Average Error: 25.4 → 16.4
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -2.1332837374323508 \cdot 10^{-27}:\\ \;\;\;\;\sqrt[3]{1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}} \cdot \left(\left(\sqrt[3]{1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}} \cdot \sqrt[3]{1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}}\right) \cdot \left(\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right)\right)\\ \mathbf{elif}\;\ell \le -9.2206008893997 \cdot 10^{-311}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{\frac{-1}{\ell}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right)\right)\\ \mathbf{elif}\;\ell \le 6.506805187540822 \cdot 10^{-39}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left(\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left({\left(\frac{1}{h}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \end{array}\]

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 l < -2.1332837374323508e-27

    1. Initial program 24.8

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

    3. Applied associate-*r/26.2

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

    5. Applied add-cube-cbrt26.2

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

    6. Applied associate-*r*26.2

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

    7. Taylor expanded around -inf 22.4

      \[\leadsto \left(\left(\color{blue}{e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{h}\right) - \log \left(\frac{-1}{d}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\sqrt[3]{1 - \frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}} \cdot \sqrt[3]{1 - \frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right)\right) \cdot \sqrt[3]{1 - \frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\]

    8. Simplified19.1

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

    if -2.1332837374323508e-27 < l < -9.2206008893997e-311

    1. Initial program 27.3

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

    3. Applied associate-*r/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 16.6

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

    5. Simplified12.6

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

    if -9.2206008893997e-311 < l < 6.506805187540822e-39

    1. Initial program 28.1

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

    3. Applied associate-*r/22.6

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

    5. Applied div-inv22.6

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

    6. Applied unpow-prod-down12.2

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

    7. Simplified12.2

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

    if 6.506805187540822e-39 < l

    1. Initial program 24.0

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

    3. Applied div-inv24.0

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

    4. Applied unpow-prod-down17.3

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

    5. Simplified17.3

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

  4. Final simplification16.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -2.1332837374323508 \cdot 10^{-27}:\\ \;\;\;\;\sqrt[3]{1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}} \cdot \left(\left(\sqrt[3]{1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}} \cdot \sqrt[3]{1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}}\right) \cdot \left(\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right)\right)\\ \mathbf{elif}\;\ell \le -9.2206008893997 \cdot 10^{-311}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{\frac{-1}{\ell}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right)\right)\\ \mathbf{elif}\;\ell \le 6.506805187540822 \cdot 10^{-39}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left(\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left({\left(\frac{1}{h}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +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: 1.2m)Debug log

sample1.4s

Algorithm
intervals

simplify222.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

222.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

prune29.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 25.9b

localize98.0ms

Local error

Found 4 expressions with local error:

12.9b
(pow (/ d l) (/ 1 2))
12.8b
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
12.4b
(pow (/ d h) (/ 1 2))
7.5b
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

rewrite37.0ms

Algorithm
rewrite-expression-head
Counts
4 → 99
Calls

4 calls. Slowest were:

17.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
15.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))
3.0ms
(pow (/ d l) (/ 1 2))

series409.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

225.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))
66.0ms
(pow (/ d l) (/ 1 2))
65.0ms
(pow (/ d h) (/ 1 2))
54.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))

simplify7.3s

Counts
71 → 111
Calls

71 calls. Slowest were:

635.0ms
(* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h)
481.0ms
(+ (+ (log (/ 1 2)) (log (pow (/ (* M D) (* 2 d)) 2))) (log (/ h l)))
339.0ms
(* (* (* (* (/ 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)))

prune2.0s

Pruning

19 alts after pruning (19 fresh and 0 done)

Merged error: 12.1b

localize12.0ms

Local error

Found 4 expressions with local error:

12.9b
(pow (/ d l) (/ 1 2))
12.4b
(pow (/ d h) (/ 1 2))
7.5b
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)))
7.1b
(/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)

rewrite46.0ms

Algorithm
rewrite-expression-head
Counts
4 → 87
Calls

4 calls. Slowest were:

25.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)))
17.0ms
(/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)
2.0ms
(pow (/ d l) (/ 1 2))

series413.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

213.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)))
74.0ms
(pow (/ d l) (/ 1 2))
63.0ms
(/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)
62.0ms
(pow (/ d h) (/ 1 2))

simplify7.2s

Counts
57 → 99
Calls

57 calls. Slowest were:

886.0ms
(* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2))
721.0ms
(* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h)
472.0ms
(* 1/8 (/ (* (pow M 2) (* (pow D 2) h)) (* l (pow d 2))))

prune2.1s

Pruning

18 alts after pruning (18 fresh and 0 done)

Merged error: 9.6b

localize35.0ms

Local error

Found 4 expressions with local error:

12.9b
(pow (/ d l) (/ 1 2))
12.4b
(pow (/ d h) (/ 1 2))
7.1b
(/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)
7.1b
(/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)

rewrite45.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

24.0ms
(/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)
15.0ms
(/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)
3.0ms
(pow (/ d l) (/ 1 2))

series274.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

80.0ms
(/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)
70.0ms
(pow (/ d l) (/ 1 2))
66.0ms
(pow (/ d h) (/ 1 2))
57.0ms
(/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)

simplify4.7s

Counts
36 → 80
Calls

36 calls. Slowest were:

849.0ms
(* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2))
728.0ms
(* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h)
726.0ms
(* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2))

prune1.9s

Pruning

18 alts after pruning (18 fresh and 0 done)

Merged error: 9.6b

localize13.0ms

Local error

Found 4 expressions with local error:

12.9b
(pow (/ d l) (/ 1 2))
12.4b
(pow (/ d h) (/ 1 2))
12.4b
(pow (/ d h) (/ 1 2))
7.5b
(* (* (* (sqrt (pow (/ d h) (/ 1 2))) (sqrt (pow (/ d h) (/ 1 2)))) (pow (/ d l) (/ 1 2))) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)))

rewrite95.0ms

Algorithm
rewrite-expression-head
Counts
4 → 91
Calls

4 calls. Slowest were:

89.0ms
(* (* (* (sqrt (pow (/ d h) (/ 1 2))) (sqrt (pow (/ d h) (/ 1 2)))) (pow (/ d l) (/ 1 2))) (- 1 (/ (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) l)))
2.0ms
(pow (/ d l) (/ 1 2))
1.0ms
(pow (/ d h) (/ 1 2))

series503.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

254.0ms
(* (* (* (sqrt (pow (/ d h) (/ 1 2))) (sqrt (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))
83.0ms
(pow (/ d l) (/ 1 2))
74.0ms
(pow (/ d h) (/ 1 2))

simplify5.3s

Counts
63 → 103
Calls

63 calls. Slowest were:

297.0ms
(* (* (* (* (* (sqrt (pow (/ d h) (/ 1 2))) (sqrt (pow (/ d h) (/ 1 2)))) (sqrt (pow (/ d h) (/ 1 2)))) (* (* (sqrt (pow (/ d h) (/ 1 2))) (sqrt (pow (/ d h) (/ 1 2)))) (sqrt (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))))
276.0ms
(* (* (* (* (* (sqrt (pow (/ d h) (/ 1 2))) (sqrt (pow (/ d h) (/ 1 2)))) (* (sqrt (pow (/ d h) (/ 1 2))) (sqrt (pow (/ d h) (/ 1 2))))) (* (sqrt (pow (/ d h) (/ 1 2))) (sqrt (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))))
257.0ms
(+ (* 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))))

prune2.2s

Pruning

19 alts after pruning (19 fresh and 0 done)

Merged error: 9.6b

regimes1.9s

Accuracy

53% (7.3b remaining)

Error of 16.4b against oracle of 9.1b and baseline of 24.7b

bsearch8.8s

end0.0ms

sample26.9s

Algorithm
intervals