Average Error: 0.4 → 0.1
Time: 9.8m
Precision: 64
Internal Precision: 128
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{1 + v \cdot \left(v \cdot -5\right)}{\pi}}{\sqrt{\left(v \cdot 2\right) \cdot \left(v \cdot -3\right) + 2}}}{t \cdot \left(1 - v \cdot v\right)}\]

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 0.4

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

  2. Initial simplification0.3

    \[\leadsto \frac{\frac{1 + \left(-5 \cdot v\right) \cdot v}{\pi}}{\sqrt{\left(-3 \cdot v\right) \cdot \left(v \cdot 2\right) + 2} \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)}\]
  3. Using strategy rm

  4. Applied associate-/r*0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2018360 
(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: 9.8m)Debug log

start170.0ms

Algorithm
intervals

setup160.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize42.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (sqrt (+ (* (* -3 v) (* v 2)) 2)) (* t (- 1 (* v v))))
0.4b
(/ (/ (+ 1 (* (* -5 v) v)) PI) (* (sqrt (+ (* (* -3 v) (* v 2)) 2)) (* t (- 1 (* v v)))))
0.2b
(* (* -5 v) v)
0.1b
(* (* -3 v) (* v 2))

rewrite44.0ms

Algorithm
rewrite-expression-head
Counts
4 → 88
Calls

4 calls. Slowest were:

20.0ms
(/ (/ (+ 1 (* (* -5 v) v)) PI) (* (sqrt (+ (* (* -3 v) (* v 2)) 2)) (* t (- 1 (* v v)))))
16.0ms
(* (sqrt (+ (* (* -3 v) (* v 2)) 2)) (* t (- 1 (* v v))))
4.0ms
(* (* -3 v) (* v 2))

series463.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

265.0ms
(/ (/ (+ 1 (* (* -5 v) v)) PI) (* (sqrt (+ (* (* -3 v) (* v 2)) 2)) (* t (- 1 (* v v)))))
157.0ms
(* (sqrt (+ (* (* -3 v) (* v 2)) 2)) (* t (- 1 (* v v))))
21.0ms
(* (* -3 v) (* v 2))
20.0ms
(* (* -5 v) v)

simplify10.2s

Counts
69 → 100
Calls

69 calls. Slowest were:

592.0ms
(* (* (* (* -3 v) (* -3 v)) (* -3 v)) (* (* (* v 2) (* v 2)) (* v 2)))
536.0ms
(* (sqrt (+ (* (* (* -3 v) (* v 2)) (* (* -3 v) (* v 2))) (- (* 2 2) (* (* (* -3 v) (* v 2)) 2)))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
483.0ms
(* (sqrt (+ (* (* (* -3 v) (* v 2)) (* (* -3 v) (* v 2))) (- (* 2 2) (* (* (* -3 v) (* v 2)) 2)))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))

prune2.4s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0b

localize6.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (* -5 v) v)
0.1b
(/ (/ (/ (+ 1 (* (* -5 v) v)) PI) (sqrt (+ (* (* -3 v) (* v 2)) 2))) (* t (- 1 (* v v))))
0.1b
(* (* -3 v) (* v 2))
0.0b
(/ (/ (+ 1 (* (* -5 v) v)) PI) (sqrt (+ (* (* -3 v) (* v 2)) 2)))

rewrite52.0ms

Algorithm
rewrite-expression-head
Counts
4 → 149
Calls

4 calls. Slowest were:

29.0ms
(/ (/ (/ (+ 1 (* (* -5 v) v)) PI) (sqrt (+ (* (* -3 v) (* v 2)) 2))) (* t (- 1 (* v v))))
12.0ms
(/ (/ (+ 1 (* (* -5 v) v)) PI) (sqrt (+ (* (* -3 v) (* v 2)) 2)))
3.0ms
(* (* -3 v) (* v 2))

series476.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

296.0ms
(/ (/ (/ (+ 1 (* (* -5 v) v)) PI) (sqrt (+ (* (* -3 v) (* v 2)) 2))) (* t (- 1 (* v v))))
141.0ms
(/ (/ (+ 1 (* (* -5 v) v)) PI) (sqrt (+ (* (* -3 v) (* v 2)) 2)))
20.0ms
(* (* -5 v) v)
19.0ms
(* (* -3 v) (* v 2))

simplify13.6s

Counts
174 → 161
Calls

174 calls. Slowest were:

645.0ms
(* (* (* (* -3 v) (* -3 v)) (* -3 v)) (* (* (* v 2) (* v 2)) (* v 2)))
559.0ms
(/ (* (* (/ (+ 1 (* (* -5 v) v)) PI) (/ (+ 1 (* (* -5 v) v)) PI)) (/ (+ 1 (* (* -5 v) v)) PI)) (* (* (sqrt (+ (* (* -3 v) (* v 2)) 2)) (sqrt (+ (* (* -3 v) (* v 2)) 2))) (sqrt (+ (* (* -3 v) (* v 2)) 2))))
504.0ms
(/ (/ (/ (+ 1 (* (* -5 v) v)) PI) (sqrt (+ (pow (* (* -3 v) (* v 2)) 3) (pow 2 3)))) t)

prune3.4s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize76.0ms

Local error

Found 4 expressions with local error:

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

rewrite188.0ms

Algorithm
rewrite-expression-head
Counts
4 → 1556
Calls

4 calls. Slowest were:

32.0ms
(/ (/ (sqrt (/ (+ 1 (* v (* v -5))) PI)) (* (cbrt (sqrt (+ (* -6 (* v v)) 2))) (cbrt (sqrt (+ (* -6 (* v v)) 2))))) (/ (* t (- 1 (* v v))) (/ (sqrt (/ (+ 1 (* (* -5 v) v)) PI)) (cbrt (sqrt (+ (* (* -3 v) (* v 2)) 2))))))
4.0ms
(* (* -5 v) v)
2.0ms
(* -6 (* v v))

series336.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

283.0ms
(/ (/ (sqrt (/ (+ 1 (* v (* v -5))) PI)) (* (cbrt (sqrt (+ (* -6 (* v v)) 2))) (cbrt (sqrt (+ (* -6 (* v v)) 2))))) (/ (* t (- 1 (* v v))) (/ (sqrt (/ (+ 1 (* (* -5 v) v)) PI)) (cbrt (sqrt (+ (* (* -3 v) (* v 2)) 2))))))
20.0ms
(* -6 (* v v))
20.0ms
(* -6 (* v v))
12.0ms
(* (* -5 v) v)

simplify8.4m

Counts
2936 → 1568
Calls

2936 calls. Slowest were:

672.0ms
(/ (cbrt (/ (sqrt (/ (+ 1 (* v (* v -5))) PI)) (* (cbrt (sqrt (+ (* -6 (* v v)) 2))) (cbrt (sqrt (+ (* -6 (* v v)) 2)))))) (sqrt (/ (* t (- 1 (* v v))) (/ (sqrt (/ (+ 1 (* (* -5 v) v)) PI)) (cbrt (sqrt (+ (* (* -3 v) (* v 2)) 2)))))))
618.0ms
(/ (sqrt (/ (sqrt (/ (+ 1 (* v (* v -5))) PI)) (* (cbrt (sqrt (+ (* -6 (* v v)) 2))) (cbrt (sqrt (+ (* -6 (* v v)) 2)))))) (/ (- 1 (* v v)) (/ (cbrt (sqrt (/ (+ 1 (* (* -5 v) v)) PI))) (cbrt (sqrt (sqrt (+ (* (* -3 v) (* v 2)) 2)))))))
611.0ms
(/ (/ (* (* (sqrt (/ (+ 1 (* v (* v -5))) PI)) (sqrt (/ (+ 1 (* v (* v -5))) PI))) (sqrt (/ (+ 1 (* v (* v -5))) PI))) (* (* (* (cbrt (sqrt (+ (* -6 (* v v)) 2))) (cbrt (sqrt (+ (* -6 (* v v)) 2)))) (* (cbrt (sqrt (+ (* -6 (* v v)) 2))) (cbrt (sqrt (+ (* -6 (* v v)) 2))))) (* (cbrt (sqrt (+ (* -6 (* v v)) 2))) (cbrt (sqrt (+ (* -6 (* v v)) 2)))))) (/ (* (* (* t (- 1 (* v v))) (* t (- 1 (* v v)))) (* t (- 1 (* v v)))) (* (* (/ (sqrt (/ (+ 1 (* (* -5 v) v)) PI)) (cbrt (sqrt (+ (* (* -3 v) (* v 2)) 2)))) (/ (sqrt (/ (+ 1 (* (* -5 v) v)) PI)) (cbrt (sqrt (+ (* (* -3 v) (* v 2)) 2))))) (/ (sqrt (/ (+ 1 (* (* -5 v) v)) PI)) (cbrt (sqrt (+ (* (* -3 v) (* v 2)) 2)))))))

prune37.3s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ (/ 1 (- 1 (* v v))) (sqrt (+ (* -6 (* v v)) 2)))
0.5b
(* (/ (/ (+ 1 (* (* -5 v) v)) PI) t) (/ (/ 1 (- 1 (* v v))) (sqrt (+ (* -6 (* v v)) 2))))
0.4b
(/ (/ (+ 1 (* (* -5 v) v)) PI) t)
0.2b
(* -6 (* v v))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 151
Calls

4 calls. Slowest were:

11.0ms
(* (/ (/ (+ 1 (* (* -5 v) v)) PI) t) (/ (/ 1 (- 1 (* v v))) (sqrt (+ (* -6 (* v v)) 2))))
8.0ms
(/ (/ (+ 1 (* (* -5 v) v)) PI) t)
6.0ms
(/ (/ 1 (- 1 (* v v))) (sqrt (+ (* -6 (* v v)) 2)))

series326.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

187.0ms
(* (/ (/ (+ 1 (* (* -5 v) v)) PI) t) (/ (/ 1 (- 1 (* v v))) (sqrt (+ (* -6 (* v v)) 2))))
71.0ms
(/ (/ 1 (- 1 (* v v))) (sqrt (+ (* -6 (* v v)) 2)))
57.0ms
(/ (/ (+ 1 (* (* -5 v) v)) PI) t)
11.0ms
(* -6 (* v v))

simplify7.8s

Counts
158 → 163
Calls

158 calls. Slowest were:

258.0ms
(- (+ (* (pow v 2) (sqrt 1/2)) (+ (* (pow v 4) (sqrt 1/2)) (+ (sqrt 1/2) (+ (* 3 (/ (pow v 4) (sqrt 1/2))) (* 3/4 (/ (pow v 2) (sqrt 1/2))))))) (* 9/32 (/ (pow v 4) (pow (sqrt 1/2) 3))))
250.0ms
(- (+ (/ (sqrt -1/6) (pow v 7)) (+ (/ (sqrt -1/6) (pow v 5)) (/ (sqrt -1/6) (pow v 3)))) (+ (* 1/36 (/ 1 (* (pow v 5) (sqrt -1/6)))) (+ (* 1/27 (/ 1 (* (pow v 7) (sqrt -1/6)))) (* 1/2592 (/ 1 (* (pow v 7) (pow (sqrt -1/6) 3)))))))
250.0ms
(- (+ (* 1/36 (/ 1 (* (pow v 5) (sqrt -1/6)))) (+ (* 1/27 (/ 1 (* (pow v 7) (sqrt -1/6)))) (* 1/2592 (/ 1 (* (pow v 7) (pow (sqrt -1/6) 3)))))) (+ (/ (sqrt -1/6) (pow v 7)) (+ (/ (sqrt -1/6) (pow v 5)) (/ (sqrt -1/6) (pow v 3)))))

prune2.1s

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

regimes91.0ms

Accuracy

0% (0.1b remaining)

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

bsearch3.0ms