Average Error: 0.5 → 0.1
Time: 2.2m
Precision: 64
Internal Precision: 128
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{1 + v \cdot \left(v \cdot -5\right)}{\pi}}{\sqrt{\left(v \cdot 2\right) \cdot \left(v \cdot -3\right) + 2}}}{t \cdot \left(1 - v \cdot v\right)}\]

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 0.5

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

  2. 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 2018362 
(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: 2.1m)Debug log

start221.0ms

Algorithm
intervals

setup148.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.3b

localize62.0ms

Local error

Found 4 expressions with local error:

0.6b
(/ (/ (+ 1 (* (* -5 v) v)) PI) (* (sqrt (+ (* (* -3 v) (* v 2)) 2)) (* t (- 1 (* v v)))))
0.4b
(* (sqrt (+ (* (* -3 v) (* v 2)) 2)) (* t (- 1 (* v v))))
0.2b
(* (* -5 v) v)
0.1b
(* (* -3 v) (* v 2))

rewrite54.0ms

Algorithm
rewrite-expression-head
Counts
4 → 88
Calls

4 calls. Slowest were:

27.0ms
(* (sqrt (+ (* (* -3 v) (* v 2)) 2)) (* t (- 1 (* v v))))
15.0ms
(/ (/ (+ 1 (* (* -5 v) v)) PI) (* (sqrt (+ (* (* -3 v) (* v 2)) 2)) (* t (- 1 (* v v)))))
6.0ms
(* (* -3 v) (* v 2))

series421.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

240.0ms
(/ (/ (+ 1 (* (* -5 v) v)) PI) (* (sqrt (+ (* (* -3 v) (* v 2)) 2)) (* t (- 1 (* v v)))))
140.0ms
(* (sqrt (+ (* (* -3 v) (* v 2)) 2)) (* t (- 1 (* v v))))
20.0ms
(* (* -3 v) (* v 2))
20.0ms
(* (* -5 v) v)

simplify10.8s

Counts
69 → 100
Calls

69 calls. Slowest were:

681.0ms
(- (+ (/ t (* v (sqrt -6))) (+ (* 1/2 (/ t (* v (pow (sqrt -6) 3)))) (* t (* v (sqrt -6))))) (+ (* t (* (pow v 3) (sqrt -6))) (/ (* t v) (sqrt -6))))
662.0ms
(* (* (* (* -3 v) (* -3 v)) (* -3 v)) (* (* (* v 2) (* v 2)) (* v 2)))
467.0ms
(* (sqrt (+ (pow (* (* -3 v) (* v 2)) 3) (pow 2 3))) (* t (- 1 (* v v))))

prune2.1s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0b

localize7.0ms

Local error

Found 4 expressions with local error:

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

rewrite59.0ms

Algorithm
rewrite-expression-head
Counts
4 → 149
Calls

4 calls. Slowest were:

23.0ms
(/ (/ (+ 1 (* (* -5 v) v)) PI) (sqrt (+ (* (* -3 v) (* v 2)) 2)))
21.0ms
(/ (/ (/ (+ 1 (* (* -5 v) v)) PI) (sqrt (+ (* (* -3 v) (* v 2)) 2))) (* t (- 1 (* v v))))
6.0ms
(* (* -3 v) (* v 2))

series414.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

235.0ms
(/ (/ (/ (+ 1 (* (* -5 v) v)) PI) (sqrt (+ (* (* -3 v) (* v 2)) 2))) (* t (- 1 (* v v))))
139.0ms
(/ (/ (+ 1 (* (* -5 v) v)) PI) (sqrt (+ (* (* -3 v) (* v 2)) 2)))
20.0ms
(* (* -5 v) v)
19.0ms
(* (* -3 v) (* v 2))

simplify16.8s

Counts
174 → 161
Calls

174 calls. Slowest were:

744.0ms
(* (* (* (* -3 v) (* -3 v)) (* -3 v)) (* (* (* v 2) (* v 2)) (* v 2)))
483.0ms
(/ (/ (/ (+ 1 (* (* -5 v) v)) PI) (sqrt (+ (pow (* (* -3 v) (* v 2)) 3) (pow 2 3)))) t)
442.0ms
(/ (/ (+ 1 (* (* -5 v) v)) PI) (sqrt (+ (pow (* (* -3 v) (* v 2)) 3) (pow 2 3))))

prune3.1s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize40.0ms

Local error

Found 4 expressions with local error:

5.3b
(* t (* v v))
0.6b
(/ (/ (+ 1 (* (* -5 v) v)) PI) (pow (* (- t (* t (* v v))) (sqrt (+ 2 (* -6 (* v v))))) 1))
0.4b
(* (- t (* t (* v v))) (sqrt (+ 2 (* -6 (* v v)))))
0.2b
(* -6 (* v v))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 86
Calls

4 calls. Slowest were:

10.0ms
(/ (/ (+ 1 (* (* -5 v) v)) PI) (pow (* (- t (* t (* v v))) (sqrt (+ 2 (* -6 (* v v))))) 1))
5.0ms
(* (- t (* t (* v v))) (sqrt (+ 2 (* -6 (* v v)))))
2.0ms
(* t (* v v))

series398.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

236.0ms
(/ (/ (+ 1 (* (* -5 v) v)) PI) (pow (* (- t (* t (* v v))) (sqrt (+ 2 (* -6 (* v v))))) 1))
129.0ms
(* (- t (* t (* v v))) (sqrt (+ 2 (* -6 (* v v)))))
17.0ms
(* -6 (* v v))
16.0ms
(* t (* v v))

simplify12.0s

Counts
80 → 98
Calls

80 calls. Slowest were:

767.0ms
(* (* (* (- t (* t (* v v))) (- t (* t (* v v)))) (- t (* t (* v v)))) (* (* (sqrt (+ 2 (* -6 (* v v)))) (sqrt (+ 2 (* -6 (* v v))))) (sqrt (+ 2 (* -6 (* v v))))))
614.0ms
(sqrt (+ (* 2 2) (- (* (* -6 (* v v)) (* -6 (* v v))) (* 2 (* -6 (* v v))))))
440.0ms
(* (- (pow t 3) (pow (* t (* v v)) 3)) (sqrt (+ (pow 2 3) (pow (* -6 (* v v)) 3))))

prune1.8s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize18.0ms

Local error

Found 4 expressions with local error:

0.6b
(/ (+ 1 (* v (* v -5))) (/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (+ (* (* -3 v) (* v 2)) 2)))))
0.2b
(* v (* v -5))
0.1b
(/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (+ (* (* -3 v) (* v 2)) 2))))
0.1b
(* (* -3 v) (* v 2))

rewrite37.0ms

Algorithm
rewrite-expression-head
Counts
4 → 262
Calls

4 calls. Slowest were:

17.0ms
(/ (+ 1 (* v (* v -5))) (/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (+ (* (* -3 v) (* v 2)) 2)))))
10.0ms
(/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (+ (* (* -3 v) (* v 2)) 2))))
3.0ms
(* (* -3 v) (* v 2))

series390.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

231.0ms
(/ (+ 1 (* v (* v -5))) (/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (+ (* (* -3 v) (* v 2)) 2)))))
130.0ms
(/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (+ (* (* -3 v) (* v 2)) 2))))
18.0ms
(* (* -3 v) (* v 2))
11.0ms
(* v (* v -5))

simplify1.2m

Counts
368 → 274
Calls

368 calls. Slowest were:

770.0ms
(/ (- 1 (* v v)) (sqrt (+ (* (* (* -3 v) (* v 2)) (* (* -3 v) (* v 2))) (- (* 2 2) (* (* (* -3 v) (* v 2)) 2)))))
708.0ms
(/ (sqrt (+ 1 (* v (* v -5)))) (/ (- 1 (* v v)) (/ (sqrt (/ 1 PI)) (sqrt (sqrt (+ (* (* -3 v) (* v 2)) 2))))))
702.0ms
(/ (cbrt (+ 1 (* v (* v -5)))) (/ (- 1 (* v v)) (/ 1 (sqrt (+ (* (* -3 v) (* v 2)) 2)))))

prune4.1s

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

regimes115.0ms

Accuracy

0% (0.1b remaining)

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

bsearch3.0ms