Error

Derivation

1. Split input into 4 regimes

2. if l < 3.923387526444069e-286

   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. Simplified25.6

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

   4. Applied times-frac24.4

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

   6. Applied div-inv24.4

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

   7. Applied *-un-lft-identity24.4

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

   8. Applied times-frac22.3

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

   10. Applied div-inv22.3

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

   12. Applied clear-num22.3

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

   if 3.923387526444069e-286 < l < 6.9895339504965e-199

   1. Initial program 30.9

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

   2. Simplified30.8

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

   4. Applied times-frac30.8

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

   6. Applied div-inv30.8

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

   7. Applied *-un-lft-identity30.8

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

   8. Applied times-frac25.7

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

   10. Applied div-inv25.6

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

   12. Applied div-inv25.6

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

   13. Applied sqrt-prod9.3

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

   if 6.9895339504965e-199 < l < 2.1100856900836157e+114

   1. Initial program 22.2

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

   2. Simplified21.8

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

   4. Applied times-frac19.7

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

   6. Applied div-inv19.7

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

   7. Applied *-un-lft-identity19.7

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

   8. Applied times-frac17.7

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

   10. Applied div-inv17.7

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

   12. Applied sqrt-div11.6

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

   if 2.1100856900836157e+114 < l

   1. Initial program 29.5

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

   2. Simplified29.5

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

   4. Applied times-frac27.8

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

   6. Applied div-inv27.8

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

   7. Applied *-un-lft-identity27.8

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

   8. Applied times-frac28.4

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

   10. Applied div-inv28.4

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

   12. Applied sqrt-div19.9

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

4. Final simplification18.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le 3.923387526444069 \cdot 10^{-286}:\\ \;\;\;\;\left(1 - \left(\frac{1}{\ell} \cdot \frac{\frac{1}{\frac{2 \cdot d}{D \cdot M}}}{\frac{1}{h}}\right) \cdot \frac{\left(D \cdot M\right) \cdot \frac{1}{2 \cdot d}}{2}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\ell \le 6.9895339504965 \cdot 10^{-199}:\\ \;\;\;\;\left(\left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(D \cdot M\right) \cdot \frac{1}{2 \cdot d}}{2} \cdot \left(\frac{1}{\ell} \cdot \frac{\frac{D \cdot M}{2 \cdot d}}{\frac{1}{h}}\right)\right)\\ \mathbf{elif}\;\ell \le 2.1100856900836157 \cdot 10^{+114}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - \frac{\left(D \cdot M\right) \cdot \frac{1}{2 \cdot d}}{2} \cdot \left(\frac{1}{\ell} \cdot \frac{\frac{D \cdot M}{2 \cdot d}}{\frac{1}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(D \cdot M\right) \cdot \frac{1}{2 \cdot d}}{2} \cdot \left(\frac{1}{\ell} \cdot \frac{\frac{D \cdot M}{2 \cdot d}}{\frac{1}{h}}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 
(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)))))