Average Error: 25.7 → 14.7
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -3.519360298233161 \cdot 10^{+217}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{\ell}}\right)\right)\\ \mathbf{elif}\;\ell \le 2.898584392085 \cdot 10^{-311}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right) \cdot \left(\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{d} \cdot \sqrt{d}\right) \cdot \left(1 - \frac{\frac{D}{d} \cdot \frac{M}{2}}{\frac{2 \cdot \ell}{h \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array}\]

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if l < -3.519360298233161e+217

    1. Initial program 31.6

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
    2. Using strategy rm

    3. Applied associate-*r/33.4

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right)\]

    4. Taylor expanded around -inf 29.2

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)}}\right) \cdot \left(1 - \frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right)\]

    5. Simplified26.4

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\frac{-1}{\ell}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right)\]

    if -3.519360298233161e+217 < l < 2.898584392085e-311

    1. Initial program 24.5

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
    2. Using strategy rm

    3. Applied associate-*r/23.0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right)\]

    4. Taylor expanded around -inf 20.8

      \[\leadsto \left(\color{blue}{e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{h}\right) - \log \left(\frac{-1}{d}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right)\]

    5. Simplified17.3

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{-1}{h}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right)\]

    if 2.898584392085e-311 < l

    1. Initial program 25.4

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
    2. Using strategy rm

    3. Applied associate-*r/24.6

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right)\]
    4. Using strategy rm

    5. Applied pow124.6

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{{\left(1 - \frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right)}^{1}}\]

    6. Applied pow124.6

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}^{1}} \cdot {\left(1 - \frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right)}^{1}\]

    7. Applied pow-prod-down24.6

      \[\leadsto \color{blue}{{\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right)\right)}^{1}}\]

    8. Simplified24.3

      \[\leadsto {\color{blue}{\left(\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{2 \cdot \frac{\ell}{h}}{\frac{M}{2} \cdot \frac{D}{d}}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\right)}}^{1}\]
    9. Using strategy rm

    10. Applied associate-*r/24.3

      \[\leadsto {\left(\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\color{blue}{\frac{2 \cdot \ell}{h}}}{\frac{M}{2} \cdot \frac{D}{d}}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\right)}^{1}\]

    11. Applied associate-/l/22.9

      \[\leadsto {\left(\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\color{blue}{\frac{2 \cdot \ell}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot h}}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\right)}^{1}\]
    12. Using strategy rm

    13. Applied sqrt-div16.4

      \[\leadsto {\left(\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{2 \cdot \ell}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot h}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right)\right)}^{1}\]

    14. Applied sqrt-div10.3

      \[\leadsto {\left(\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{2 \cdot \ell}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot h}}\right) \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\right)}^{1}\]

    15. Applied frac-times10.3

      \[\leadsto {\left(\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{2 \cdot \ell}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot h}}\right) \cdot \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}}\right)}^{1}\]

    16. Applied associate-*r/10.1

      \[\leadsto {\color{blue}{\left(\frac{\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{2 \cdot \ell}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot h}}\right) \cdot \left(\sqrt{d} \cdot \sqrt{d}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\right)}}^{1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification14.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -3.519360298233161 \cdot 10^{+217}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{\ell}}\right)\right)\\ \mathbf{elif}\;\ell \le 2.898584392085 \cdot 10^{-311}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right) \cdot \left(\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{d} \cdot \sqrt{d}\right) \cdot \left(1 - \frac{\frac{D}{d} \cdot \frac{M}{2}}{\frac{2 \cdot \ell}{h \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))

Details

Time bar (total: 1.3m)Debug log

sample1.1s

Algorithm
intervals
Results
1729×(pre true 80)
1314×(body nan 80)
415×(body real 80)

simplify219.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
218.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

prune36.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 26.5b

localize76.0ms

Local error

Found 4 expressions with local error:

13.7b
(pow (/ d h) (/ 1 2))
13.6b
(pow (/ d l) (/ 1 2))
10.9b
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
9.0b
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

rewrite31.0ms

Algorithm
rewrite-expression-head
Rules
43×add-exp-log
28×prod-exp
14×add-cbrt-cube
10×add-cube-cbrt
10×pow-exp
10×*-un-lft-identity
10×pow1
10×add-sqr-sqrt
9×pow-to-exp
8×unpow-prod-down
8×pow-unpow
7×associate-*r*
6×cbrt-unprod
5×div-inv
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-*r/
3×associate-*l/
2×*-commutative
2×pow-prod-down
2×associate-*l*
2×distribute-lft-in
2×pow-pow
2×sub-neg
2×distribute-rgt-in
1×flip--
1×div-exp
1×flip3--
1×frac-times
Counts
4 → 99
Calls
4 calls:
Slowest
16.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))
11.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
2.0ms
(pow (/ d h) (/ 1 2))
1.0ms
(pow (/ d l) (/ 1 2))

series509.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
253.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))
96.0ms
(pow (/ d l) (/ 1 2))
90.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
70.0ms
(pow (/ d h) (/ 1 2))

simplify8.3s

Counts
71 → 111
Calls
71 calls:
Slowest
827.0ms
(* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h)
550.0ms
(+ (+ (log (/ 1 2)) (log (pow (/ (* M D) (* 2 d)) 2))) (log (/ h l)))
451.0ms
(sqrt (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))
327.0ms
(* (* (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2))) (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2))) (* (* (/ h l) (/ h l)) (/ h l)))
246.0ms
(* (* (* (* (pow (/ d h) (/ 1 2)) (pow (/ d h) (/ 1 2))) (pow (/ d h) (/ 1 2))) (* (* (pow (/ d l) (/ 1 2)) (pow (/ d l) (/ 1 2))) (pow (/ d l) (/ 1 2)))) (* (* (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))

prune1.9s

Pruning

18 alts after pruning (18 fresh and 0 done)

Merged error: 13.7b

localize24.0ms

Local error

Found 4 expressions with local error:

13.7b
(pow (/ d h) (/ 1 2))
13.6b
(pow (/ d l) (/ 1 2))
9.0b
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)))
6.0b
(/ (* M D) (* 2 d))

rewrite42.0ms

Algorithm
rewrite-expression-head
Rules
31×add-exp-log
19×prod-exp
11×add-cbrt-cube
9×add-cube-cbrt
9×*-un-lft-identity
9×add-sqr-sqrt
8×pow-exp
8×unpow-prod-down
8×pow-unpow
8×pow1
8×pow-to-exp
5×div-inv
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×cbrt-unprod
3×associate-*r*
2×associate-*r/
2×distribute-lft-in
2×pow-pow
2×sub-neg
2×distribute-rgt-in
1×associate-/l*
1×flip--
1×*-commutative
1×pow-prod-down
1×associate-*l*
1×div-exp
1×flip3--
1×times-frac
1×frac-2neg
1×associate-/r*
1×clear-num
1×cbrt-undiv
Counts
4 → 90
Calls
4 calls:
Slowest
30.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)))
5.0ms
(/ (* M D) (* 2 d))
3.0ms
(pow (/ d h) (/ 1 2))
3.0ms
(pow (/ d l) (/ 1 2))

series394.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
202.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)))
89.0ms
(pow (/ d h) (/ 1 2))
75.0ms
(pow (/ d l) (/ 1 2))
28.0ms
(/ (* M D) (* 2 d))

simplify4.9s

Counts
59 → 102
Calls
59 calls:
Slowest
259.0ms
(+ (* 1 1) (+ (* (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l) (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)) (* 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l))))
244.0ms
(exp (* 1/2 (- (log (/ 1 h)) (log (/ 1 d)))))
240.0ms
(* (* (* (* (pow (/ d h) (/ 1 2)) (pow (/ d h) (/ 1 2))) (pow (/ d h) (/ 1 2))) (* (* (pow (/ d l) (/ 1 2)) (pow (/ d l) (/ 1 2))) (pow (/ d l) (/ 1 2)))) (* (* (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l))) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l))))
218.0ms
(* (* (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2)))) (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2)))) (* (* (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l))) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l))))
210.0ms
(+ (+ (* (log (/ d h)) (/ 1 2)) (* (log (/ d l)) (/ 1 2))) (log (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l))))

prune2.0s

Pruning

17 alts after pruning (17 fresh and 0 done)

Merged error: 12.0b

localize55.0ms

Local error

Found 4 expressions with local error:

13.7b
(sqrt (/ d h))
13.6b
(sqrt (/ d l))
9.2b
(/ (* 2 (/ l h)) (* (/ M 2) (/ D d)))
9.0b
(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (* 2 (/ l h)) (* (/ M 2) (/ D d))))) (* (sqrt (/ d l)) (sqrt (/ d h))))

rewrite39.0ms

Algorithm
rewrite-expression-head
Rules
14×add-exp-log
14×sqrt-div
14×add-cbrt-cube
10×frac-times
9×pow1
9×add-sqr-sqrt
8×sqrt-prod
8×associate-*r/
7×add-cube-cbrt
7×*-un-lft-identity
6×associate-*l/
4×add-log-exp
4×flip--
4×log1p-expm1-u
4×cbrt-unprod
4×prod-exp
4×flip3--
4×expm1-log1p-u
3×div-inv
3×associate-/r/
3×pow-prod-down
3×associate-*l*
2×div-exp
2×pow1/2
2×rem-sqrt-square
2×cbrt-undiv
1×associate-/l/
1×associate-/l*
1×*-commutative
1×associate-*r*
1×times-frac
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
4 → 86
Calls
4 calls:
Slowest
30.0ms
(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (* 2 (/ l h)) (* (/ M 2) (/ D d))))) (* (sqrt (/ d l)) (sqrt (/ d h))))
6.0ms
(/ (* 2 (/ l h)) (* (/ M 2) (/ D d)))
1.0ms
(sqrt (/ d h))
1.0ms
(sqrt (/ d l))

series369.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
231.0ms
(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (* 2 (/ l h)) (* (/ M 2) (/ D d))))) (* (sqrt (/ d l)) (sqrt (/ d h))))
71.0ms
(/ (* 2 (/ l h)) (* (/ M 2) (/ D d)))
38.0ms
(sqrt (/ d h))
29.0ms
(sqrt (/ d l))

simplify7.9s

Counts
59 → 98
Calls
59 calls:
Slowest
642.0ms
(* (+ (* 1 1) (+ (* (/ (* (/ M 2) (/ D d)) (/ (* 2 (/ l h)) (* (/ M 2) (/ D d)))) (/ (* (/ M 2) (/ D d)) (/ (* 2 (/ l h)) (* (/ M 2) (/ D d))))) (* 1 (/ (* (/ M 2) (/ D d)) (/ (* 2 (/ l h)) (* (/ M 2) (/ D d))))))) (sqrt h))
593.0ms
(* (+ (* 1 1) (+ (* (/ (* (/ M 2) (/ D d)) (/ (* 2 (/ l h)) (* (/ M 2) (/ D d)))) (/ (* (/ M 2) (/ D d)) (/ (* 2 (/ l h)) (* (/ M 2) (/ D d))))) (* 1 (/ (* (/ M 2) (/ D d)) (/ (* 2 (/ l h)) (* (/ M 2) (/ D d))))))) (sqrt l))
584.0ms
(* (+ (* 1 1) (+ (* (/ (* (/ M 2) (/ D d)) (/ (* 2 (/ l h)) (* (/ M 2) (/ D d)))) (/ (* (/ M 2) (/ D d)) (/ (* 2 (/ l h)) (* (/ M 2) (/ D d))))) (* 1 (/ (* (/ M 2) (/ D d)) (/ (* 2 (/ l h)) (* (/ M 2) (/ D d))))))) (* (sqrt l) (sqrt h)))
428.0ms
(* 4 (/ (* l d) (* h (* M D))))
384.0ms
(* 4 (/ (* l d) (* h (* M D))))

prune2.3s

Pruning

19 alts after pruning (19 fresh and 0 done)

Merged error: 9.3b

localize21.0ms

Local error

Found 4 expressions with local error:

13.7b
(sqrt (/ d h))
13.6b
(sqrt (/ d l))
9.0b
(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (* 2 l) (* (* (/ M 2) (/ D d)) h)))) (* (sqrt (/ d l)) (sqrt (/ d h))))
6.5b
(/ (* 2 l) (* (* (/ M 2) (/ D d)) h))

rewrite32.0ms

Algorithm
rewrite-expression-head
Rules
14×sqrt-div
11×add-exp-log
11×add-cbrt-cube
10×frac-times
9×associate-*l/
9×pow1
9×add-sqr-sqrt
8×sqrt-prod
7×add-cube-cbrt
7×associate-*r/
7×*-un-lft-identity
4×add-log-exp
4×flip--
4×log1p-expm1-u
4×flip3--
4×expm1-log1p-u
3×div-inv
3×cbrt-unprod
3×associate-/r/
3×prod-exp
3×pow-prod-down
3×associate-*l*
2×pow1/2
2×rem-sqrt-square
1×associate-/l*
1×*-commutative
1×associate-*r*
1×div-exp
1×times-frac
1×frac-2neg
1×associate-/r*
1×clear-num
1×cbrt-undiv
Counts
4 → 83
Calls
4 calls:
Slowest
19.0ms
(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (* 2 l) (* (* (/ M 2) (/ D d)) h)))) (* (sqrt (/ d l)) (sqrt (/ d h))))
10.0ms
(/ (* 2 l) (* (* (/ M 2) (/ D d)) h))
1.0ms
(sqrt (/ d h))
1.0ms
(sqrt (/ d l))

series325.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
202.0ms
(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (* 2 l) (* (* (/ M 2) (/ D d)) h)))) (* (sqrt (/ d l)) (sqrt (/ d h))))
71.0ms
(/ (* 2 l) (* (* (/ M 2) (/ D d)) h))
27.0ms
(sqrt (/ d h))
25.0ms
(sqrt (/ d l))

simplify8.5s

Counts
55 → 95
Calls
55 calls:
Slowest
779.0ms
(* (+ (* 1 1) (+ (* (/ (* (/ M 2) (/ D d)) (/ (* 2 l) (* (* (/ M 2) (/ D d)) h))) (/ (* (/ M 2) (/ D d)) (/ (* 2 l) (* (* (/ M 2) (/ D d)) h)))) (* 1 (/ (* (/ M 2) (/ D d)) (/ (* 2 l) (* (* (/ M 2) (/ D d)) h)))))) (sqrt l))
748.0ms
(* (+ (* 1 1) (+ (* (/ (* (/ M 2) (/ D d)) (/ (* 2 l) (* (* (/ M 2) (/ D d)) h))) (/ (* (/ M 2) (/ D d)) (/ (* 2 l) (* (* (/ M 2) (/ D d)) h)))) (* 1 (/ (* (/ M 2) (/ D d)) (/ (* 2 l) (* (* (/ M 2) (/ D d)) h)))))) (* (sqrt l) (sqrt h)))
594.0ms
(* (+ (* 1 1) (+ (* (/ (* (/ M 2) (/ D d)) (/ (* 2 l) (* (* (/ M 2) (/ D d)) h))) (/ (* (/ M 2) (/ D d)) (/ (* 2 l) (* (* (/ M 2) (/ D d)) h)))) (* 1 (/ (* (/ M 2) (/ D d)) (/ (* 2 l) (* (* (/ M 2) (/ D d)) h)))))) (sqrt h))
446.0ms
(* (- (* 1 1) (* (/ (* (/ M 2) (/ D d)) (/ (* 2 l) (* (* (/ M 2) (/ D d)) h))) (/ (* (/ M 2) (/ D d)) (/ (* 2 l) (* (* (/ M 2) (/ D d)) h))))) (* (sqrt d) (sqrt d)))
403.0ms
(/ (* (* (* 2 l) (* 2 l)) (* 2 l)) (* (* (* (* (/ M 2) (/ D d)) h) (* (* (/ M 2) (/ D d)) h)) (* (* (/ M 2) (/ D d)) h)))

prune1.8s

Pruning

20 alts after pruning (20 fresh and 0 done)

Merged error: 8.8b

regimes1.7s

Accuracy

56.5% (6.5b remaining)

Error of 14.7b against oracle of 8.2b and baseline of 23.1b

bsearch8.1s

end0.0ms

sample26.3s

Algorithm
intervals
Results
51895×(pre true 80)
39082×(body nan 80)
12813×(body real 80)