Average Error: 13.2 → 8.7
Time: 41.0s
Precision: 64
Internal Precision: 128
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(h \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{1}{\ell}}\]

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.2

    \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
  2. Using strategy rm

  3. Applied div-inv13.2

    \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}}\]

  4. Applied associate-*r*10.1

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}}\]
  5. Using strategy rm

  6. Applied unpow210.1

    \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot h\right) \cdot \frac{1}{\ell}}\]

  7. Applied associate-*l*8.7

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)\right)} \cdot \frac{1}{\ell}}\]

  8. Final simplification8.7

    \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(h \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{1}{\ell}}\]

Reproduce

herbie shell --seed 2019007 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))

Details

Time bar (total: 39.8s)Debug log

sample123.0ms

Algorithm
intervals

simplify65.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

65.0ms
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 14.3b

localize30.0ms

Local error

Found 4 expressions with local error:

12.4b
(* (pow (/ (* M D) (* 2 d)) 2) (/ h l))
4.2b
(sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))
3.8b
(/ (* M D) (* 2 d))
1.0b
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

7.0ms
(* (pow (/ (* M D) (* 2 d)) 2) (/ h l))
3.0ms
(/ (* M D) (* 2 d))
2.0ms
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))

series436.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

207.0ms
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))
130.0ms
(sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))
70.0ms
(* (pow (/ (* M D) (* 2 d)) 2) (/ h l))
29.0ms
(/ (* M D) (* 2 d))

simplify4.3s

Counts
47 → 84
Calls

47 calls. Slowest were:

702.0ms
(sqrt (+ 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))
413.0ms
(+ (log (pow (/ (* M D) (* 2 d)) 2)) (log (/ h l)))
292.0ms
(+ (* (- (log (* M D)) (log (* 2 d))) 2) (log (/ h l)))

prune1.4s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 7.0b

localize15.0ms

Local error

Found 4 expressions with local error:

5.3b
(* (pow (/ (* M D) (* 2 d)) 2) h)
5.0b
(* (* (pow (/ (* M D) (* 2 d)) 2) h) (/ 1 l))
4.2b
(sqrt (- 1 (* (* (pow (/ (* M D) (* 2 d)) 2) h) (/ 1 l))))
3.8b
(/ (* M D) (* 2 d))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

14.0ms
(* (* (pow (/ (* M D) (* 2 d)) 2) h) (/ 1 l))
7.0ms
(* (pow (/ (* M D) (* 2 d)) 2) h)
5.0ms
(/ (* M D) (* 2 d))

series290.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

131.0ms
(sqrt (- 1 (* (* (pow (/ (* M D) (* 2 d)) 2) h) (/ 1 l))))
73.0ms
(* (* (pow (/ (* M D) (* 2 d)) 2) h) (/ 1 l))
67.0ms
(* (pow (/ (* M D) (* 2 d)) 2) h)
19.0ms
(/ (* M D) (* 2 d))

simplify3.9s

Counts
38 → 76
Calls

38 calls. Slowest were:

563.0ms
(+ (log (* (pow (/ (* M D) (* 2 d)) 2) h)) (log (/ 1 l)))
375.0ms
(* 1/4 (/ (* (pow M 2) (* (pow D 2) h)) (pow d 2)))
285.0ms
(pow (/ (* M D) (* 2 d)) 2)

prune1.2s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 6.2b

localize31.0ms

Local error

Found 4 expressions with local error:

5.2b
(* (/ (* M D) (* 2 d)) h)
5.0b
(* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (/ 1 l))
4.2b
(sqrt (- 1 (* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (/ 1 l))))
3.8b
(/ (* M D) (* 2 d))

rewrite52.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

40.0ms
(* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (/ 1 l))
8.0ms
(* (/ (* M D) (* 2 d)) h)
2.0ms
(/ (* M D) (* 2 d))

series254.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

139.0ms
(sqrt (- 1 (* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (/ 1 l))))
54.0ms
(* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (/ 1 l))
38.0ms
(* (/ (* M D) (* 2 d)) h)
23.0ms
(/ (* M D) (* 2 d))

simplify7.6s

Counts
49 → 84
Calls

49 calls. Slowest were:

1.4s
(* (* (/ (* (* (* M D) (* M D)) (* M D)) (* (* (* 2 d) (* 2 d)) (* 2 d))) (* (* (* (/ (* M D) (* 2 d)) h) (* (/ (* M D) (* 2 d)) h)) (* (/ (* M D) (* 2 d)) h))) (* (* (/ 1 l) (/ 1 l)) (/ 1 l)))
575.0ms
(* (* (* (* (/ (* M D) (* 2 d)) (/ (* M D) (* 2 d))) (/ (* M D) (* 2 d))) (* (* (* (/ (* M D) (* 2 d)) h) (* (/ (* M D) (* 2 d)) h)) (* (/ (* M D) (* 2 d)) h))) (* (* (/ 1 l) (/ 1 l)) (/ 1 l)))
539.0ms
(* (* (* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h))) (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h))) (* (* (/ 1 l) (/ 1 l)) (/ 1 l)))

prune1.1s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 6.2b

localize21.0ms

Local error

Found 4 expressions with local error:

5.2b
(* (/ (* M D) (* 2 d)) h)
5.2b
(* (/ (* M D) (* 2 d)) h)
5.0b
(* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (/ 1 l))
5.0b
(* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (/ 1 l))

rewrite103.0ms

Algorithm
rewrite-expression-head
Counts
4 → 86
Calls

4 calls. Slowest were:

45.0ms
(* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (/ 1 l))
42.0ms
(* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (/ 1 l))
7.0ms
(* (/ (* M D) (* 2 d)) h)

series173.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

54.0ms
(* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (/ 1 l))
52.0ms
(* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (/ 1 l))
35.0ms
(* (/ (* M D) (* 2 d)) h)
31.0ms
(* (/ (* M D) (* 2 d)) h)

simplify10.5s

Counts
66 → 98
Calls

66 calls. Slowest were:

916.0ms
(* (* (/ (* (* (* M D) (* M D)) (* M D)) (* (* (* 2 d) (* 2 d)) (* 2 d))) (* (* (* (/ (* M D) (* 2 d)) h) (* (/ (* M D) (* 2 d)) h)) (* (/ (* M D) (* 2 d)) h))) (* (* (/ 1 l) (/ 1 l)) (/ 1 l)))
833.0ms
(* (* (/ (* (* (* M D) (* M D)) (* M D)) (* (* (* 2 d) (* 2 d)) (* 2 d))) (* (* (* (/ (* M D) (* 2 d)) h) (* (/ (* M D) (* 2 d)) h)) (* (/ (* M D) (* 2 d)) h))) (* (* (/ 1 l) (/ 1 l)) (/ 1 l)))
772.0ms
(* (* (* (* (/ (* M D) (* 2 d)) (/ (* M D) (* 2 d))) (/ (* M D) (* 2 d))) (* (* (* (/ (* M D) (* 2 d)) h) (* (/ (* M D) (* 2 d)) h)) (* (/ (* M D) (* 2 d)) h))) (* (* (/ 1 l) (/ 1 l)) (/ 1 l)))

prune2.7s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 6.2b

regimes1.9s

Accuracy

0% (1.6b remaining)

Error of 8.7b against oracle of 7.1b and baseline of 8.7b

bsearch4.0ms

end0.0ms

sample3.5s

Algorithm
intervals