Average Error: 58.0 → 33.4
Time: 49.6s
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 -inf 33.4

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

  4. Final simplification33.4

    \[\leadsto 0\]

Reproduce

herbie shell --seed 2019022 +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: 48.0s)Debug log

sample464.0ms

Algorithm
intervals
Results
114.0ms433×body80nan
96.0ms57×body2560valid
87.0ms60×body5120valid
67.0ms241×body80valid
27.0ms32×body1280valid
24.0ms19×body10240exit
11.0ms15×body640valid
1.0ms4×body160valid
1.0ms2×body320valid

simplify321.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
321.0ms
(* (/ 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)))))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.1b

localize59.0ms

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)))

rewrite87.0ms

Algorithm
rewrite-expression-head
Rules
19×add-exp-log
19×add-cbrt-cube
7×*-un-lft-identity
6×add-log-exp
6×div-exp
6×cbrt-undiv
5×add-cube-cbrt
5×fma-def
5×add-sqr-sqrt
4×div-inv
4×log1p-expm1-u
4×times-frac
4×pow1
4×expm1-log1p-u
3×associate-/l*
3×cbrt-unprod
3×prod-exp
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 → 78
Calls
4 calls:
Slowest
62.0ms
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
8.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
8.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
8.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))

series471.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
210.0ms
(+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))
99.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
90.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))
72.0ms
(/ (* c0 (* d d)) (* (* w h) (* D D)))

simplify12.9s

Counts
43 → 90
Calls
43 calls:
Slowest
880.0ms
(/ (* (* (* c0 (* d d)) (* c0 (* d d))) (* c0 (* d d))) (* (* (* (* w h) (* w h)) (* w h)) (* (* (* D D) (* D D)) (* D D))))
870.0ms
(+ (* (* 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)))))
719.0ms
(/ (* (* (* c0 (* d d)) (* c0 (* d d))) (* c0 (* d d))) (* (* (* (* w h) (* D D)) (* (* w h) (* D D))) (* (* w h) (* D D))))
694.0ms
(/ (* (* (* c0 (* d d)) (* c0 (* d d))) (* c0 (* d d))) (* (* (* (* w h) (* D D)) (* (* w h) (* D D))) (* (* w h) (* D D))))
638.0ms
(/ (* (* (* c0 (* d d)) (* c0 (* d d))) (* c0 (* d d))) (* (* (* (* w h) (* w h)) (* w h)) (* (* (* D D) (* D D)) (* D D))))

prune1.5s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 32.6b

localize3.0ms

Local error

Found 1 expressions with local error:

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

rewrite3.0ms

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×log1p-expm1-u
1×*-commutative
1×add-exp-log
1×associate-*l/
1×mul0
1×add-cbrt-cube
1×pow1
1×expm1-log1p-u
Counts
1 → 16
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)

simplify8.0ms

Counts
6 → 19
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

prune220.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 31.3b

localize0.0ms

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: 31.3b

localize67.0ms

Local error

Found 4 expressions with local error:

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

rewrite64.0ms

Algorithm
rewrite-expression-head
Rules
14×add-exp-log
14×add-cbrt-cube
14×pow1
10×associate-*l/
10×frac-times
8×associate-*r*
8×add-sqr-sqrt
7×add-cube-cbrt
7×*-un-lft-identity
6×add-log-exp
6×cbrt-unprod
6×associate-*r/
6×prod-exp
6×pow-prod-down
4×log1p-expm1-u
4×expm1-log1p-u
3×sqrt-prod
2×div-inv
2×*-commutative
2×associate-*l*
1×flip-+
1×pow1/2
1×flip3-+
1×fma-def
1×sum-log
1×rem-sqrt-square
1×+-commutative
Counts
4 → 84
Calls
4 calls:
Slowest
27.0ms
(+ (* (* (/ 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))))
17.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
17.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
2.0ms
(sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (- M) M)))

series545.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
223.0ms
(sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (- M) M)))
207.0ms
(+ (* (* (/ 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))))
60.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))
55.0ms
(* (* (/ d D) (/ d D)) (/ (/ c0 h) w))

simplify9.1s

Counts
58 → 96
Calls
58 calls:
Slowest
715.0ms
(/ (* c0 (pow d 2)) (* w (* (pow D 2) h)))
561.0ms
(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)))))
552.0ms
(* (exp (* (* (/ d D) (/ d D)) (/ (/ c0 h) w))) (exp (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (- M) M)))))
505.0ms
(sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (- M) M)))
501.0ms
(sqrt (sqrt (fma (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (* (/ d D) (/ d D)) (/ (/ c0 h) w)) (* (- M) M))))

prune1.7s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 31.3b

regimes1.1s

Accuracy

0% (3.0b remaining)

Error of 33.4b against oracle of 30.4b and baseline of 33.4b

bsearch0.0ms

end0.0ms

sample19.3s

Algorithm
intervals
Results
5.3s2353×body5120valid
4.0s13029×body80nan
3.7s1717×body2560valid
1.8s6814×body80valid
1.6s587×body10240exit
1.2s867×body1280valid
514.0ms459×body640valid
219.0ms217×body320valid
53.0ms121×body160valid