Average Error: 25.8 → 15.8
Time: 1.0m
Precision: 64
Internal Precision: 128
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -4.59385696066452 \cdot 10^{-34}:\\ \;\;\;\;\left(1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left(\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right)\\ \mathbf{elif}\;\ell \le 9.1617595904819 \cdot 10^{-311}:\\ \;\;\;\;\left(\left(\sqrt{\frac{-1}{\ell}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{1}{\ell} \cdot \left(h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \le 1.146046982289444 \cdot 10^{-55}:\\ \;\;\;\;\left(1 - \frac{1}{\ell} \cdot \left(h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{-1}{2} \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right)\right) \cdot \left(\frac{h}{\ell}\right) + 1)_* \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array}\]

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

  1. Split input into 4 regimes

  2. if l < -4.59385696066452e-34

    1. Initial program 25.0

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

    2. Taylor expanded around -inf 20.7

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

    3. Simplified17.2

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

    if -4.59385696066452e-34 < l < 9.1617595904819e-311

    1. Initial program 26.9

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

    3. Applied div-inv26.9

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

    4. Applied associate-*r*22.2

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

    5. Taylor expanded around -inf 15.9

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

    6. Simplified11.6

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

    if 9.1617595904819e-311 < l < 1.146046982289444e-55

    1. Initial program 27.8

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

    3. Applied div-inv27.9

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

    4. Applied associate-*r*21.6

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

    6. Applied div-inv21.7

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

    7. Applied unpow-prod-down11.1

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

    8. Simplified11.1

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

    if 1.146046982289444e-55 < 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. Simplified25.5

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

    4. Applied sqrt-div17.9

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

  4. Final simplification15.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -4.59385696066452 \cdot 10^{-34}:\\ \;\;\;\;\left(1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left(\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right)\\ \mathbf{elif}\;\ell \le 9.1617595904819 \cdot 10^{-311}:\\ \;\;\;\;\left(\left(\sqrt{\frac{-1}{\ell}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{1}{\ell} \cdot \left(h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \le 1.146046982289444 \cdot 10^{-55}:\\ \;\;\;\;\left(1 - \frac{1}{\ell} \cdot \left(h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{-1}{2} \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right)\right) \cdot \left(\frac{h}{\ell}\right) + 1)_* \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +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: 57.0s)Debug log

sample826.0ms

Algorithm
intervals

simplify126.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

126.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

prune28.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 25.6b

localize61.0ms

Local error

Found 4 expressions with local error:

14.8b
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
12.2b
(pow (/ d h) (/ 1 2))
11.2b
(pow (/ d l) (/ 1 2))
4.8b
(/ (* M D) (* 2 d))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 83
Calls

4 calls. Slowest were:

11.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
5.0ms
(/ (* M D) (* 2 d))
3.0ms
(pow (/ d h) (/ 1 2))

series282.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

92.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
92.0ms
(pow (/ d l) (/ 1 2))
82.0ms
(pow (/ d h) (/ 1 2))
16.0ms
(/ (* M D) (* 2 d))

simplify3.6s

Counts
52 → 95
Calls

52 calls. Slowest were:

745.0ms
(* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h)
384.0ms
(+ (+ (log (/ 1 2)) (log (pow (/ (* M D) (* 2 d)) 2))) (log (/ h l)))
264.0ms
(* (* (* (* (/ 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)))

prune1.6s

Pruning

17 alts after pruning (17 fresh and 0 done)

Merged error: 12.6b

localize17.0ms

Local error

Found 4 expressions with local error:

12.2b
(pow (/ d h) (/ 1 2))
11.2b
(pow (/ d l) (/ 1 2))
7.7b
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h)
7.7b
(* (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) (/ 1 l))

rewrite72.0ms

Algorithm
rewrite-expression-head
Counts
4 → 74
Calls

4 calls. Slowest were:

46.0ms
(* (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) (/ 1 l))
19.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h)
3.0ms
(pow (/ d l) (/ 1 2))

series308.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

102.0ms
(pow (/ d h) (/ 1 2))
78.0ms
(pow (/ d l) (/ 1 2))
64.0ms
(* (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h) (/ 1 l))
63.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) h)

simplify5.0s

Counts
45 → 86
Calls

45 calls. Slowest were:

981.0ms
(* (* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h) 1)
748.0ms
(* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2))
616.0ms
(* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h)

prune1.7s

Pruning

18 alts after pruning (18 fresh and 0 done)

Merged error: 10.5b

localize22.0ms

Local error

Found 4 expressions with local error:

17.4b
(sqrt (/ h l))
17.4b
(sqrt (/ h l))
12.2b
(pow (/ d h) (/ 1 2))
11.2b
(pow (/ d l) (/ 1 2))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

2.0ms
(pow (/ d h) (/ 1 2))
1.0ms
(sqrt (/ h l))
1.0ms
(pow (/ d l) (/ 1 2))

series243.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

99.0ms
(pow (/ d h) (/ 1 2))
92.0ms
(pow (/ d l) (/ 1 2))
28.0ms
(sqrt (/ h l))
24.0ms
(sqrt (/ h l))

simplify1.1s

Counts
44 → 84
Calls

44 calls. Slowest were:

246.0ms
(exp (* 1/2 (- (log (/ 1 h)) (log (/ 1 d)))))
190.0ms
(exp (* 1/2 (- (log (/ 1 l)) (log (/ 1 d)))))
168.0ms
(exp (* 1/2 (- (log (/ -1 h)) (log (/ -1 d)))))

prune1.7s

Pruning

18 alts after pruning (17 fresh and 1 done)

Merged error: 10.5b

localize47.0ms

Local error

Found 4 expressions with local error:

12.2b
(sqrt (/ d h))
11.2b
(sqrt (/ d l))
11.2b
(fma (* (* (* (/ M 2) (/ D d)) (* (/ M 2) (/ D d))) -1/2) (/ h l) 1)
5.0b
(* (/ M 2) (/ D d))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

9.0ms
(* (/ M 2) (/ D d))
1.0ms
(sqrt (/ d h))
1.0ms
(sqrt (/ d l))

series141.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

77.0ms
(fma (* (* (* (/ M 2) (/ D d)) (* (/ M 2) (/ D d))) -1/2) (/ h l) 1)
23.0ms
(sqrt (/ d h))
22.0ms
(sqrt (/ d l))
19.0ms
(* (/ M 2) (/ D d))

simplify636.0ms

Counts
36 → 78
Calls

36 calls. Slowest were:

272.0ms
(* (* (* (/ M 2) (/ M 2)) (/ M 2)) (* (* (/ D d) (/ D d)) (/ D d)))
129.0ms
(- 1 (* 1/8 (/ (* (pow M 2) (* (pow D 2) h)) (* l (pow d 2)))))
62.0ms
(+ (log (/ M 2)) (log (/ D d)))

prune1.2s

Pruning

19 alts after pruning (18 fresh and 1 done)

Merged error: 10.4b

regimes1.4s

Accuracy

64.7% (5.5b remaining)

Error of 15.8b against oracle of 10.3b and baseline of 25.8b

bsearch11.6s

end0.0ms

sample25.5s

Algorithm
intervals