Average Error: 25.1 → 18.9
Time: 50.3s
Precision: 64
Internal Precision: 128
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -2.3072532229286938 \cdot 10^{+76}:\\ \;\;\;\;\left(1 - \left(\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right) \cdot \sqrt{\frac{h}{\ell}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right)\right)\\ \mathbf{elif}\;d \le -3.946078378106929 \cdot 10^{-292}:\\ \;\;\;\;\left(\left(\sqrt{\frac{-1}{\ell}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(\frac{1}{h}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array}\]

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 3 regimes

  2. if d < -2.3072532229286938e+76

    1. Initial program 23.6

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

    3. Applied add-sqr-sqrt23.7

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

    4. Applied associate-*r*23.7

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

    5. Taylor expanded around -inf 16.5

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

    6. Simplified12.2

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

    if -2.3072532229286938e+76 < d < -3.946078378106929e-292

    1. Initial program 24.7

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

    3. Applied add-sqr-sqrt24.7

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

    4. Applied associate-*r*24.7

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

    5. Taylor expanded around -inf 23.5

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

    6. Simplified20.4

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

    if -3.946078378106929e-292 < d

    1. Initial program 25.8

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

    3. Applied div-inv25.9

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

    4. Applied unpow-prod-down20.6

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

    5. Simplified20.6

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

  4. Final simplification18.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -2.3072532229286938 \cdot 10^{+76}:\\ \;\;\;\;\left(1 - \left(\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right) \cdot \sqrt{\frac{h}{\ell}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right)\right)\\ \mathbf{elif}\;d \le -3.946078378106929 \cdot 10^{-292}:\\ \;\;\;\;\left(\left(\sqrt{\frac{-1}{\ell}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(\frac{1}{h}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 +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: 47.0s)Debug log

sample1.3s

Algorithm
intervals

simplify177.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
177.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

prune42.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 25.4b

localize74.0ms

Local error

Found 4 expressions with local error:

14.3b
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
13.4b
(pow (/ d l) (/ 1 2))
12.5b
(pow (/ d h) (/ 1 2))
6.4b
(/ (* M D) (* 2 d))

rewrite45.0ms

Algorithm
rewrite-expression-head
Rules
22×add-exp-log
11×add-cbrt-cube
9×add-cube-cbrt
9×prod-exp
9×*-un-lft-identity
9×add-sqr-sqrt
8×unpow-prod-down
8×pow-unpow
8×pow1
6×div-inv
4×add-log-exp
4×log1p-expm1-u
4×pow-exp
4×associate-*r*
4×expm1-log1p-u
3×cbrt-unprod
3×associate-*l/
3×pow-to-exp
2×div-exp
2×pow-pow
1×associate-/l*
1×*-commutative
1×associate-*r/
1×pow-prod-down
1×associate-*l*
1×times-frac
1×frac-2neg
1×associate-/r*
1×clear-num
1×cbrt-undiv
1×frac-times
Counts
4 → 83
Calls
4 calls:
Slowest
24.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
14.0ms
(/ (* M D) (* 2 d))
3.0ms
(pow (/ d h) (/ 1 2))
3.0ms
(pow (/ d l) (/ 1 2))

series248.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
88.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
79.0ms
(pow (/ d h) (/ 1 2))
65.0ms
(pow (/ d l) (/ 1 2))
16.0ms
(/ (* M D) (* 2 d))

simplify3.9s

Counts
52 → 95
Calls
52 calls:
Slowest
633.0ms
(* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h)
436.0ms
(+ (+ (log (/ 1 2)) (log (pow (/ (* M D) (* 2 d)) 2))) (log (/ h l)))
306.0ms
(* (* (* (* (/ 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)))
220.0ms
(* (* (* (* (/ 1 2) (/ 1 2)) (/ 1 2)) (* (* (pow (/ (* M D) (* 2 d)) 2) (pow (/ (* M D) (* 2 d)) 2)) (pow (/ (* M D) (* 2 d)) 2))) (* (* (/ h l) (/ h l)) (/ h l)))
220.0ms
(exp (* 1/2 (- (log (/ 1 l)) (log (/ 1 d)))))

prune1.8s

Pruning

15 alts after pruning (15 fresh and 0 done)

Merged error: 13.2b

localize14.0ms

Local error

Found 4 expressions with local error:

13.7b
(sqrt (/ h l))
13.7b
(sqrt (/ h l))
13.4b
(pow (/ d l) (/ 1 2))
12.5b
(pow (/ d h) (/ 1 2))

rewrite5.0ms

Algorithm
rewrite-expression-head
Rules
12×add-sqr-sqrt
10×add-cube-cbrt
10×*-un-lft-identity
8×sqrt-prod
8×unpow-prod-down
8×pow-unpow
6×div-inv
6×add-exp-log
6×pow1
4×add-log-exp
4×log1p-expm1-u
4×add-cbrt-cube
4×expm1-log1p-u
2×pow-exp
2×sqrt-div
2×pow1/2
2×pow-pow
2×rem-sqrt-square
2×pow-to-exp
Counts
4 → 72
Calls
4 calls:
Slowest
1.0ms
(pow (/ d l) (/ 1 2))
1.0ms
(pow (/ d h) (/ 1 2))
1.0ms
(sqrt (/ h l))
1.0ms
(sqrt (/ h l))

series189.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
62.0ms
(pow (/ d l) (/ 1 2))
61.0ms
(pow (/ d h) (/ 1 2))
36.0ms
(sqrt (/ h l))
29.0ms
(sqrt (/ h l))

simplify846.0ms

Counts
44 → 84
Calls
44 calls:
Slowest
160.0ms
(exp (* 1/2 (- (log (/ 1 h)) (log (/ 1 d)))))
150.0ms
(exp (* 1/2 (- (log (/ 1 l)) (log (/ 1 d)))))
138.0ms
(exp (* 1/2 (- (log (/ -1 l)) (log (/ -1 d)))))
122.0ms
(exp (* 1/2 (- (log (/ -1 h)) (log (/ -1 d)))))
51.0ms
(* (log (/ d l)) (/ 1 2))

prune1.7s

Pruning

15 alts after pruning (15 fresh and 0 done)

Merged error: 13.2b

localize19.0ms

Local error

Found 4 expressions with local error:

13.7b
(sqrt (/ h l))
13.7b
(sqrt (/ h l))
13.4b
(pow (/ d l) (/ 1 2))
12.5b
(pow (/ d h) (/ 1 2))

rewrite5.0ms

Algorithm
rewrite-expression-head
Rules
12×add-sqr-sqrt
10×add-cube-cbrt
10×*-un-lft-identity
8×sqrt-prod
8×unpow-prod-down
8×pow-unpow
6×div-inv
6×add-exp-log
6×pow1
4×add-log-exp
4×log1p-expm1-u
4×add-cbrt-cube
4×expm1-log1p-u
2×pow-exp
2×sqrt-div
2×pow1/2
2×pow-pow
2×rem-sqrt-square
2×pow-to-exp
Counts
4 → 72
Calls
4 calls:
Slowest
1.0ms
(pow (/ d l) (/ 1 2))
1.0ms
(pow (/ d h) (/ 1 2))
1.0ms
(sqrt (/ h l))
1.0ms
(sqrt (/ h l))

series235.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
84.0ms
(pow (/ d l) (/ 1 2))
83.0ms
(pow (/ d h) (/ 1 2))
36.0ms
(sqrt (/ h l))
31.0ms
(sqrt (/ h l))

simplify951.0ms

Counts
44 → 84
Calls
44 calls:
Slowest
204.0ms
(exp (* 1/2 (- (log (/ 1 h)) (log (/ 1 d)))))
192.0ms
(exp (* 1/2 (- (log (/ 1 l)) (log (/ 1 d)))))
137.0ms
(exp (* 1/2 (- (log (/ -1 l)) (log (/ -1 d)))))
125.0ms
(exp (* 1/2 (- (log (/ -1 h)) (log (/ -1 d)))))
66.0ms
(* (log (/ d l)) (/ 1 2))

prune2.0s

Pruning

16 alts after pruning (16 fresh and 0 done)

Merged error: 13.1b

localize23.0ms

Local error

Found 4 expressions with local error:

13.7b
(sqrt (/ h l))
13.7b
(sqrt (/ h l))
13.4b
(pow (/ d l) (/ 1 2))
13.4b
(pow (/ d l) (/ 1 2))

rewrite5.0ms

Algorithm
rewrite-expression-head
Rules
12×add-sqr-sqrt
10×add-cube-cbrt
10×*-un-lft-identity
8×sqrt-prod
8×unpow-prod-down
8×pow-unpow
6×div-inv
6×add-exp-log
6×pow1
4×add-log-exp
4×log1p-expm1-u
4×add-cbrt-cube
4×expm1-log1p-u
2×pow-exp
2×sqrt-div
2×pow1/2
2×pow-pow
2×rem-sqrt-square
2×pow-to-exp
Counts
4 → 72
Calls
4 calls:
Slowest
1.0ms
(pow (/ d l) (/ 1 2))
1.0ms
(pow (/ d l) (/ 1 2))
1.0ms
(sqrt (/ h l))
1.0ms
(sqrt (/ h l))

series246.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
97.0ms
(pow (/ d l) (/ 1 2))
91.0ms
(pow (/ d l) (/ 1 2))
31.0ms
(sqrt (/ h l))
28.0ms
(sqrt (/ h l))

simplify973.0ms

Counts
44 → 84
Calls
44 calls:
Slowest
206.0ms
(exp (* 1/2 (- (log (/ 1 l)) (log (/ 1 d)))))
172.0ms
(exp (* 1/2 (- (log (/ 1 l)) (log (/ 1 d)))))
150.0ms
(exp (* 1/2 (- (log (/ -1 l)) (log (/ -1 d)))))
133.0ms
(exp (* 1/2 (- (log (/ -1 l)) (log (/ -1 d)))))
72.0ms
(* (log (/ d l)) (/ 1 2))

prune2.1s

Pruning

18 alts after pruning (17 fresh and 1 done)

Merged error: 13.1b

regimes1.4s

Accuracy

48% (6.8b remaining)

Error of 18.9b against oracle of 12.1b and baseline of 25.2b

bsearch3.0s

end0.0ms

sample25.8s

Algorithm
intervals