Average Error: 58.1 → 33.7
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 35.4

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

  3. Taylor expanded around 0 33.7

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

  4. Final simplification33.7

    \[\leadsto 0\]

Reproduce

herbie shell --seed 2019016 
(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

sample655.0ms

Algorithm
intervals
Results
843×(pre true 80)
431×(body nan 80)
208×(body real 80)
70×(body real 5120)
51×(body real 2560)
35×(body real 1280)
20×(body exit 10240)
13×(body real 640)
10×(body real 320)
5×(body real 160)

simplify453.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
453.0ms
(* (/ 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: 57.2b

localize60.0ms

Local error

Found 4 expressions with local error:

25.3b
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
17.4b
(/ (* c0 (* d d)) (* (* w h) (* D D)))
17.4b
(/ (* c0 (* d d)) (* (* w h) (* D D)))
17.4b
(/ (* c0 (* d d)) (* (* w h) (* D D)))

rewrite64.0ms

Algorithm
rewrite-expression-head
Rules
19×add-exp-log
19×add-cbrt-cube
6×add-log-exp
6×div-exp
6×*-un-lft-identity
6×cbrt-undiv
4×add-cube-cbrt
4×pow1
4×add-sqr-sqrt
3×associate-/l*
3×div-inv
3×cbrt-unprod
3×prod-exp
3×times-frac
3×frac-2neg
3×associate-/r*
3×clear-num
2×frac-add
2×sqrt-div
1×distribute-lft-out
1×flip--
1×flip-+
1×flip3--
1×flip3-+
1×sum-log
1×+-commutative
Counts
4 → 65
Calls
4 calls:
Slowest
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))))
6.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
5.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
5.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))

series497.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
214.0ms
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
97.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
95.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
91.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))

simplify13.1s

Counts
39 → 77
Calls
39 calls:
Slowest
1.0s
(/ (* (* (* c0 (* d d)) (* c0 (* d d))) (* c0 (* d d))) (* (* (* (* w h) (* w h)) (* w h)) (* (* (* D D) (* D D)) (* D D))))
961.0ms
(+ (* (* 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)))))
753.0ms
(* (* (* 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))))))
695.0ms
(/ (* (* (* c0 (* d d)) (* c0 (* d d))) (* c0 (* d d))) (* (* (* (* w h) (* D D)) (* (* w h) (* D D))) (* (* w h) (* D D))))
660.0ms
(+ (* (* c0 (* d d)) (sqrt (+ (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M)))) (* (* (* 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))))))

prune1.4s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 30.7b

localize3.0ms

Local error

Found 1 expressions with local error:

6.2b
(* (/ c0 (* 2 w)) 0)

rewrite2.0ms

Algorithm
rewrite-expression-head
Rules
4×associate-*l*
2×add-cube-cbrt
2×*-un-lft-identity
2×add-sqr-sqrt
1×add-log-exp
1×div-inv
1×*-commutative
1×add-exp-log
1×associate-*l/
1×mul0
1×add-cbrt-cube
1×pow1
Counts
1 → 14
Calls
1 calls:
Slowest
2.0ms
(* (/ c0 (* 2 w)) 0)

series2.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
2.0ms
(* (/ c0 (* 2 w)) 0)

simplify7.0ms

Counts
6 → 17
Calls
6 calls:
Slowest
3.0ms
(* (cbrt (/ c0 (* 2 w))) (cbrt (/ c0 (* 2 w))))
2.0ms
(sqrt (/ c0 (* 2 w)))
1.0ms
(* c0 0)
0.0ms
0
0.0ms
0

prune279.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 29.2b

localize1.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Rules
Counts
0 → 0
Calls
0 calls:
Slowest

series0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

simplify0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

prune86.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 29.2b

localize48.0ms

Local error

Found 4 expressions with local error:

25.3b
(+ (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (sqrt (* (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M))))
18.1b
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
18.1b
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
18.1b
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))

rewrite58.0ms

Algorithm
rewrite-expression-head
Rules
83×frac-times
69×associate-*r/
56×frac-add
56×sqrt-div
47×associate-*l/
22×flip-+
22×flip3-+
21×flip--
21×flip3--
19×add-exp-log
19×add-cbrt-cube
19×pow1
12×associate-*l*
9×cbrt-unprod
9×prod-exp
9×pow-prod-down
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×add-log-exp
3×div-inv
3×*-commutative
3×associate-*r*
1×sum-log
1×+-commutative
Counts
4 → 145
Calls
4 calls:
Slowest
22.0ms
(+ (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (sqrt (* (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M))))
13.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
11.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
9.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))

series820.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
458.0ms
(+ (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (sqrt (* (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M))))
129.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
128.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
104.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))

simplify35.2s

Counts
179 → 157
Calls
179 calls:
Slowest
781.0ms
(+ (* (* (/ 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))) (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (+ (* M M) (* (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M)))))) (* w (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))))))
518.0ms
(+ (* (* (/ c0 h) (* 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))) (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (+ (* M M) (* (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M)))))) (* (* w 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))))))
501.0ms
(* (* w D) (sqrt (+ (* (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (+ (* M M) (* (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M)))))
483.0ms
(+ (log (/ (/ c0 h) w)) (+ (log (/ d D)) (log (/ d D))))
462.0ms
(/ (* c0 (pow d 2)) (* (pow D 2) (* w h)))

prune3.5s

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 29.2b

regimes1.2s

Accuracy

0% (2.8b remaining)

Error of 33.7b against oracle of 30.9b and baseline of 33.7b

bsearch0.0ms

end0.0ms

sample21.1s

Algorithm
intervals
Results
25958×(pre true 80)
12968×(body nan 80)
6669×(body real 80)
2309×(body real 5120)
1748×(body real 2560)
883×(body real 1280)
547×(body exit 10240)
446×(body real 640)
271×(body real 320)
117×(body real 160)