Average Error: 0.5 → 0.6
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1}{\left(\sqrt{2} \cdot \pi\right) \cdot t} - \left(\frac{5}{2} \cdot \frac{{v}^{2}}{\left(\sqrt{2} \cdot \pi\right) \cdot t} + \frac{{v}^{4}}{\left(\sqrt{2} \cdot \pi\right) \cdot t} \cdot \frac{53}{8}\right)\]

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \left(\frac{53}{8} \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{5}{2} \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)}\]

  3. Final simplification0.6

    \[\leadsto \frac{1}{\left(\sqrt{2} \cdot \pi\right) \cdot t} - \left(\frac{5}{2} \cdot \frac{{v}^{2}}{\left(\sqrt{2} \cdot \pi\right) \cdot t} + \frac{{v}^{4}}{\left(\sqrt{2} \cdot \pi\right) \cdot t} \cdot \frac{53}{8}\right)\]

Reproduce

herbie shell --seed 2019004 
(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

sample189.0ms

Algorithm
intervals

simplify109.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

109.0ms
(/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v))))

prune15.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.4b

localize56.0ms

Local error

Found 4 expressions with local error:

0.4b
(* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v))))))
0.3b
(* PI t)
0.2b
(/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v))))
0.2b
(* 5 (* v v))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

18.0ms
(/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v))))
5.0ms
(* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v))))))
1.0ms
(* 5 (* v v))

series569.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

393.0ms
(/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v))))
125.0ms
(* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v))))))
33.0ms
(* PI t)
17.0ms
(* 5 (* v v))

simplify6.3s

Counts
47 → 81
Calls

47 calls. Slowest were:

598.0ms
(- (* 1 1) (* (* 5 (* v v)) (* 5 (* v v))))
425.0ms
(* (* (* (* PI t) (* PI t)) (* PI t)) (* (* (sqrt (* 2 (- 1 (* 3 (* v v))))) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (sqrt (* 2 (- 1 (* 3 (* v v)))))))
353.0ms
(* (sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v)))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))

prune1.2s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize51.0ms

Local error

Found 4 expressions with local error:

5.5b
(/ (pow v 2) (* t (* (sqrt 2) PI)))
1.8b
(/ (pow v 4) (* t (* (sqrt 2) PI)))
0.2b
(* t (* (sqrt 2) PI))
0.2b
(* t (* (sqrt 2) PI))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

4.0ms
(/ (pow v 2) (* t (* (sqrt 2) PI)))
4.0ms
(/ (pow v 4) (* t (* (sqrt 2) PI)))
3.0ms
(* t (* (sqrt 2) PI))

series399.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

114.0ms
(/ (pow v 2) (* t (* (sqrt 2) PI)))
106.0ms
(/ (pow v 4) (* t (* (sqrt 2) PI)))
96.0ms
(* t (* (sqrt 2) PI))
83.0ms
(* t (* (sqrt 2) PI))

simplify1.8s

Counts
38 → 72
Calls

38 calls. Slowest were:

313.0ms
(- (log (pow v 2)) (log (* t (* (sqrt 2) PI))))
211.0ms
(/ (* (* (pow v 2) (pow v 2)) (pow v 2)) (* (* (* t (* (sqrt 2) PI)) (* t (* (sqrt 2) PI))) (* t (* (sqrt 2) PI))))
178.0ms
(- (log (pow v 4)) (log (* t (* (sqrt 2) PI))))

prune1.1s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize33.0ms

Local error

Found 4 expressions with local error:

0.4b
(/ (/ (sqrt (+ 1 (* -5 (* v v)))) (* PI t)) (sqrt (+ (* (* v v) -6) 2)))
0.3b
(* PI t)
0.2b
(/ (sqrt (+ 1 (* -5 (* v v)))) (* PI t))
0.2b
(* 5 (* v v))

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
4 → 137
Calls

4 calls. Slowest were:

18.0ms
(/ (/ (sqrt (+ 1 (* -5 (* v v)))) (* PI t)) (sqrt (+ (* (* v v) -6) 2)))
7.0ms
(/ (sqrt (+ 1 (* -5 (* v v)))) (* PI t))
2.0ms
(* 5 (* v v))

series346.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

188.0ms
(/ (/ (sqrt (+ 1 (* -5 (* v v)))) (* PI t)) (sqrt (+ (* (* v v) -6) 2)))
104.0ms
(/ (sqrt (+ 1 (* -5 (* v v)))) (* PI t))
35.0ms
(* PI t)
19.0ms
(* 5 (* v v))

simplify8.1s

Counts
175 → 149
Calls

175 calls. Slowest were:

705.0ms
(/ (* (* (sqrt (+ 1 (* -5 (* v v)))) (sqrt (+ 1 (* -5 (* v v))))) (sqrt (+ 1 (* -5 (* v v))))) (* (* (* PI t) (* PI t)) (* PI t)))
479.0ms
(+ (* 1/18 (/ 1 (* t (* (sqrt 5/6) (* (pow v 2) PI))))) (/ (sqrt 5/6) (* t PI)))
449.0ms
(sqrt (+ (pow 1 3) (pow (* -5 (* v v)) 3)))

prune2.5s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize52.0ms

Local error

Found 4 expressions with local error:

0.4b
(/ (/ (sqrt (+ 1 (* -5 (* v v)))) (* PI t)) (sqrt (+ (pow (* (* v v) -6) 3) (pow 2 3))))
0.3b
(* PI t)
0.2b
(/ (sqrt (+ 1 (* -5 (* v v)))) (* PI t))
0.2b
(* 5 (* v v))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 159
Calls

4 calls. Slowest were:

14.0ms
(/ (/ (sqrt (+ 1 (* -5 (* v v)))) (* PI t)) (sqrt (+ (pow (* (* v v) -6) 3) (pow 2 3))))
7.0ms
(/ (sqrt (+ 1 (* -5 (* v v)))) (* PI t))
2.0ms
(* 5 (* v v))

series332.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

178.0ms
(/ (/ (sqrt (+ 1 (* -5 (* v v)))) (* PI t)) (sqrt (+ (pow (* (* v v) -6) 3) (pow 2 3))))
102.0ms
(/ (sqrt (+ 1 (* -5 (* v v)))) (* PI t))
41.0ms
(* PI t)
11.0ms
(* 5 (* v v))

simplify48.2s

Counts
217 → 171
Calls

217 calls. Slowest were:

695.0ms
(/ (/ (sqrt (sqrt (+ 1 (* -5 (* v v))))) t) (cbrt (sqrt (+ (pow (* (* v v) -6) 3) (pow 2 3)))))
646.0ms
(/ (* (* (sqrt (+ 1 (* -5 (* v v)))) (sqrt (+ 1 (* -5 (* v v))))) (sqrt (+ 1 (* -5 (* v v))))) (* (* (* PI t) (* PI t)) (* PI t)))
511.0ms
(/ (cbrt (/ (sqrt (+ 1 (* -5 (* v v)))) (* PI t))) (sqrt (cbrt (+ (pow (* (* v v) -6) 3) (pow 2 3)))))

prune4.9s

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0b

regimes337.0ms

Accuracy

0% (0.6b remaining)

Error of 0.6b against oracle of 0.0b and baseline of 0.6b

bsearch5.0ms

end0.0ms

sample3.3s

Algorithm
intervals