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

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

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

    if -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

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

    if -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

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

    10. Applied sqrt-div17.6

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

    11. Applied sqrt-div12.4

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

    12. Applied frac-times12.4

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

    13. Applied associate-*r/12.6

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

    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

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

    10. Applied associate-*r/29.8

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

    11. Applied associate-/l/26.6

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

    13. Applied sqrt-div20.9

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

  4. Final simplification15.5

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

Reproduce

herbie shell --seed 2019002 +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

sample1.5s

Algorithm
intervals

simplify160.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

160.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

prune33.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 24.0b

localize82.0ms

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))))

rewrite41.0ms

Algorithm
rewrite-expression-head
Counts
4 → 99
Calls

4 calls. Slowest were:

18.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))
16.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
3.0ms
(pow (/ d l) (/ 1 2))

series516.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

266.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))
103.0ms
(pow (/ d h) (/ 1 2))
76.0ms
(pow (/ d l) (/ 1 2))
70.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))

simplify8.8s

Counts
71 → 111
Calls

71 calls. Slowest were:

723.0ms
(* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h)
443.0ms
(+ (+ (log (/ 1 2)) (log (pow (/ (* M D) (* 2 d)) 2))) (log (/ h l)))
419.0ms
(+ (* 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)))))

prune2.3s

Pruning

21 alts after pruning (21 fresh and 0 done)

Merged error: 9.8b

localize11.0ms

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)))

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
4 → 85
Calls

4 calls. Slowest were:

17.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)))
10.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h)
2.0ms
(pow (/ d l) (/ 1 2))

series451.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

198.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)))
99.0ms
(pow (/ d l) (/ 1 2))
96.0ms
(pow (/ d h) (/ 1 2))
58.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h)

simplify6.9s

Counts
57 → 97
Calls

57 calls. Slowest were:

765.0ms
(* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h)
343.0ms
(* 1/8 (/ (* (pow M 2) (* (pow D 2) h)) (pow d 2)))
307.0ms
(* 1/8 (/ (* (pow M 2) (* (pow D 2) h)) (pow d 2)))

prune2.2s

Pruning

16 alts after pruning (16 fresh and 0 done)

Merged error: 8.1b

localize47.0ms

Local error

Found 4 expressions with local error:

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

rewrite45.0ms

Algorithm
rewrite-expression-head
Counts
4 → 86
Calls

4 calls. Slowest were:

29.0ms
(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (* 2 (/ l h)) (* (/ M 2) (/ D d))))) (* (sqrt (/ d l)) (sqrt (/ d h))))
11.0ms
(/ (* 2 (/ l h)) (* (/ M 2) (/ D d)))
1.0ms
(sqrt (/ d h))

series332.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

211.0ms
(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (* 2 (/ l h)) (* (/ M 2) (/ D d))))) (* (sqrt (/ d l)) (sqrt (/ d h))))
58.0ms
(/ (* 2 (/ l h)) (* (/ M 2) (/ D d)))
37.0ms
(sqrt (/ d h))
26.0ms
(sqrt (/ d l))

simplify8.5s

Counts
59 → 98
Calls

59 calls. Slowest were:

738.0ms
(* (+ (* 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))
670.0ms
(* (+ (* 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))
543.0ms
(* (+ (* 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)))

prune1.9s

Pruning

18 alts after pruning (18 fresh and 0 done)

Merged error: 6.8b

localize41.0ms

Local error

Found 4 expressions with local error:

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

rewrite53.0ms

Algorithm
rewrite-expression-head
Counts
4 → 77
Calls

4 calls. Slowest were:

35.0ms
(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (* 2 l) (* (* (/ M 2) (/ D d)) h)))) (* (sqrt (/ d l)) (sqrt (/ d h))))
14.0ms
(* (* (/ M 2) (/ D d)) h)
2.0ms
(sqrt (/ d l))

series331.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

232.0ms
(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (* 2 l) (* (* (/ M 2) (/ D d)) h)))) (* (sqrt (/ d l)) (sqrt (/ d h))))
37.0ms
(sqrt (/ d h))
35.0ms
(* (* (/ M 2) (/ D d)) h)
27.0ms
(sqrt (/ d l))

simplify8.1s

Counts
54 → 89
Calls

54 calls. Slowest were:

877.0ms
(* (+ (* 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)))
803.0ms
(* (+ (* 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))
685.0ms
(* (+ (* 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))

prune1.5s

Pruning

20 alts after pruning (20 fresh and 0 done)

Merged error: 6.4b

regimes1.7s

Accuracy

52.3% (7.1b remaining)

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

bsearch7.0s

end0.0ms

sample25.2s

Algorithm
intervals