Average Error: 25.5 → 24.9
Time: 2.7m
Precision: 64
Internal Precision: 576
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le 1.113648115348954 \cdot 10^{-309}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;d \le 2.1357792220822114 \cdot 10^{-153}:\\ \;\;\;\;\frac{\left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \frac{h}{\frac{\ell}{d}}\right) \cdot \left(\frac{M}{2} \cdot \frac{-1}{2}\right)}{\frac{d}{D} \cdot \left(\sqrt{\ell} \cdot \sqrt{h}\right)} + \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \le 2.0911954549093263 \cdot 10^{-48}:\\ \;\;\;\;\left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} + \left(\frac{h}{\ell} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right)\right)\\ \mathbf{elif}\;d \le 1.5462672678959554 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} + \frac{\left(h \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(\frac{M}{2} \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right)\right)}{\frac{d}{D} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array}\]

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 5 regimes

  2. if d < 1.113648115348954e-309

    1. Initial program 25.2

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

    if 1.113648115348954e-309 < d < 2.1357792220822114e-153

    1. Initial program 39.7

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

    2. Initial simplification39.6

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

    4. Applied sqrt-div37.7

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

    5. Applied sqrt-div37.7

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

    6. Applied frac-times37.7

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

    7. Applied associate-*r/36.6

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

    8. Applied associate-*l/37.3

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

    9. Applied associate-*r/37.3

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

    10. Applied frac-times38.2

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

    11. Simplified36.0

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

    if 2.1357792220822114e-153 < d < 2.0911954549093263e-48

    1. Initial program 27.1

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

    2. Initial simplification27.5

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

    4. Applied div-inv27.5

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

    5. Applied sqrt-prod22.7

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

    if 2.0911954549093263e-48 < d < 1.5462672678959554e+92

    1. Initial program 14.1

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

    2. Initial simplification16.3

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

    4. Applied associate-*l/15.8

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

    5. Applied associate-*l/17.3

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

    6. Applied associate-*r/17.3

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

    7. Applied frac-times12.7

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

    if 1.5462672678959554e+92 < d

    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. Initial simplification28.2

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

    3. Taylor expanded around 0 27.4

      \[\leadsto \color{blue}{0} + \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification24.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 1.113648115348954 \cdot 10^{-309}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;d \le 2.1357792220822114 \cdot 10^{-153}:\\ \;\;\;\;\frac{\left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \frac{h}{\frac{\ell}{d}}\right) \cdot \left(\frac{M}{2} \cdot \frac{-1}{2}\right)}{\frac{d}{D} \cdot \left(\sqrt{\ell} \cdot \sqrt{h}\right)} + \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \le 2.0911954549093263 \cdot 10^{-48}:\\ \;\;\;\;\left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} + \left(\frac{h}{\ell} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right)\right)\\ \mathbf{elif}\;d \le 1.5462672678959554 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} + \frac{\left(h \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(\frac{M}{2} \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right)\right)}{\frac{d}{D} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018254 
(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)))))