Average Error: 58.1 → 32.1
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) \le 3.5838182145780534 \cdot 10^{+231}:\\ \;\;\;\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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. Split input into 2 regimes

  2. if (* (/ 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))))) < 3.5838182145780534e+231

    1. Initial program 34.2

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

    if 3.5838182145780534e+231 < (* (/ 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)))))

    1. Initial program 62.6

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

    2. Taylor expanded around inf 33.8

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

    3. Taylor expanded around -inf 31.7

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification32.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) \le 3.5838182145780534 \cdot 10^{+231}:\\ \;\;\;\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(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

sample592.0ms

Algorithm
intervals

simplify461.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

461.0ms
(* (/ 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)))))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 56.8b

localize44.0ms

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)))

rewrite64.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

46.0ms
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
6.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
5.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))

series376.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

162.0ms
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
78.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
75.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
60.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))

simplify12.6s

Counts
39 → 77
Calls

39 calls. Slowest were:

945.0ms
(/ (* (* (* c0 (* d d)) (* c0 (* d d))) (* c0 (* d d))) (* (* (* (* w h) (* w h)) (* w h)) (* (* (* D D) (* D D)) (* D D))))
876.0ms
(+ (* (* 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)))))
809.0ms
(* (* (* 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

8 alts after pruning (7 fresh and 1 done)

Merged error: 31.1b

localize3.0ms

Local error

Found 1 expressions with local error:

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

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
1 → 14
Calls

1 calls. Slowest were:

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

series2.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

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

simplify9.0ms

Counts
6 → 17
Calls

6 calls. Slowest were:

4.0ms
(sqrt (/ c0 (* 2 w)))
3.0ms
(* (cbrt (/ c0 (* 2 w))) (cbrt (/ c0 (* 2 w))))
1.0ms
(* c0 0)

prune196.0ms

Pruning

7 alts after pruning (6 fresh and 1 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:

prune30.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 30.2b

localize49.0ms

Local error

Found 4 expressions with local error:

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

rewrite57.0ms

Algorithm
rewrite-expression-head
Counts
4 → 145
Calls

4 calls. Slowest were:

26.0ms
(+ (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (sqrt (* (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M))))
9.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
9.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))

series688.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

370.0ms
(+ (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (sqrt (* (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M))))
118.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
103.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
98.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))

simplify35.4s

Counts
179 → 157
Calls

179 calls. Slowest were:

898.0ms
(+ (log (/ (/ c0 h) w)) (+ (log (/ d D)) (log (/ d D))))
802.0ms
(+ (* (* (/ (/ c0 h) w) (* 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))) (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (+ (* M M) (* (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M)))))) (* (* D D) (sqrt (* (+ (pow M 3) (pow (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) 3)) (- (pow (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) 3) (pow M 3))))))
555.0ms
(+ (log (/ (/ c0 h) w)) (+ (log (/ d D)) (log (/ d D))))

prune3.7s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 30.2b

regimes1.1s

Accuracy

55.4% (1.2b remaining)

Error of 32.1b against oracle of 30.9b and baseline of 33.6b

bsearch11.0ms

end0.0ms

sample21.0s

Algorithm
intervals