Average Error: 13.4 → 7.9
Time: 34.6s
Precision: 64
Internal Precision: 128
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{h \cdot \frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}} \cdot w0\]

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 unpow213.4

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

  4. Applied associate-*l*11.9

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

  6. Applied associate-*r/7.9

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

  7. Final simplification7.9

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

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(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: 28.1s)Debug log

start148.0ms

Algorithm
intervals

setup83.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 13.2b

localize26.0ms

Local error

Found 4 expressions with local error:

6.4b
(fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1)
4.7b
(sqrt (fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1))
4.6b
(/ (/ M 2) (/ d D))
4.6b
(/ (/ M 2) (/ d D))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 104
Calls

4 calls. Slowest were:

4.0ms
(/ (/ M 2) (/ d D))
3.0ms
(/ (/ M 2) (/ d D))
1.0ms
(sqrt (fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1))

series328.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

180.0ms
(sqrt (fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1))
108.0ms
(fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1)
22.0ms
(/ (/ M 2) (/ d D))
17.0ms
(/ (/ M 2) (/ d D))

simplify4.1s

Counts
96 → 116
Calls

96 calls. Slowest were:

718.0ms
(sqrt (fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1))
622.0ms
(/ (* (* (/ M 2) (/ M 2)) (/ M 2)) (* (* (/ d D) (/ d D)) (/ d D)))
598.0ms
(/ (* (* (/ M 2) (/ M 2)) (/ M 2)) (* (* (/ d D) (/ d D)) (/ d D)))

prune2.2s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 9.9b

localize21.0ms

Local error

Found 4 expressions with local error:

6.4b
(fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1)
6.4b
(fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1)
4.7b
(sqrt (fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1))
4.7b
(sqrt (fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

2.0ms
(sqrt (fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1))
1.0ms
(sqrt (fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1))
0.0ms
(fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1)

series651.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

212.0ms
(sqrt (fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1))
206.0ms
(sqrt (fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1))
142.0ms
(fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1)
90.0ms
(fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1)

simplify2.9s

Counts
20 → 60
Calls

20 calls. Slowest were:

457.0ms
(sqrt (fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1))
391.0ms
(sqrt (fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1))
387.0ms
(sqrt (sqrt (fma (* (/ (/ M 2) (/ d D)) (/ (/ M 2) (/ d D))) (- (/ h l)) 1)))

prune1.2s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 9.9b

localize26.0ms

Local error

Found 4 expressions with local error:

10.7b
(* (pow (/ (* M D) (* 2 d)) 2) (/ h l))
5.6b
(/ (* M D) (* 2 d))
4.7b
(sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))
2.4b
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 80
Calls

4 calls. Slowest were:

10.0ms
(* (pow (/ (* M D) (* 2 d)) 2) (/ h l))
2.0ms
(/ (* M D) (* 2 d))
2.0ms
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))

series410.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

201.0ms
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))
127.0ms
(sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))
59.0ms
(* (pow (/ (* M D) (* 2 d)) 2) (/ h l))
24.0ms
(/ (* M D) (* 2 d))

simplify5.2s

Counts
47 → 92
Calls

47 calls. Slowest were:

879.0ms
(sqrt (+ 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))
513.0ms
(+ (* (- (log (* M D)) (log (* 2 d))) 2) (log (/ h l)))
441.0ms
(+ (log (pow (/ (* M D) (* 2 d)) 2)) (log (/ h l)))

prune1.7s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 8.4b

localize30.0ms

Local error

Found 4 expressions with local error:

8.8b
(* (/ (* M D) (* 2 d)) (/ h l))
5.6b
(/ (* M D) (* 2 d))
5.6b
(/ (* M D) (* 2 d))
4.7b
(sqrt (- 1 (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) (/ h l)))))

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 77
Calls

4 calls. Slowest were:

10.0ms
(* (/ (* M D) (* 2 d)) (/ h l))
5.0ms
(/ (* M D) (* 2 d))
5.0ms
(/ (* M D) (* 2 d))

series286.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

155.0ms
(sqrt (- 1 (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) (/ h l)))))
72.0ms
(* (/ (* M D) (* 2 d)) (/ h l))
30.0ms
(/ (* M D) (* 2 d))
29.0ms
(/ (* M D) (* 2 d))

simplify4.8s

Counts
42 → 89
Calls

42 calls. Slowest were:

463.0ms
(sqrt (- (pow 1 3) (pow (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) (/ h l))) 3)))
450.0ms
(* 1/2 (/ (* M (* D h)) (* l d)))
443.0ms
(* (/ (* (* (* M D) (* M D)) (* M D)) (* (* (* 2 d) (* 2 d)) (* 2 d))) (* (* (/ h l) (/ h l)) (/ h l)))

prune1.8s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 7.8b

regimes2.2s

Accuracy

0% (1.3b remaining)

Error of 7.9b against oracle of 6.6b and baseline of 7.9b

bsearch4.0ms