Average Error: 58.0 → 33.8
Time: 1.6m
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 58.0

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

  2. Initial simplification55.5

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

  3. Taylor expanded around -inf 33.8

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

  4. Final simplification33.8

    \[\leadsto 0\]

Reproduce

herbie shell --seed 2018365 +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: 1.0m)Debug log

start1.2s

Algorithm
intervals

setup503.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 53.5b

localize60.0ms

Local error

Found 4 expressions with local error:

47.6b
(fma (/ c0 (* w 2)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M)))) (* (/ c0 (* w 2)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w))))
14.8b
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
14.8b
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
14.8b
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))

rewrite43.0ms

Algorithm
rewrite-expression-head
Counts
4 → 94
Calls

4 calls. Slowest were:

18.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
15.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
8.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))

series594.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

442.0ms
(fma (/ c0 (* w 2)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M)))) (* (/ c0 (* w 2)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w))))
67.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
42.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
42.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))

simplify9.4s

Counts
72 → 106
Calls

72 calls. Slowest were:

413.0ms
(+ (+ (log (/ d D)) (log (/ d D))) (log (/ (/ c0 h) w)))
411.0ms
(/ (* c0 (pow d 2)) (* w (* (pow D 2) h)))
394.0ms
(* (* d (/ d D)) (/ (/ c0 h) w))

prune2.3s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 25.3b

localize1.0ms

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:

prune102.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 25.3b

localize16.0ms

Local error

Found 4 expressions with local error:

30.0b
(+ (* (/ c0 (* w 2)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M))))) (* (/ c0 (* w 2)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w))))
14.8b
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
14.8b
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
14.8b
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))

rewrite79.0ms

Algorithm
rewrite-expression-head
Counts
4 → 114
Calls

4 calls. Slowest were:

25.0ms
(+ (* (/ c0 (* w 2)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M))))) (* (/ c0 (* w 2)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w))))
18.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
17.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))

series668.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

506.0ms
(+ (* (/ c0 (* w 2)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M))))) (* (/ c0 (* w 2)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w))))
65.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
49.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
48.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))

simplify15.9s

Counts
106 → 126
Calls

106 calls. Slowest were:

607.0ms
(+ (log (* (/ d D) (/ d D))) (log (/ (/ c0 h) w)))
525.0ms
(+ (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M)))) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)))
445.0ms
(/ (* c0 (pow d 2)) (* w (* (pow D 2) h)))

prune2.7s

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 24.8b

localize40.0ms

Local error

Found 4 expressions with local error:

30.0b
(+ (* (/ c0 (* w 2)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M))))) (* (/ c0 (* w 2)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w))))
30.0b
(+ (* (/ c0 (* w 2)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M))))) (* (/ c0 (* w 2)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w))))
30.0b
(+ (* (/ c0 (* w 2)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M))))) (* (/ c0 (* w 2)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w))))
14.8b
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))

rewrite58.0ms

Algorithm
rewrite-expression-head
Counts
4 → 118
Calls

4 calls. Slowest were:

21.0ms
(+ (* (/ c0 (* w 2)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M))))) (* (/ c0 (* w 2)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w))))
14.0ms
(+ (* (/ c0 (* w 2)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M))))) (* (/ c0 (* w 2)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w))))
12.0ms
(+ (* (/ c0 (* w 2)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M))))) (* (/ c0 (* w 2)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w))))

series1.5s

Counts
4 → 12
Calls

4 calls. Slowest were:

522.0ms
(+ (* (/ c0 (* w 2)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M))))) (* (/ c0 (* w 2)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w))))
485.0ms
(+ (* (/ c0 (* w 2)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M))))) (* (/ c0 (* w 2)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w))))
451.0ms
(+ (* (/ c0 (* w 2)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M))))) (* (/ c0 (* w 2)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w))))
65.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))

simplify20.1s

Counts
134 → 130
Calls

134 calls. Slowest were:

662.0ms
(+ (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M)))) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)))
626.0ms
(+ (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M)))) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)))
594.0ms
(+ (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* M (- M)))) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)))

prune3.8s

Pruning

13 alts after pruning (11 fresh and 2 done)

Merged error: 24.8b

regimes1.8s

Accuracy

0% (6.2b remaining)

Error of 33.8b against oracle of 27.7b and baseline of 33.8b

bsearch0.0ms