Average Error: 13.2 → 8.1
Time: 32.5s
Precision: 64
Internal Precision: 128
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left(\frac{1}{\ell} \cdot \left(h \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}}\]

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. Using strategy rm

  9. Applied associate-*l*8.1

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

  10. Final simplification8.1

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

Reproduce

herbie shell --seed 2019010 
(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: 31.6s)Debug log

sample163.0ms

Algorithm
intervals

simplify63.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
63.0ms
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))

prune20.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 11.6b

localize29.0ms

Local error

Found 4 expressions with local error:

9.1b
(* (pow (/ (* M D) (* 2 d)) 2) (/ h l))
4.5b
(/ (* M D) (* 2 d))
2.8b
(sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))
2.6b
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
14×add-exp-log
11×add-sqr-sqrt
10×add-cube-cbrt
10×associate-*r*
10×*-un-lft-identity
9×associate-*l*
8×add-cbrt-cube
6×sqrt-prod
5×unpow-prod-down
4×add-log-exp
4×prod-exp
4×sqrt-div
4×pow1
3×div-inv
3×associate-*r/
2×flip--
2×*-commutative
2×pow-exp
2×div-exp
2×flip3--
2×times-frac
1×unpow2
1×associate-/l*
1×cbrt-unprod
1×pow1/2
1×frac-2neg
1×associate-/r*
1×clear-num
1×rem-sqrt-square
1×cbrt-undiv
1×pow-to-exp
Counts
4 → 72
Calls
4 calls:
Slowest
10.0ms
(* (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)))))
1.0ms
(sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))

series860.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
509.0ms
(sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))
253.0ms
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))
77.0ms
(* (pow (/ (* M D) (* 2 d)) 2) (/ h l))
20.0ms
(/ (* M D) (* 2 d))

simplify4.7s

Counts
47 → 84
Calls
47 calls:
Slowest
828.0ms
(sqrt (+ 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))
437.0ms
(+ (log (pow (/ (* M D) (* 2 d)) 2)) (log (/ h l)))
312.0ms
(+ (* (- (log (* M D)) (log (* 2 d))) 2) (log (/ h l)))
297.0ms
(* (cbrt (pow (/ (* M D) (* 2 d)) 2)) (cbrt (pow (/ (* M D) (* 2 d)) 2)))
293.0ms
(+ (* (log (/ (* M D) (* 2 d))) 2) (log (/ h l)))

prune1.3s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 5.7b

localize32.0ms

Local error

Found 4 expressions with local error:

4.5b
(/ (* M D) (* 2 d))
4.2b
(* (* (pow (/ (* M D) (* 2 d)) 2) h) (/ 1 l))
3.2b
(* (pow (/ (* M D) (* 2 d)) 2) h)
2.8b
(sqrt (- 1 (* (* (pow (/ (* M D) (* 2 d)) 2) h) (/ 1 l))))

rewrite32.0ms

Algorithm
rewrite-expression-head
Rules
10×associate-*l*
9×add-sqr-sqrt
8×add-cube-cbrt
8×add-exp-log
8×add-cbrt-cube
8×*-un-lft-identity
6×pow1
5×unpow-prod-down
4×add-log-exp
4×associate-*r*
3×div-inv
3×sqrt-prod
2×*-commutative
2×sqrt-div
2×times-frac
1×unpow2
1×associate-/l*
1×flip--
1×cbrt-unprod
1×un-div-inv
1×associate-*r/
1×prod-exp
1×pow-prod-down
1×div-exp
1×pow1/2
1×flip3--
1×frac-2neg
1×associate-/r*
1×clear-num
1×rem-sqrt-square
1×cbrt-undiv
Counts
4 → 64
Calls
4 calls:
Slowest
16.0ms
(* (* (pow (/ (* M D) (* 2 d)) 2) h) (/ 1 l))
8.0ms
(* (pow (/ (* M D) (* 2 d)) 2) h)
5.0ms
(/ (* M D) (* 2 d))
2.0ms
(sqrt (- 1 (* (* (pow (/ (* M D) (* 2 d)) 2) h) (/ 1 l))))

series272.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
120.0ms
(sqrt (- 1 (* (* (pow (/ (* M D) (* 2 d)) 2) h) (/ 1 l))))
61.0ms
(* (* (pow (/ (* M D) (* 2 d)) 2) h) (/ 1 l))
59.0ms
(* (pow (/ (* M D) (* 2 d)) 2) h)
32.0ms
(/ (* M D) (* 2 d))

simplify4.2s

Counts
38 → 76
Calls
38 calls:
Slowest
560.0ms
(+ (log (* (pow (/ (* M D) (* 2 d)) 2) h)) (log (/ 1 l)))
336.0ms
(* 1/4 (/ (* (pow M 2) (* (pow D 2) h)) (pow d 2)))
328.0ms
(sqrt (pow (/ (* M D) (* 2 d)) 2))
317.0ms
(* 1/4 (/ (* (pow M 2) (* (pow D 2) h)) (pow d 2)))
316.0ms
(* 1/4 (/ (* (pow M 2) (* (pow D 2) h)) (pow d 2)))

prune1.3s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 5.5b

localize21.0ms

Local error

Found 4 expressions with local error:

5.4b
(* (/ (* M D) (* 2 d)) h)
4.5b
(/ (* M D) (* 2 d))
4.5b
(/ (* M D) (* 2 d))
4.2b
(* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (/ 1 l))

rewrite38.0ms

Algorithm
rewrite-expression-head
Rules
17×add-exp-log
17×add-cbrt-cube
10×associate-*l/
9×pow1
6×add-cube-cbrt
6×associate-*l*
6×*-un-lft-identity
6×add-sqr-sqrt
5×cbrt-unprod
5×prod-exp
5×frac-times
4×add-log-exp
4×div-inv
4×associate-*r*
3×associate-*r/
3×pow-prod-down
3×div-exp
3×times-frac
3×cbrt-undiv
2×associate-/l*
2×*-commutative
2×frac-2neg
2×associate-/r*
2×clear-num
1×un-div-inv
Counts
4 → 73
Calls
4 calls:
Slowest
25.0ms
(* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (/ 1 l))
7.0ms
(* (/ (* M D) (* 2 d)) h)
2.0ms
(/ (* M D) (* 2 d))
2.0ms
(/ (* M D) (* 2 d))

series202.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
87.0ms
(* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (/ 1 l))
60.0ms
(* (/ (* M D) (* 2 d)) h)
28.0ms
(/ (* M D) (* 2 d))
28.0ms
(/ (* M D) (* 2 d))

simplify6.6s

Counts
47 → 85
Calls
47 calls:
Slowest
993.0ms
(* (* (/ (* (* (* 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)))
745.0ms
(* (* (* (* (/ (* 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)))
533.0ms
(* (* (* (* (/ (* 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)))
468.0ms
(+ (+ (log (/ (* M D) (* 2 d))) (log (* (/ (* M D) (* 2 d)) h))) (log (/ 1 l)))
309.0ms
(* 1/2 (/ (* M (* D h)) d))

prune1.5s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 5.4b

localize14.0ms

Local error

Found 4 expressions with local error:

5.8b
(* (* (/ (* M D) (* 2 d)) h) (/ 1 l))
5.4b
(* (/ (* M D) (* 2 d)) h)
4.5b
(/ (* M D) (* 2 d))
4.5b
(/ (* M D) (* 2 d))

rewrite34.0ms

Algorithm
rewrite-expression-head
Rules
10×add-exp-log
10×add-cbrt-cube
6×add-cube-cbrt
6×associate-*l*
6×*-un-lft-identity
6×pow1
6×add-sqr-sqrt
4×add-log-exp
4×div-inv
4×associate-*l/
4×associate-*r*
3×times-frac
2×associate-/l*
2×*-commutative
2×div-exp
2×frac-2neg
2×associate-/r*
2×clear-num
2×cbrt-undiv
1×cbrt-unprod
1×un-div-inv
1×associate-*r/
1×prod-exp
1×pow-prod-down
1×frac-times
Counts
4 → 64
Calls
4 calls:
Slowest
18.0ms
(* (* (/ (* M D) (* 2 d)) h) (/ 1 l))
7.0ms
(* (/ (* M D) (* 2 d)) h)
4.0ms
(/ (* M D) (* 2 d))
4.0ms
(/ (* M D) (* 2 d))

series173.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
64.0ms
(* (* (/ (* M D) (* 2 d)) h) (/ 1 l))
53.0ms
(* (/ (* M D) (* 2 d)) h)
29.0ms
(/ (* M D) (* 2 d))
27.0ms
(/ (* M D) (* 2 d))

simplify4.0s

Counts
36 → 76
Calls
36 calls:
Slowest
969.0ms
(* (* (/ (* M D) (* 2 d)) h) (/ 1 l))
306.0ms
(* 1/2 (/ (* M (* D h)) (* l d)))
295.0ms
(* 1/2 (/ (* M (* D h)) (* l d)))
279.0ms
(* 1/2 (/ (* M (* D h)) (* l d)))
247.0ms
(* (* (* (* (/ (* M D) (* 2 d)) h) (* (/ (* M D) (* 2 d)) h)) (* (/ (* M D) (* 2 d)) h)) (* (* (/ 1 l) (/ 1 l)) (/ 1 l)))

prune1.0s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 5.4b

regimes1.4s

Accuracy

0% (0.9b remaining)

Error of 8.1b against oracle of 7.2b and baseline of 8.1b

bsearch4.0ms

end0.0ms

sample3.7s

Algorithm
intervals