Average Error: 58.1 → 33.8
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[0\]

Error

Bits error versus c0

Bits error versus w

Bits error versus h

Bits error versus D

Bits error versus d

Bits error versus M

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.1

    \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

  2. Taylor expanded around inf 35.8

    \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0}\]

  3. Taylor expanded around -inf 33.8

    \[\leadsto \color{blue}{0}\]

  4. Final simplification33.8

    \[\leadsto 0\]

Reproduce

herbie shell --seed 2019005 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  (* (/ c0 (* 2 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))

Details

Time bar (total: 1.3m)Debug log

sample578.0ms

Algorithm
intervals

simplify387.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

387.0ms
(* (/ c0 (* 2 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M)))))

prune26.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 59.4b

localize71.0ms

Local error

Found 4 expressions with local error:

29.7b
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
15.6b
(/ (* c0 (* d d)) (* (* w h) (* D D)))
15.6b
(/ (* c0 (* d d)) (* (* w h) (* D D)))
15.6b
(/ (* c0 (* d d)) (* (* w h) (* D D)))

rewrite118.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

87.0ms
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
12.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
12.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))

series525.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

249.0ms
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
101.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
98.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
77.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))

simplify12.8s

Counts
39 → 77
Calls

39 calls. Slowest were:

1.1s
(/ (* (* (* c0 (* d d)) (* c0 (* d d))) (* c0 (* d d))) (* (* (* (* w h) (* D D)) (* (* w h) (* D D))) (* (* w h) (* D D))))
974.0ms
(+ (* (* c0 (* d d)) (sqrt (+ (* (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D))))) (+ (* (* M M) (* M M)) (* (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M)))))) (* (* (* w h) (* D D)) (sqrt (- (pow (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) 3) (pow (* M M) 3)))))
848.0ms
(* (* (* w h) (* D D)) (sqrt (+ (* (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D))))) (+ (* (* M M) (* M M)) (* (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))

prune1.3s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 36.3b

localize3.0ms

Local error

Found 1 expressions with local error:

6.6b
(* (/ c0 (* 2 w)) 0)

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
1 → 14
Calls

1 calls. Slowest were:

3.0ms
(* (/ c0 (* 2 w)) 0)

series2.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

2.0ms
(* (/ c0 (* 2 w)) 0)

simplify13.0ms

Counts
6 → 17
Calls

6 calls. Slowest were:

5.0ms
(* (cbrt (/ c0 (* 2 w))) (cbrt (/ c0 (* 2 w))))
4.0ms
(sqrt (/ c0 (* 2 w)))
1.0ms
(* c0 0)

prune300.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 34.1b

localize0.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Counts
0 → 0
Calls

0 calls. Slowest were:

series0.0ms

Counts
0 → 0
Calls

0 calls. Slowest were:

simplify0.0ms

Counts
0 → 0
Calls

0 calls. Slowest were:

prune57.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 34.1b

localize49.0ms

Local error

Found 4 expressions with local error:

29.7b
(+ (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (sqrt (* (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M))))
13.7b
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
13.7b
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
13.7b
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))

rewrite75.0ms

Algorithm
rewrite-expression-head
Counts
4 → 145
Calls

4 calls. Slowest were:

33.0ms
(+ (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (sqrt (* (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M))))
14.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
11.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))

series745.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

469.0ms
(+ (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (sqrt (* (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M))))
111.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
89.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
75.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))

simplify32.4s

Counts
179 → 157
Calls

179 calls. Slowest were:

715.0ms
(+ (* (* (/ c0 h) (* (/ d D) (/ d D))) (sqrt (* (+ (* M M) (- (* (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (* M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))))) (+ (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M)))) (* w (sqrt (* (+ (pow M 3) (pow (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) 3)) (- (* (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (* M M))))))
565.0ms
(+ (log (/ (/ c0 h) w)) (+ (log (/ d D)) (log (/ d D))))
517.0ms
(+ (log (/ (/ c0 h) w)) (+ (log (/ d D)) (log (/ d D))))

prune3.7s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 34.1b

regimes1.2s

Accuracy

0% (2.9b remaining)

Error of 33.8b against oracle of 30.9b and baseline of 33.8b

bsearch0.0ms

end0.0ms

sample20.8s

Algorithm
intervals