Average Error: 13.4 → 8.6
Time: 29.8s
Precision: 64
Internal Precision: 128
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(h \cdot \frac{M \cdot D}{2 \cdot d}\right)}{\ell}} \cdot w0\]

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 13.4

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

  3. Applied associate-*r/10.0

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

  5. Applied unpow210.0

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

  6. Applied associate-*l*8.6

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

  7. Final simplification8.6

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

Reproduce

herbie shell --seed 2019005 
(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: 28.7s)Debug log

sample143.0ms

Algorithm
intervals

simplify97.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

97.0ms
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))

prune14.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 12.0b

localize28.0ms

Local error

Found 4 expressions with local error:

8.5b
(* (pow (/ (* M D) (* 2 d)) 2) (/ h l))
5.0b
(/ (* M D) (* 2 d))
5.0b
(sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))
1.9b
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

9.0ms
(/ (* M D) (* 2 d))
6.0ms
(* (pow (/ (* M D) (* 2 d)) 2) (/ h l))
3.0ms
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))

series457.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

226.0ms
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))
133.0ms
(sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))
77.0ms
(* (pow (/ (* M D) (* 2 d)) 2) (/ h l))
21.0ms
(/ (* M D) (* 2 d))

simplify4.8s

Counts
47 → 84
Calls

47 calls. Slowest were:

850.0ms
(sqrt (+ 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))
398.0ms
(+ (log (pow (/ (* M D) (* 2 d)) 2)) (log (/ h l)))
314.0ms
(+ (* (- (log (* M D)) (log (* 2 d))) 2) (log (/ h l)))

prune1.4s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 7.3b

localize18.0ms

Local error

Found 4 expressions with local error:

5.0b
(/ (* M D) (* 2 d))
5.0b
(sqrt (- 1 (/ (* (pow (/ (* M D) (* 2 d)) 2) h) l)))
4.5b
(* (pow (/ (* M D) (* 2 d)) 2) h)
3.0b
(/ (* (pow (/ (* M D) (* 2 d)) 2) h) l)

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 57
Calls

4 calls. Slowest were:

7.0ms
(/ (* (pow (/ (* M D) (* 2 d)) 2) h) l)
5.0ms
(* (pow (/ (* M D) (* 2 d)) 2) h)
3.0ms
(/ (* M D) (* 2 d))

series308.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

144.0ms
(sqrt (- 1 (/ (* (pow (/ (* M D) (* 2 d)) 2) h) l)))
80.0ms
(/ (* (pow (/ (* M D) (* 2 d)) 2) h) l)
58.0ms
(* (pow (/ (* M D) (* 2 d)) 2) h)
26.0ms
(/ (* M D) (* 2 d))

simplify5.3s

Counts
31 → 69
Calls

31 calls. Slowest were:

1.0s
(sqrt (* (cbrt (- 1 (/ (* (pow (/ (* M D) (* 2 d)) 2) h) l))) (cbrt (- 1 (/ (* (pow (/ (* M D) (* 2 d)) 2) h) l)))))
759.0ms
(sqrt (sqrt (- 1 (/ (* (pow (/ (* M D) (* 2 d)) 2) h) l))))
672.0ms
(sqrt (- 1 (/ (* (pow (/ (* M D) (* 2 d)) 2) h) l)))

prune1.1s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 7.1b

localize33.0ms

Local error

Found 4 expressions with local error:

6.0b
(* (/ (* M D) (* 2 d)) h)
5.0b
(/ (* M D) (* 2 d))
5.0b
(/ (* M D) (* 2 d))
5.0b
(sqrt (- 1 (/ (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) l)))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 58
Calls

4 calls. Slowest were:

9.0ms
(* (/ (* M D) (* 2 d)) h)
6.0ms
(/ (* M D) (* 2 d))
4.0ms
(/ (* M D) (* 2 d))

series185.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

116.0ms
(sqrt (- 1 (/ (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) l)))
32.0ms
(* (/ (* M D) (* 2 d)) h)
20.0ms
(/ (* M D) (* 2 d))
17.0ms
(/ (* M D) (* 2 d))

simplify2.7s

Counts
31 → 70
Calls

31 calls. Slowest were:

288.0ms
(sqrt (- (* 1 1) (* (/ (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) l) (/ (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) l))))
255.0ms
(* 1/2 (/ (* M (* D h)) d))
238.0ms
(* 1/2 (/ (* M (* D h)) d))

prune1.2s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 7.1b

localize23.0ms

Local error

Found 4 expressions with local error:

18.7b
(cbrt (/ (* M D) (* 2 d)))
18.7b
(cbrt (/ (* M D) (* 2 d)))
18.7b
(cbrt (/ (* M D) (* 2 d)))
6.0b
(* (/ (* M D) (* 2 d)) h)

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

8.0ms
(* (/ (* M D) (* 2 d)) h)
1.0ms
(cbrt (/ (* M D) (* 2 d)))
1.0ms
(cbrt (/ (* M D) (* 2 d)))

series402.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

144.0ms
(cbrt (/ (* M D) (* 2 d)))
115.0ms
(cbrt (/ (* M D) (* 2 d)))
98.0ms
(cbrt (/ (* M D) (* 2 d)))
44.0ms
(* (/ (* M D) (* 2 d)) h)

simplify3.0s

Counts
35 → 68
Calls

35 calls. Slowest were:

342.0ms
(* (cbrt 1/2) (exp (* 1/3 (- (log (/ 1 d)) (+ (log (/ 1 M)) (log (/ 1 D)))))))
328.0ms
(* (cbrt 1/2) (exp (* 1/3 (- (log (/ 1 d)) (+ (log (/ 1 M)) (log (/ 1 D)))))))
272.0ms
(* (cbrt 1/2) (exp (* 1/3 (- (log (/ 1 d)) (+ (log (/ 1 M)) (log (/ 1 D)))))))

prune1.4s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 7.1b

regimes2.0s

Accuracy

0% (0.9b remaining)

Error of 8.6b against oracle of 7.8b and baseline of 8.6b

bsearch2.0ms

end0.0ms

sample4.1s

Algorithm
intervals