Derivation

Split input into 3 regimes
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
  \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right)\]
if -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
  \[\leadsto \left(\color{blue}{\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right)\]
if 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
  \[\leadsto {\color{blue}{\left(\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\frac{\ell \cdot 2}{h}}{\frac{M}{2} \cdot \frac{D}{d}}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\right)}}^{1}\]
9. Using strategy rm
10. Applied associate-/l/22.9
  \[\leadsto {\left(\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\color{blue}{\frac{\ell \cdot 2}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot h}}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\right)}^{1}\]
11. Using strategy rm
12. Applied sqrt-div16.4
  \[\leadsto {\left(\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\ell \cdot 2}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot h}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right)\right)}^{1}\]
13. Applied sqrt-div10.3
  \[\leadsto {\left(\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\ell \cdot 2}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot h}}\right) \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\right)}^{1}\]
14. Applied frac-times10.3
  \[\leadsto {\left(\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\ell \cdot 2}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot h}}\right) \cdot \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}}\right)}^{1}\]
15. Applied associate-*r/10.1
  \[\leadsto {\color{blue}{\left(\frac{\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\ell \cdot 2}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot h}}\right) \cdot \left(\sqrt{d} \cdot \sqrt{d}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\right)}}^{1}\]
Recombined 3 regimes into one program.
Final simplification14.7
\[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -3.519360298233161 \cdot 10^{+217}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{\ell}}\right)\right)\\ \mathbf{elif}\;\ell \le 2.898584392085 \cdot 10^{-311}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right) \cdot \left(\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{d} \cdot \sqrt{d}\right) \cdot \left(1 - \frac{\frac{D}{d} \cdot \frac{M}{2}}{\frac{2 \cdot \ell}{h \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array}\]

Details

sample1.4s

Algorithm

intervals

Results

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

simplify151.0ms

Counts

1 → 1

Calls

1 calls:

Slowest

151.0ms

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

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 26.8b

localize50.0ms

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))))

rewrite65.0ms

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`
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 → 91

Calls

4 calls:

Slowest

29.0ms	(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))
29.0ms	(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
3.0ms	(pow (/ d l) (/ 1 2))
2.0ms	(pow (/ d h) (/ 1 2))

series493.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

241.0ms	(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))
87.0ms	(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
83.0ms	(pow (/ d l) (/ 1 2))
83.0ms	(pow (/ d h) (/ 1 2))

simplify7.0s

Counts

71 → 103

Calls

71 calls:

Slowest

668.0ms	(* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h)
419.0ms	(* (* (* (* (/ 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)))
396.0ms	(+ (+ (log (/ 1 2)) (log (pow (/ (* M D) (* 2 d)) 2))) (log (/ h l)))
213.0ms	(* (* (* (* (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)))))
212.0ms	(+ (* 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)))))

prune2.0s

Pruning

18 alts after pruning (18 fresh and 0 done)

Merged error: 13.7b

localize18.0ms

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))

rewrite45.0ms

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`
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 → 82

Calls

4 calls:

Slowest

32.0ms	(* (* (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))

series435.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

248.0ms	(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)))
86.0ms	(pow (/ d h) (/ 1 2))
78.0ms	(pow (/ d l) (/ 1 2))
23.0ms	(/ (* M D) (* 2 d))

simplify4.5s

Counts

59 → 94

Calls

59 calls:

Slowest

205.0ms	(* (* (* (* (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))))
194.0ms	(+ (* 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))))
190.0ms	(+ (+ (* (log (/ d h)) (/ 1 2)) (* (log (/ d l)) (/ 1 2))) (log (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l))))
186.0ms	(+ (+ (log (pow (/ d h) (/ 1 2))) (* (log (/ d l)) (/ 1 2))) (log (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l))))
185.0ms	(exp (* 1/2 (- (log (/ 1 l)) (log (/ 1 d)))))

prune1.7s

Pruning

16 alts after pruning (16 fresh and 0 done)

Merged error: 12.0b

localize35.0ms

Local error

Found 4 expressions with local error:

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

rewrite45.0ms

Algorithm

rewrite-expression-head

Rules

14×	`add-exp-log`
14×	`sqrt-div`
14×	`add-cbrt-cube`
11×	`add-sqr-sqrt`
10×	`frac-times`
9×	`add-cube-cbrt`
9×	`*-un-lft-identity`
9×	`pow1`
8×	`sqrt-prod`
7×	`associate-*r/`
6×	`associate-*l/`
5×	`div-inv`
4×	`add-log-exp`
4×	`associate-/l*`
4×	`flip--`
4×	`cbrt-unprod`
4×	`prod-exp`
4×	`flip3--`
4×	`times-frac`
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×	`*-commutative`
1×	`associate-r`
1×	`frac-2neg`
1×	`associate-/r*`
1×	`clear-num`

Counts

4 → 84

Calls

4 calls:

Slowest

29.0ms	(* (- 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 h))
1.0ms	(sqrt (/ d l))

series339.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

232.0ms	(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (/ (* l 2) h) (* (/ M 2) (/ D d))))) (* (sqrt (/ d l)) (sqrt (/ d h))))
48.0ms	(/ (/ (* l 2) h) (* (/ M 2) (/ D d)))
34.0ms	(sqrt (/ d l))
24.0ms	(sqrt (/ d h))

simplify6.9s

Counts

68 → 96

Calls

68 calls:

Slowest

545.0ms	(* (- (* 1 1) (* (/ (* (/ M 2) (/ D d)) (/ (/ (* l 2) h) (* (/ M 2) (/ D d)))) (/ (* (/ M 2) (/ D d)) (/ (/ (* l 2) h) (* (/ M 2) (/ D d)))))) (* (sqrt (/ d l)) (sqrt d)))
431.0ms	(* (+ (* 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)))
404.0ms	(* (+ (* 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))
375.0ms	(* 4 (/ (* l d) (* h (* M D))))
356.0ms	(* (+ (* 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 h))

prune1.7s

Pruning

19 alts after pruning (19 fresh and 0 done)

Merged error: 9.3b

localize23.0ms

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)) (/ (* l 2) (* (* (/ M 2) (/ D d)) h)))) (* (sqrt (/ d l)) (sqrt (/ d h))))
6.5b	(/ (* l 2) (* (* (/ M 2) (/ D d)) h))

rewrite28.0ms

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×	`flip3--`
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 → 75

Calls

4 calls:

Slowest

19.0ms	(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (* l 2) (* (* (/ M 2) (/ D d)) h)))) (* (sqrt (/ d l)) (sqrt (/ d h))))
8.0ms	(/ (* l 2) (* (* (/ M 2) (/ D d)) h))
1.0ms	(sqrt (/ d h))
0.0ms	(sqrt (/ d l))

series319.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

185.0ms	(* (- 1 (/ (* (/ M 2) (/ D d)) (/ (* l 2) (* (* (/ M 2) (/ D d)) h)))) (* (sqrt (/ d l)) (sqrt (/ d h))))
68.0ms	(/ (* l 2) (* (* (/ M 2) (/ D d)) h))
36.0ms	(sqrt (/ d h))
30.0ms	(sqrt (/ d l))

simplify7.8s

Counts

55 → 87

Calls

55 calls:

Slowest

667.0ms	(* (+ (* 1 1) (+ (* (/ (* (/ M 2) (/ D d)) (/ (* l 2) (* (* (/ M 2) (/ D d)) h))) (/ (* (/ M 2) (/ D d)) (/ (* l 2) (* (* (/ M 2) (/ D d)) h)))) (* 1 (/ (* (/ M 2) (/ D d)) (/ (* l 2) (* (* (/ M 2) (/ D d)) h)))))) (sqrt h))
656.0ms	(* (+ (* 1 1) (+ (* (/ (* (/ M 2) (/ D d)) (/ (* l 2) (* (* (/ M 2) (/ D d)) h))) (/ (* (/ M 2) (/ D d)) (/ (* l 2) (* (* (/ M 2) (/ D d)) h)))) (* 1 (/ (* (/ M 2) (/ D d)) (/ (* l 2) (* (* (/ M 2) (/ D d)) h)))))) (* (sqrt l) (sqrt h)))
580.0ms	(* (+ (* 1 1) (+ (* (/ (* (/ M 2) (/ D d)) (/ (* l 2) (* (* (/ M 2) (/ D d)) h))) (/ (* (/ M 2) (/ D d)) (/ (* l 2) (* (* (/ M 2) (/ D d)) h)))) (* 1 (/ (* (/ M 2) (/ D d)) (/ (* l 2) (* (* (/ M 2) (/ D d)) h)))))) (sqrt l))
422.0ms	(* 4 (/ (* l d) (* h (* M D))))
419.0ms	(* 4 (/ (* l d) (* h (* M D))))

prune1.4s

Pruning

19 alts after pruning (19 fresh and 0 done)

Merged error: 8.8b

regimes1.6s

Accuracy

57.5% (6.6b remaining)

Error of 14.7b against oracle of 8.1b and baseline of 23.6b

bsearch8.3s

end0.0ms

sample26.7s

Algorithm

intervals

Results

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

Error

Try it out

Derivation

`if l < -3.519360298233161e+217`

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

`if 2.898584392085e-311 < l`

Reproduce

Details

sample1.4s

simplify151.0ms

prune11.0ms

localize50.0ms

rewrite65.0ms

series493.0ms

simplify7.0s

prune2.0s

localize18.0ms

rewrite45.0ms

series435.0ms

simplify4.5s

prune1.7s

localize35.0ms

rewrite45.0ms

series339.0ms

simplify6.9s

prune1.7s

localize23.0ms

rewrite28.0ms

series319.0ms

simplify7.8s

prune1.4s

regimes1.6s

bsearch8.3s

end0.0ms

sample26.7s

In
Out