/ (/ (cbrt (fma (* v v) -5 1)) (sqrt (/ PI (cbrt (fma (* v v) -5 1))))) (* t (- 1 (* v v)))))
204.0ms
(/ (cbrt (fma v (* v -5) 1)) (sqrt (fma -6 (* v v) 2)))
172.0ms
(sqrt (/ PI (cbrt (fma v (* v -5) 1))))
165.0ms
(sqrt (/ PI (cbrt (fma (* v v) -5 1))))

simplify28.5s

Counts
469 → 476
Calls

469 calls. Slowest were:

723.0ms
(+ (- (- (log (cbrt (fma v (* v -5) 1))) (log (sqrt (fma -6 (* v v) 2)))) (log (sqrt (/ PI (cbrt (fma v (* v -5) 1)))))) (- (log (/ (cbrt (fma (* v v) -5 1)) (sqrt (/ PI (cbrt (fma (* v v) -5 1)))))) (log (* t (- 1 (* v v))))))
705.0ms
(* (* (* (/ (/ (cbrt (fma v (* v -5) 1)) (sqrt (fma -6 (* v v) 2))) (sqrt (/ PI (cbrt (fma v (* v -5) 1))))) (/ (/ (cbrt (fma v (* v -5) 1)) (sqrt (fma -6 (* v v) 2))) (sqrt (/ PI (cbrt (fma v (* v -5) 1)))))) (/ (/ (cbrt (fma v (* v -5) 1)) (sqrt (fma -6 (* v v) 2))) (sqrt (/ PI (cbrt (fma v (* v -5) 1)))))) (/ (* (* (/ (cbrt (fma (* v v) -5 1)) (sqrt (/ PI (cbrt (fma (* v v) -5 1))))) (/ (cbrt (fma (* v v) -5 1)) (sqrt (/ PI (cbrt (fma (* v v) -5 1)))))) (/ (cbrt (fma (* v v) -5 1)) (sqrt (/ PI (cbrt (fma (* v v) -5 1)))))) (* (* (* t (- 1 (* v v))) (* t (- 1 (* v v)))) (* t (- 1 (* v v))))))
589.0ms
(+ (- (log (/ (cbrt (fma v (* v -5) 1)) (sqrt (fma -6 (* v v) 2)))) (log (sqrt (/ PI (cbrt (fma v (* v -5) 1)))))) (- (- (log (cbrt (fma (* v v) -5 1))) (log (sqrt (/ PI (cbrt (fma (* v v) -5 1)))))) (log (* t (- 1 (* v v))))))

prune11.6s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes177.0ms

Accuracy

0% (0.1b remaining)

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

bsearch2.0ms

end0.0ms

sample3.7s

Algorithm
intervals