Average Error: 58.0 → 33.4
Time: 1.2m
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. Taylor expanded around inf 35.1

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

  3. Taylor expanded around 0 33.4

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

  4. Final simplification33.4

    \[\leadsto 0\]

Reproduce

herbie shell --seed 2019022 
(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: 1.2m)Debug log

sample754.0ms

Algorithm
intervals
Results
379.0ms241×body80valid
103.0ms60×body5120valid
99.0ms433×body80nan
65.0ms57×body2560valid
33.0ms19×body10240exit
29.0ms32×body1280valid
10.0ms15×body640valid
2.0ms4×body160valid
1.0ms2×body320valid

simplify402.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
402.0ms
(* (/ 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)))))

prune14.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.1b

localize40.0ms

Local error

Found 4 expressions with local error:

24.5b
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
20.2b
(/ (* c0 (* d d)) (* (* w h) (* D D)))
20.2b
(/ (* c0 (* d d)) (* (* w h) (* D D)))
20.2b
(/ (* c0 (* d d)) (* (* w h) (* D D)))

rewrite74.0ms

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
51.0ms
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
7.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
7.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
7.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))

series382.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
183.0ms
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
71.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
69.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
58.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))

simplify10.1s

Counts
39 → 77
Calls
39 calls:
Slowest
720.0ms
(+ (* (* 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)))))
597.0ms
(* (* (* 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))))))
530.0ms
(/ (* (* (* c0 (* d d)) (* c0 (* d d))) (* c0 (* d d))) (* (* (* (* w h) (* w h)) (* w h)) (* (* (* D D) (* D D)) (* D D))))
518.0ms
(/ (* (* (* c0 (* d d)) (* c0 (* d d))) (* c0 (* d d))) (* (* (* (* w h) (* D D)) (* (* w h) (* D D))) (* (* w h) (* D D))))
504.0ms
(/ (* (* (* c0 (* d d)) (* c0 (* d d))) (* c0 (* d d))) (* (* (* (* w h) (* D D)) (* (* w h) (* D D))) (* (* w h) (* D D))))

prune1.3s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 32.5b

localize3.0ms

Local error

Found 1 expressions with local error:

4.9b
(* (/ c0 (* 2 w)) 0)

rewrite4.0ms

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
4.0ms
(* (/ c0 (* 2 w)) 0)

series2.0ms

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

simplify15.0ms

Counts
6 → 17
Calls
6 calls:
Slowest
5.0ms
(* (cbrt (/ c0 (* 2 w))) (cbrt (/ c0 (* 2 w))))
4.0ms
(sqrt (/ c0 (* 2 w)))
4.0ms
0
1.0ms
(* c0 0)
0.0ms
0

prune231.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 31.2b

localize2.0ms

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

prune53.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 31.2b

localize36.0ms

Local error

Found 4 expressions with local error:

24.5b
(+ (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (sqrt (* (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M))))
15.1b
(sqrt (* (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M)))
14.3b
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
14.3b
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))

rewrite61.0ms

Algorithm
rewrite-expression-head
Rules
82×frac-times
66×associate-*r/
64×sqrt-div
56×frac-add
46×associate-*l/
25×flip-+
25×flip3-+
24×flip--
24×flip3--
14×add-exp-log
14×add-cbrt-cube
14×pow1
8×associate-*l*
6×add-log-exp
6×cbrt-unprod
6×add-cube-cbrt
6×prod-exp
6×pow-prod-down
6×*-un-lft-identity
6×add-sqr-sqrt
2×div-inv
2×*-commutative
2×associate-*r*
1×sqrt-prod
1×pow1/2
1×sum-log
1×+-commutative
Counts
4 → 136
Calls
4 calls:
Slowest
21.0ms
(+ (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (sqrt (* (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M))))
12.0ms
(sqrt (* (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M)))
11.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
11.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))

series861.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
412.0ms
(+ (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (sqrt (* (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M))))
288.0ms
(sqrt (* (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M)))
83.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))
77.0ms
(* (/ (/ c0 h) w) (* (/ d D) (/ d D)))

simplify32.6s

Counts
170 → 148
Calls
170 calls:
Slowest
667.0ms
(+ (* (* (/ c0 h) (* 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 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))))))
511.0ms
(* 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)))))
465.0ms
(/ (* c0 (pow d 2)) (* (pow D 2) (* w h)))
447.0ms
(+ (log (/ (/ c0 h) w)) (+ (log (/ d D)) (log (/ d D))))
435.0ms
(+ (* (* (/ (/ c0 h) w) (* (/ 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)))))) (* 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))))))

prune3.0s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 31.2b

regimes1.3s

Accuracy

0% (3.1b remaining)

Error of 33.4b against oracle of 30.3b and baseline of 33.4b

bsearch0.0ms

end0.0ms

sample20.2s

Algorithm
intervals
Results
5.4s2353×body5120valid
4.7s13029×body80nan
3.8s1717×body2560valid
1.8s6814×body80valid
1.6s867×body1280valid
1.3s587×body10240exit
550.0ms459×body640valid
158.0ms217×body320valid
75.0ms121×body160valid