Average Error: 26.1 → 15.5
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;h \le -2.2462022782129031 \cdot 10^{-116}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{\frac{-1}{\ell}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right)\right)\\ \mathbf{elif}\;h \le -3.3946239036053276 \cdot 10^{-300}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left(\left(\sqrt{\frac{-1}{h}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right)\\ \mathbf{elif}\;h \le 3.795599479384339 \cdot 10^{+168} \lor \neg \left(h \le 6.389442324558605 \cdot 10^{+290}\right):\\ \;\;\;\;\frac{\left(\sqrt{d} \cdot \sqrt{d}\right) \cdot \left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\frac{\ell \cdot 2}{h}}{\frac{M}{2} \cdot \frac{D}{d}}}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\ell \cdot 2}{\frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{1}{h}}}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array}\]

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 4 regimes

  2. if h < -2.2462022782129031e-116

    1. Initial program 22.9

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

    3. Applied associate-*r/22.0

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

    4. Taylor expanded around -inf 19.7

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

    5. Simplified16.1

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

    if -2.2462022782129031e-116 < h < -3.3946239036053276e-300

    1. Initial program 32.8

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

    3. Applied associate-*r/32.7

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

    4. Taylor expanded around -inf 22.8

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

    5. Simplified18.8

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

    if -3.3946239036053276e-300 < h < 3.795599479384339e+168 or 6.389442324558605e+290 < h

    1. Initial program 25.3

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

    3. Applied associate-*r/24.8

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

    5. Applied pow124.8

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

    6. Applied pow124.8

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

    7. Applied pow-prod-down24.8

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

    8. Simplified23.8

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

    10. Applied sqrt-div17.6

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

    11. Applied sqrt-div12.4

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

    12. Applied frac-times12.4

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

    13. Applied associate-*r/12.5

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

    if 3.795599479384339e+168 < h < 6.389442324558605e+290

    1. Initial program 30.6

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

    3. Applied associate-*r/27.8

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

    5. Applied pow127.8

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

    6. Applied pow127.8

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

    7. Applied pow-prod-down27.8

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

    8. Simplified29.8

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

    10. Applied div-inv29.8

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

    11. Applied associate-/l*26.6

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

    13. Applied sqrt-div20.9

      \[\leadsto {\left(\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\ell \cdot 2}{\frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{1}{h}}}}\right) \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right)\right)}^{1}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;h \le -2.2462022782129031 \cdot 10^{-116}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{\frac{-1}{\ell}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right)\right)\\ \mathbf{elif}\;h \le -3.3946239036053276 \cdot 10^{-300}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left(\left(\sqrt{\frac{-1}{h}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right)\\ \mathbf{elif}\;h \le 3.795599479384339 \cdot 10^{+168} \lor \neg \left(h \le 6.389442324558605 \cdot 10^{+290}\right):\\ \;\;\;\;\frac{\left(\sqrt{d} \cdot \sqrt{d}\right) \cdot \left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\frac{\ell \cdot 2}{h}}{\frac{M}{2} \cdot \frac{D}{d}}}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\ell \cdot 2}{\frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{1}{h}}}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(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

sample787.0ms

Algorithm
intervals

simplify174.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

174.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

prune21.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 24.2b

localize84.0ms

Local error

Found 4 expressions with local error:

12.0b
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
10.8b
(pow (/ d l) (/ 1 2))
10.2b
(pow (/ d h) (/ 1 2))
5.6b
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
4 → 91
Calls

4 calls. Slowest were:

24.0ms
(* (* (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))
1.0ms
(pow (/ d l) (/ 1 2))

series459.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

187.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))
94.0ms
(pow (/ d l) (/ 1 2))
92.0ms
(pow (/ d h) (/ 1 2))
85.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))

simplify8.1s

Counts
71 → 103
Calls

71 calls. Slowest were:

761.0ms
(* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h)
563.0ms
(+ (+ (log (/ 1 2)) (log (pow (/ (* M D) (* 2 d)) 2))) (log (/ h l)))
510.0ms
(+ (+ (* (log (/ d h)) (/ 1 2)) (log (pow (/ d l) (/ 1 2)))) (log (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))

prune2.1s

Pruning

20 alts after pruning (20 fresh and 0 done)

Merged error: 9.6b

localize15.0ms

Local error

Found 4 expressions with local error:

10.8b
(pow (/ d l) (/ 1 2))
10.2b
(pow (/ d h) (/ 1 2))
7.2b
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h)
5.6b
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)))

rewrite60.0ms

Algorithm
rewrite-expression-head
Counts
4 → 77
Calls

4 calls. Slowest were:

33.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)))
20.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h)
3.0ms
(pow (/ d h) (/ 1 2))

series491.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

249.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)))
91.0ms
(pow (/ d h) (/ 1 2))
79.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h)
71.0ms
(pow (/ d l) (/ 1 2))

simplify6.1s

Counts
57 → 89
Calls

57 calls. Slowest were:

646.0ms
(* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h)
353.0ms
(* 1/8 (/ (* (pow M 2) (* (pow D 2) h)) (pow d 2)))
317.0ms
(* 1/8 (/ (* (pow M 2) (* (pow D 2) h)) (pow d 2)))

prune1.7s

Pruning

16 alts after pruning (16 fresh and 0 done)

Merged error: 8.1b

localize54.0ms

Local error

Found 4 expressions with local error:

10.8b
(sqrt (/ d l))
10.5b
(/ (/ (* l 2) h) (* (/ M 2) (/ D d)))
10.2b
(sqrt (/ d h))
5.6b
(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (/ (* l 2) h) (* (/ M 2) (/ D d))))) (* (sqrt (/ d l)) (sqrt (/ d h))))

rewrite46.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

31.0ms
(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (/ (* l 2) h) (* (/ M 2) (/ D d))))) (* (sqrt (/ d l)) (sqrt (/ d h))))
12.0ms
(/ (/ (* l 2) h) (* (/ M 2) (/ D d)))
1.0ms
(sqrt (/ d l))

series379.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

237.0ms
(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (/ (* l 2) h) (* (/ M 2) (/ D d))))) (* (sqrt (/ d l)) (sqrt (/ d h))))
68.0ms
(/ (/ (* l 2) h) (* (/ M 2) (/ D d)))
38.0ms
(sqrt (/ d h))
37.0ms
(sqrt (/ d l))

simplify7.4s

Counts
68 → 96
Calls

68 calls. Slowest were:

862.0ms
(* 4 (/ (* l d) (* h (* M D))))
471.0ms
(* (+ (* 1 1) (+ (* (/ (* (/ M 2) (/ D d)) (/ (/ (* l 2) h) (* (/ M 2) (/ D d)))) (/ (* (/ M 2) (/ D d)) (/ (/ (* l 2) h) (* (/ M 2) (/ D d))))) (* 1 (/ (* (/ M 2) (/ D d)) (/ (/ (* l 2) h) (* (/ M 2) (/ D d))))))) (* (sqrt l) (sqrt h)))
451.0ms
(* (+ (* 1 1) (+ (* (/ (* (/ M 2) (/ D d)) (/ (/ (* l 2) h) (* (/ M 2) (/ D d)))) (/ (* (/ M 2) (/ D d)) (/ (/ (* l 2) h) (* (/ M 2) (/ D d))))) (* 1 (/ (* (/ M 2) (/ D d)) (/ (/ (* l 2) h) (* (/ M 2) (/ D d))))))) (sqrt l))

prune2.0s

Pruning

19 alts after pruning (19 fresh and 0 done)

Merged error: 6.8b

localize33.0ms

Local error

Found 4 expressions with local error:

10.8b
(sqrt (/ d l))
10.2b
(sqrt (/ d h))
7.0b
(/ (* (/ M 2) (/ D d)) (/ 1 h))
5.6b
(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (* l 2) (/ (* (/ M 2) (/ D d)) (/ 1 h))))) (* (sqrt (/ d l)) (sqrt (/ d h))))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

14.0ms
(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (* l 2) (/ (* (/ M 2) (/ D d)) (/ 1 h))))) (* (sqrt (/ d l)) (sqrt (/ d h))))
6.0ms
(/ (* (/ M 2) (/ D d)) (/ 1 h))
1.0ms
(sqrt (/ d l))

series311.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

183.0ms
(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (* l 2) (/ (* (/ M 2) (/ D d)) (/ 1 h))))) (* (sqrt (/ d l)) (sqrt (/ d h))))
50.0ms
(/ (* (/ M 2) (/ D d)) (/ 1 h))
39.0ms
(sqrt (/ d l))
39.0ms
(sqrt (/ d h))

simplify9.5s

Counts
70 → 96
Calls

70 calls. Slowest were:

666.0ms
(* (+ (* 1 1) (+ (* (/ (* (/ M 2) (/ D d)) (/ (* l 2) (/ (* (/ M 2) (/ D d)) (/ 1 h)))) (/ (* (/ M 2) (/ D d)) (/ (* l 2) (/ (* (/ M 2) (/ D d)) (/ 1 h))))) (* 1 (/ (* (/ M 2) (/ D d)) (/ (* l 2) (/ (* (/ M 2) (/ D d)) (/ 1 h))))))) (sqrt h))
576.0ms
(* (+ (* 1 1) (+ (* (/ (* (/ M 2) (/ D d)) (/ (* l 2) (/ (* (/ M 2) (/ D d)) (/ 1 h)))) (/ (* (/ M 2) (/ D d)) (/ (* l 2) (/ (* (/ M 2) (/ D d)) (/ 1 h))))) (* 1 (/ (* (/ M 2) (/ D d)) (/ (* l 2) (/ (* (/ M 2) (/ D d)) (/ 1 h))))))) (* (sqrt l) (sqrt h)))
547.0ms
(* (+ (* 1 1) (+ (* (/ (* (/ M 2) (/ D d)) (/ (* l 2) (/ (* (/ M 2) (/ D d)) (/ 1 h)))) (/ (* (/ M 2) (/ D d)) (/ (* l 2) (/ (* (/ M 2) (/ D d)) (/ 1 h))))) (* 1 (/ (* (/ M 2) (/ D d)) (/ (* l 2) (/ (* (/ M 2) (/ D d)) (/ 1 h))))))) (sqrt l))

prune1.7s

Pruning

20 alts after pruning (20 fresh and 0 done)

Merged error: 6.5b

regimes1.7s

Accuracy

52% (7.2b remaining)

Error of 15.5b against oracle of 8.4b and baseline of 23.3b

bsearch7.4s

end0.0ms

sample24.4s

Algorithm
intervals