Average Error: 58.5 → 33.5
Time: 51.4s
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 58.5

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

  2. Taylor expanded around inf 35.1

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

  3. Taylor expanded around -inf 33.5

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

  4. Final simplification33.5

    \[\leadsto 0\]

Reproduce

herbie shell --seed 2019008 +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: 50.1s)Debug log

sample573.0ms

Algorithm
intervals

simplify392.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

392.0ms
(* (/ 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: 57.5b

localize57.0ms

Local error

Found 4 expressions with local error:

28.1b
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
17.7b
(/ (* c0 (* d d)) (* (* w h) (* D D)))
17.7b
(/ (* c0 (* d d)) (* (* w h) (* D D)))
17.7b
(/ (* c0 (* d d)) (* (* w h) (* D D)))

rewrite74.0ms

Algorithm
rewrite-expression-head
Counts
4 → 78
Calls

4 calls. Slowest were:

47.0ms
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
11.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
8.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))

series495.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

228.0ms
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
104.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
84.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
79.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))

simplify12.9s

Counts
43 → 90
Calls

43 calls. Slowest were:

957.0ms
(/ (* (* (* c0 (* d d)) (* c0 (* d d))) (* c0 (* d d))) (* (* (* (* w h) (* w h)) (* w h)) (* (* (* D D) (* D D)) (* D D))))
885.0ms
(+ (* (* 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)))))
861.0ms
(* (* (* 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))))))

prune1.6s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 31.9b

localize6.0ms

Local error

Found 1 expressions with local error:

4.7b
(* (/ c0 (* 2 w)) 0)

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
1 → 16
Calls

1 calls. Slowest were:

5.0ms
(* (/ c0 (* 2 w)) 0)

series3.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

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

simplify18.0ms

Counts
6 → 19
Calls

6 calls. Slowest were:

6.0ms
(* (cbrt (/ c0 (* 2 w))) (cbrt (/ c0 (* 2 w))))
5.0ms
(sqrt (/ c0 (* 2 w)))
1.0ms
(* c0 0)

prune302.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 30.1b

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:

prune36.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 30.1b

localize59.0ms

Local error

Found 4 expressions with local error:

28.1b
(+ (* (* (/ 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))))
14.4b
(sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (- M) M)))
11.5b
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
11.5b
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))

rewrite46.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

17.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
17.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
10.0ms
(+ (* (* (/ 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))))

series471.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

211.0ms
(+ (* (* (/ 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))))
176.0ms
(sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (- M) M)))
42.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
42.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))

simplify9.1s

Counts
58 → 96
Calls

58 calls. Slowest were:

575.0ms
(sqrt (* (cbrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (- M) M))) (cbrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (- M) M)))))
555.0ms
(sqrt (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (- M) M))))
506.0ms
(sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (- M) M)))

prune1.8s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 30.1b

regimes964.0ms

Accuracy

0% (2.7b remaining)

Error of 33.5b against oracle of 30.8b and baseline of 33.5b

bsearch0.0ms

end0.0ms

sample21.2s

Algorithm
intervals