Average Error: 1.0 → 0.0
Time: 46.2s
Precision: 64
Internal Precision: 128
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3}}{\pi}}{\left(1 - v \cdot v\right) \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}}\]

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  2. Initial simplification0.0

    \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}\]
  3. Using strategy rm

  4. Applied associate-/l/0.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{4}{3}}{\pi}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(1 - v \cdot v\right)}}\]

  5. Final simplification0.0

    \[\leadsto \frac{\frac{\frac{4}{3}}{\pi}}{\left(1 - v \cdot v\right) \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}}\]

Reproduce

herbie shell --seed 2018365 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))

Details

Time bar (total: 40.4s)Debug log

start180.0ms

Algorithm
intervals

setup211.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize28.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ 4/3 PI)
0.2b
(* (* v v) -6)
0.0b
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (+ 2 (* (* v v) -6))))
0.0b
(/ (/ 4/3 PI) (- 1 (* v v)))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 202
Calls

4 calls. Slowest were:

7.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (+ 2 (* (* v v) -6))))
4.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
3.0ms
(* (* v v) -6)

series317.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

212.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (+ 2 (* (* v v) -6))))
86.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
19.0ms
(* (* v v) -6)
0.0ms
(/ 4/3 PI)

simplify21.5s

Counts
289 → 212
Calls

289 calls. Slowest were:

779.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
626.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
410.0ms
(/ (/ (/ 4/3 PI) (- (pow 1 3) (pow (* v v) 3))) (sqrt (sqrt (+ 2 (* (* v v) -6)))))

prune3.4s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize9.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ 4/3 PI)
0.2b
(* (* v v) -6)
0.0b
(sqrt (+ 2 (* (* v v) -6)))
0.0b
(/ (/ 4/3 PI) (* (sqrt (+ 2 (* (* v v) -6))) (- 1 (* v v))))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

5.0ms
(/ (/ 4/3 PI) (* (sqrt (+ 2 (* (* v v) -6))) (- 1 (* v v))))
5.0ms
(* (* v v) -6)
1.0ms
(sqrt (+ 2 (* (* v v) -6)))

series234.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

196.0ms
(/ (/ 4/3 PI) (* (sqrt (+ 2 (* (* v v) -6))) (- 1 (* v v))))
27.0ms
(sqrt (+ 2 (* (* v v) -6)))
11.0ms
(* (* v v) -6)
0.0ms
(/ 4/3 PI)

simplify4.2s

Counts
39 → 75
Calls

39 calls. Slowest were:

450.0ms
(sqrt (+ (* 2 2) (- (* (* (* v v) -6) (* (* v v) -6)) (* 2 (* (* v v) -6)))))
443.0ms
(sqrt (+ (pow 2 3) (pow (* (* v v) -6) 3)))
362.0ms
(* (sqrt (- 2 (* (* v v) -6))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))

prune1.1s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize17.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ 4/3 PI)
1.0b
(log (/ 4/3 PI))
1.0b
(log (* (sqrt (+ 2 (* (* v v) -6))) (- 1 (* v v))))
1.0b
(exp (- (log (/ 4/3 PI)) (log (* (sqrt (+ 2 (* (* v v) -6))) (- 1 (* v v))))))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
4 → 78
Calls

4 calls. Slowest were:

26.0ms
(exp (- (log (/ 4/3 PI)) (log (* (sqrt (+ 2 (* (* v v) -6))) (- 1 (* v v))))))
4.0ms
(log (* (sqrt (+ 2 (* (* v v) -6))) (- 1 (* v v))))
1.0ms
(log (/ 4/3 PI))

series537.0ms

Counts
4 → 8
Calls

4 calls. Slowest were:

385.0ms
(exp (- (log (/ 4/3 PI)) (log (* (sqrt (+ 2 (* (* v v) -6))) (- 1 (* v v))))))
152.0ms
(log (* (sqrt (+ 2 (* (* v v) -6))) (- 1 (* v v))))
0.0ms
(/ 4/3 PI)
0.0ms
(log (/ 4/3 PI))

simplify7.0s

Counts
52 → 86
Calls

52 calls. Slowest were:

477.0ms
(- (+ (log (* -1 (sqrt -6))) (/ 1 (* (pow v 2) (pow (sqrt -6) 2)))) (+ (* 3 (log (/ -1 v))) (+ (* 1/2 (/ 1 (pow v 4))) (+ (/ 1 (* (pow v 4) (pow (sqrt -6) 4))) (/ 1 (pow v 2))))))
452.0ms
(- (+ (log (* -1 (sqrt -6))) (/ 1 (* (pow v 2) (pow (sqrt -6) 2)))) (+ (* 3 (log (/ 1 v))) (+ (* 1/2 (/ 1 (pow v 4))) (+ (/ 1 (* (pow v 4) (pow (sqrt -6) 4))) (/ 1 (pow v 2))))))
383.0ms
(- (log (/ 4/3 PI)) (log (* (sqrt (+ 2 (* (* v v) -6))) (- 1 (* v v)))))

prune1.5s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes46.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms