Error

Derivation

1. Split input into 4 regimes

2. if d < -4.273486988078354e+85

   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. Taylor expanded around -inf 17.2

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

   3. Simplified12.8

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

   if -4.273486988078354e+85 < d < -1.410505834504642e-85

   1. Initial program 16.0

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

   3. Applied associate-*r/14.3

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

   5. Applied pow114.3

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

   6. Applied pow114.3

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

   7. Applied pow-prod-down14.3

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

   8. Simplified14.1

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

   10. Applied associate-*r/14.1

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

   11. Applied associate-/l/11.3

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

   13. Applied pow111.3

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

   14. Applied pow111.3

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

   15. Applied pow-prod-down11.3

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

   16. Applied pow111.3

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

   17. Applied pow-prod-down11.3

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

   18. Simplified12.5

       \[\leadsto {\left({\color{blue}{\left((\left(\frac{M}{\frac{\ell \cdot 4}{\frac{D}{d}}}\right) \cdot \left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \left(-h\right)\right) + 1)_* \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\right)}}^{1}\right)}^{1}\]

   if -1.410505834504642e-85 < d < 4.05717666479264e-310

   1. Initial program 37.2

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

   3. Applied associate-*r/37.0

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

   4. Taylor expanded around -inf 32.8

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

   5. Simplified30.5

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

   if 4.05717666479264e-310 < d

   1. Initial program 25.3

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

   3. Applied associate-*r/24.6

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

   5. Applied pow124.6

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

   6. Applied pow124.6

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

   7. Applied pow-prod-down24.6

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

   8. Simplified24.4

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

   10. Applied associate-*r/24.4

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

   11. Applied associate-/l/22.8

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

   13. Applied sqrt-div16.4

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

   14. Applied sqrt-div10.2

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

   15. Applied frac-times10.2

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

   16. Applied associate-*r/10.1

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

4. Final simplification14.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -4.273486988078354 \cdot 10^{+85}:\\ \;\;\;\;\left(1 - \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right)\right)\\ \mathbf{elif}\;d \le -1.410505834504642 \cdot 10^{-85}:\\ \;\;\;\;(\left(\frac{M}{\frac{\ell \cdot 4}{\frac{D}{d}}}\right) \cdot \left(\left(-h\right) \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) + 1)_* \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \le 4.05717666479264 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{\frac{-1}{\ell}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{2 \cdot \ell}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot h}}\right) \cdot \left(\sqrt{d} \cdot \sqrt{d}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 +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

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