Average Error: 13.6 → 8.3
Time: 27.5s
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 13.6

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

  3. Applied unpow213.6

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

  4. Applied associate-*l*12.2

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

  6. Applied div-inv12.2

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

  7. Applied associate-*r*8.3

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

  8. Final simplification8.3

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

Reproduce

herbie shell --seed 2019022 +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: 26.2s)Debug log

sample86.0ms

Algorithm
intervals
Results
44.0ms298×body80valid
19.0ms118×body80nan

simplify72.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
72.0ms
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))

prune17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 10.2b

localize34.0ms

Local error

Found 4 expressions with local error:

10.2b
(* (pow (/ (* M D) (* 2 d)) 2) (/ h l))
4.3b
(/ (* M D) (* 2 d))
2.8b
(sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))
1.5b
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))

rewrite23.0ms

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×log1p-expm1-u
4×prod-exp
4×sqrt-div
4×pow1
4×expm1-log1p-u
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 → 80
Calls
4 calls:
Slowest
9.0ms
(* (pow (/ (* M D) (* 2 d)) 2) (/ h l))
5.0ms
(/ (* M D) (* 2 d))
4.0ms
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))
3.0ms
(sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))

series432.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
231.0ms
(* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))
121.0ms
(sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))
53.0ms
(* (pow (/ (* M D) (* 2 d)) 2) (/ h l))
26.0ms
(/ (* M D) (* 2 d))

simplify4.5s

Counts
47 → 92
Calls
47 calls:
Slowest
712.0ms
(sqrt (+ 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))
417.0ms
(+ (* (- (log (* M D)) (log (* 2 d))) 2) (log (/ h l)))
348.0ms
(+ (log (pow (/ (* M D) (* 2 d)) 2)) (log (/ h l)))
291.0ms
(sqrt (pow (/ (* M D) (* 2 d)) 2))
241.0ms
(+ (* (log (/ (* M D) (* 2 d))) 2) (log (/ h l)))

prune1.4s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 5.0b

localize12.0ms

Local error

Found 4 expressions with local error:

7.1b
(* (/ (* M D) (* 2 d)) (/ h l))
4.3b
(/ (* M D) (* 2 d))
4.3b
(/ (* M D) (* 2 d))
2.8b
(sqrt (- 1 (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) (/ h l)))))

rewrite23.0ms

Algorithm
rewrite-expression-head
Rules
13×add-exp-log
13×add-cbrt-cube
8×add-sqr-sqrt
7×add-cube-cbrt
7×*-un-lft-identity
6×pow1
5×associate-*l*
4×add-log-exp
4×div-inv
4×log1p-expm1-u
4×associate-*r*
4×expm1-log1p-u
3×sqrt-prod
3×div-exp
3×times-frac
3×cbrt-undiv
2×associate-/l*
2×cbrt-unprod
2×prod-exp
2×sqrt-div
2×frac-2neg
2×associate-/r*
2×clear-num
1×flip--
1×*-commutative
1×associate-*r/
1×associate-*l/
1×pow-prod-down
1×pow1/2
1×flip3--
1×rem-sqrt-square
1×frac-times
Counts
4 → 77
Calls
4 calls:
Slowest
8.0ms
(* (/ (* M D) (* 2 d)) (/ h l))
5.0ms
(/ (* M D) (* 2 d))
5.0ms
(/ (* M D) (* 2 d))
3.0ms
(sqrt (- 1 (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) (/ h l)))))

series252.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
161.0ms
(sqrt (- 1 (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) (/ h l)))))
48.0ms
(* (/ (* M D) (* 2 d)) (/ h l))
25.0ms
(/ (* M D) (* 2 d))
17.0ms
(/ (* M D) (* 2 d))

simplify4.5s

Counts
42 → 89
Calls
42 calls:
Slowest
610.0ms
(sqrt (* (cbrt (- 1 (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) (/ h l))))) (cbrt (- 1 (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) (/ h l)))))))
415.0ms
(sqrt (- (* 1 1) (* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) (/ h l))) (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) (/ h l))))))
358.0ms
(sqrt (- (pow 1 3) (pow (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) (/ h l))) 3)))
353.0ms
(* (/ (* (* (* M D) (* M D)) (* M D)) (* (* (* 2 d) (* 2 d)) (* 2 d))) (* (* (/ h l) (/ h l)) (/ h l)))
306.0ms
(* 1/2 (/ (* M (* D h)) (* l d)))

prune1.3s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 4.8b

localize18.0ms

Local error

Found 4 expressions with local error:

4.9b
(* (* (/ (* M D) (* 2 d)) h) (/ 1 l))
4.8b
(* (/ (* M D) (* 2 d)) h)
4.3b
(/ (* M D) (* 2 d))
4.3b
(/ (* M D) (* 2 d))

rewrite20.0ms

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×log1p-expm1-u
4×associate-*l/
4×associate-*r*
4×expm1-log1p-u
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 → 72
Calls
4 calls:
Slowest
9.0ms
(* (* (/ (* M D) (* 2 d)) h) (/ 1 l))
5.0ms
(* (/ (* M D) (* 2 d)) h)
3.0ms
(/ (* M D) (* 2 d))
3.0ms
(/ (* M D) (* 2 d))

series131.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
55.0ms
(* (* (/ (* M D) (* 2 d)) h) (/ 1 l))
44.0ms
(* (/ (* M D) (* 2 d)) h)
16.0ms
(/ (* M D) (* 2 d))
16.0ms
(/ (* M D) (* 2 d))

simplify3.7s

Counts
36 → 84
Calls
36 calls:
Slowest
486.0ms
(* (* (/ (* M D) (* 2 d)) h) (/ 1 l))
337.0ms
(* 1/2 (/ (* M (* D h)) (* l d)))
326.0ms
(* (* (* M D) h) (/ 1 l))
323.0ms
(* 1/2 (/ (* M (* D h)) (* l d)))
317.0ms
(* 1/2 (/ (* M (* D h)) (* l d)))

prune1.2s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 4.8b

localize26.0ms

Local error

Found 4 expressions with local error:

5.2b
(* M (* D h))
4.9b
(* (* 1/2 (/ (* M (* D h)) d)) (/ 1 l))
4.6b
(/ (* M (* D h)) d)
4.3b
(/ (* M D) (* 2 d))

rewrite20.0ms

Algorithm
rewrite-expression-head
Rules
8×add-exp-log
8×add-cbrt-cube
6×pow1
5×add-cube-cbrt
5×associate-*r*
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×div-inv
3×associate-*r/
2×associate-/l*
2×*-commutative
2×frac-2neg
2×clear-num
1×cbrt-unprod
1×un-div-inv
1×prod-exp
1×associate-*l/
1×pow-prod-down
1×associate-*l*
1×div-exp
1×times-frac
1×associate-/r*
1×cbrt-undiv
1×frac-times
Counts
4 → 63
Calls
4 calls:
Slowest
7.0ms
(* (* 1/2 (/ (* M (* D h)) d)) (/ 1 l))
6.0ms
(/ (* M (* D h)) d)
5.0ms
(/ (* M D) (* 2 d))
1.0ms
(* M (* D h))

series140.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
53.0ms
(* (* 1/2 (/ (* M (* D h)) d)) (/ 1 l))
33.0ms
(/ (* M (* D h)) d)
28.0ms
(* M (* D h))
26.0ms
(/ (* M D) (* 2 d))

simplify2.4s

Counts
26 → 75
Calls
26 calls:
Slowest
375.0ms
(* 1/2 (/ (* M (* D h)) (* l d)))
310.0ms
(* 1/2 (/ (* M (* D h)) (* l d)))
308.0ms
(* 1/2 (/ (* M (* D h)) (* l d)))
284.0ms
(* (* 1/2 (* M (* D h))) (/ 1 l))
201.0ms
(* (* 1/2 (/ (* M (* D h)) d)) (/ 1 l))

prune1.2s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 4.3b

regimes1.7s

Accuracy

0% (1.2b remaining)

Error of 8.3b against oracle of 7.1b and baseline of 8.3b

bsearch4.0ms

end0.0ms

sample3.1s

Algorithm
intervals
Results
2.0s9306×body80valid
608.0ms3187×body80nan