Average Error: 58.1 → 33.6
Time: 54.1s
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 35.3

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

  3. Taylor expanded around inf 33.6

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

  4. Final simplification33.6

    \[\leadsto 0\]

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(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: 52.7s)Debug log

sample514.0ms

Algorithm
intervals

simplify517.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

517.0ms
(* (/ 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)))))

prune19.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 56.8b

localize66.0ms

Local error

Found 4 expressions with local error:

25.5b
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
19.7b
(/ (* c0 (* d d)) (* (* w h) (* D D)))
19.7b
(/ (* c0 (* d d)) (* (* w h) (* D D)))
19.7b
(/ (* c0 (* d d)) (* (* w h) (* D D)))

rewrite68.0ms

Algorithm
rewrite-expression-head
Counts
4 → 78
Calls

4 calls. Slowest were:

45.0ms
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
10.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
7.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))

series432.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

199.0ms
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
83.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
80.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
70.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))

simplify13.1s

Counts
43 → 90
Calls

43 calls. Slowest were:

1.0s
(/ (* (* (* c0 (* d d)) (* c0 (* d d))) (* c0 (* d d))) (* (* (* (* w h) (* w h)) (* w h)) (* (* (* D D) (* D D)) (* D D))))
918.0ms
(+ (* (* 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)))))
736.0ms
(* (* (* 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))))))

prune1.9s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 31.1b

localize3.0ms

Local error

Found 1 expressions with local error:

5.7b
(* (/ c0 (* 2 w)) 0)

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
1 → 16
Calls

1 calls. Slowest were:

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

series2.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

1.0ms
(* (/ c0 (* 2 w)) 0)

simplify7.0ms

Counts
6 → 19
Calls

6 calls. Slowest were:

3.0ms
(* (cbrt (/ c0 (* 2 w))) (cbrt (/ c0 (* 2 w))))
2.0ms
(sqrt (/ c0 (* 2 w)))
1.0ms
(* c0 0)

prune223.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 30.2b

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:

prune65.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 30.2b

localize72.0ms

Local error

Found 4 expressions with local error:

25.5b
(+ (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (- M) M))))
15.6b
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
15.6b
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
15.6b
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))

rewrite84.0ms

Algorithm
rewrite-expression-head
Counts
4 → 98
Calls

4 calls. Slowest were:

26.0ms
(+ (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (- M) M))))
19.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
18.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))

series408.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

225.0ms
(+ (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (- M) M))))
69.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
67.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
47.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))

simplify11.3s

Counts
74 → 110
Calls

74 calls. Slowest were:

487.0ms
(* (exp (* (* (/ d D) (/ d D)) (/ (/ c0 h) w))) (exp (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (- M) M)))))
441.0ms
(+ (+ (log (/ d D)) (log (/ d D))) (log (/ (/ c0 h) w)))
440.0ms
(+ (+ (log (/ d D)) (log (/ d D))) (log (/ (/ c0 h) w)))

prune2.3s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 30.2b

regimes956.0ms

Accuracy

0% (2.6b remaining)

Error of 33.6b against oracle of 30.9b and baseline of 33.6b

bsearch0.0ms

end0.0ms

sample20.7s

Algorithm
intervals