Average Error: 0.5 → 0.1
Time: 51.8s
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 2019005 +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: 51.0s)Debug log

sample179.0ms

Algorithm
intervals

simplify135.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

135.0ms
(/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v))))

prune34.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.3b

localize56.0ms

Local error

Found 4 expressions with local error:

0.6b
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))))
0.5b
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))
0.0b
(/ (fma (* v v) -5 1) PI)
0.0b
(- 1 (* v v))

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

15.0ms
(/ (/ (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)

series663.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

301.0ms
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))))
185.0ms
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))
145.0ms
(/ (fma (* v v) -5 1) PI)
31.0ms
(- 1 (* v v))

simplify5.2s

Counts
50 → 96
Calls

50 calls. Slowest were:

371.0ms
(- (log (/ (fma (* v v) -5 1) PI)) (+ (log (sqrt (* 2 (fma (* v v) -3 1)))) (log (* t (- 1 (* v v))))))
318.0ms
(- (* t (sqrt 2)) (* 5/2 (* t (* (sqrt 2) (pow v 2)))))
311.0ms
(- (/ 1 (* t (* (sqrt 2) PI))) (+ (* 53/8 (/ (pow v 4) (* t (* (sqrt 2) PI)))) (* 5/2 (/ (pow v 2) (* t (* (sqrt 2) PI))))))

prune1.4s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0b

localize6.0ms

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
(- 1 (* v v))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 127
Calls

4 calls. Slowest were:

9.0ms
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))) (* t (- 1 (* v v))))
8.0ms
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))
2.0ms
(/ (fma (* v v) -5 1) PI)

series655.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

317.0ms
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))) (* t (- 1 (* v v))))
204.0ms
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))
103.0ms
(/ (fma (* v v) -5 1) PI)
31.0ms
(- 1 (* v v))

simplify9.4s

Counts
123 → 139
Calls

123 calls. Slowest were:

521.0ms
(- (/ 1 (* (sqrt 2) PI)) (+ (* 33/8 (/ (pow v 4) (* (sqrt 2) PI))) (* 7/2 (/ (pow v 2) (* (sqrt 2) PI)))))
373.0ms
(- (/ 1 (* t (* (sqrt 2) PI))) (+ (* 53/8 (/ (pow v 4) (* t (* (sqrt 2) PI)))) (* 5/2 (/ (pow v 2) (* t (* (sqrt 2) PI))))))
272.0ms
(/ (/ (* (* (/ (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))))) (* (* (* t (- 1 (* v v))) (* t (- 1 (* v v)))) (* t (- 1 (* v v)))))

prune1.9s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize20.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ (fma (* v v) -5 1) (sqrt 2))
0.5b
(* (/ (/ (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))))
6.0ms
(/ (fma (* v v) -5 1) (sqrt 2))
6.0ms
(/ (/ (fma (* v v) -5 1) (sqrt 2)) t)

series572.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

308.0ms
(* (/ (/ (fma (* v v) -5 1) (sqrt 2)) t) (/ (/ (/ 1 PI) (sqrt (fma (* v v) -3 1))) (- 1 (* v v))))
125.0ms
(/ (/ 1 PI) (sqrt (fma (* v v) -3 1)))
73.0ms
(/ (/ (fma (* v v) -5 1) (sqrt 2)) t)
66.0ms
(/ (fma (* v v) -5 1) (sqrt 2))

simplify12.4s

Counts
212 → 230
Calls

212 calls. Slowest were:

436.0ms
(* (/ (fma (* v v) -5 1) (sqrt 2)) (/ (/ 1 PI) (sqrt (fma (* v v) -3 1))))
353.0ms
(- (+ (* 1/54 (/ 1 (* (pow v 5) (* (sqrt -1/3) PI)))) (+ (* 1/18 (/ 1 (* (pow v 3) (* PI (sqrt -1/3))))) (* 1/648 (/ 1 (* (pow v 5) (* (pow (sqrt -1/3) 3) PI)))))) (/ (sqrt -1/3) (* PI v)))
318.0ms
(/ (* (* (/ 1 PI) (/ 1 PI)) (/ 1 PI)) (* (* (sqrt (fma (* v v) -3 1)) (sqrt (fma (* v v) -3 1))) (sqrt (fma (* v v) -3 1))))

prune3.5s

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0b

localize25.0ms

Local error

Found 4 expressions with local error:

0.6b
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- (* 1 1) (* (* v v) (* v v))))))
0.4b
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- (* 1 1) (* (* v v) (* v v)))))
0.1b
(* (* v v) (* v v))
0.0b
(/ (fma (* v v) -5 1) PI)

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
4 → 95
Calls

4 calls. Slowest were:

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

series600.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

296.0ms
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- (* 1 1) (* (* v v) (* v v))))))
166.0ms
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- (* 1 1) (* (* v v) (* v v)))))
125.0ms
(/ (fma (* v v) -5 1) PI)
14.0ms
(* (* v v) (* v v))

simplify7.8s

Counts
62 → 107
Calls

62 calls. Slowest were:

450.0ms
(/ (/ 1 PI) (* t (- (* 1 1) (* (* v v) (* v v)))))
383.0ms
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- (* 1 1) (* (* v v) (* v v)))))
371.0ms
(* (* (- (* (* v v) (* v v))) t) (sqrt (* 2 (fma (* v v) -3 1))))

prune1.7s

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0b

regimes190.0ms

Accuracy

0% (0.1b remaining)

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

bsearch4.0ms

end0.0ms

sample4.4s

Algorithm
intervals