Average Error: 25.2 → 16.4
Time: 1.9m
Precision: 64
Internal Precision: 128
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) = -\infty:\\ \;\;\;\;\left(1 - \frac{\frac{\left(M \cdot \frac{D}{2}\right) \cdot h}{\frac{d}{M \cdot \frac{D}{2}}}}{d \cdot \left(\ell \cdot 2\right)}\right) \cdot \left(\left(\left|\frac{\sqrt[3]{\frac{-1}{\ell}}}{\sqrt[3]{\frac{-1}{d}}}\right| \cdot \sqrt{\frac{\sqrt[3]{\frac{-1}{\ell}}}{\sqrt[3]{\frac{-1}{d}}}}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le 2.2174029682308495 \cdot 10^{+182}:\\ \;\;\;\;\left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{2}{\frac{h}{\ell}}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{\frac{-1}{\ell}}}{\sqrt[3]{\frac{-1}{d}}}} \cdot \left(\left|\frac{\sqrt[3]{\frac{-1}{\ell}}}{\sqrt[3]{\frac{-1}{d}}}\right| \cdot \sqrt{\frac{d}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\frac{\left(M \cdot \frac{D}{2}\right) \cdot h}{\frac{d}{M \cdot \frac{D}{2}}}}{d \cdot \left(\ell \cdot 2\right)}\right) \cdot \left(\left(\left|\frac{\sqrt[3]{\frac{-1}{\ell}}}{\sqrt[3]{\frac{-1}{d}}}\right| \cdot \sqrt{\frac{\sqrt[3]{\frac{-1}{\ell}}}{\sqrt[3]{\frac{-1}{d}}}}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \end{array}\]

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 2 regimes

  2. if (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) < -inf.0 or 2.2174029682308495e+182 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

    1. Initial program 53.2

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

    3. Applied associate-*l/53.2

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

    4. Applied frac-times49.2

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

    5. Simplified48.7

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

    6. Taylor expanded around -inf 52.8

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

    7. Simplified48.8

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

    9. Applied add-cube-cbrt48.8

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

    10. Applied add-cube-cbrt48.9

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

    11. Applied times-frac48.9

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

    12. Applied sqrt-prod42.3

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

    13. Simplified41.5

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

    15. Applied associate-*l/42.0

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

    16. Applied associate-*l/42.2

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

    17. Applied associate-*l/39.6

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

    18. Applied associate-/l/40.1

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

    19. Simplified36.5

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

    if -inf.0 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) < 2.2174029682308495e+182

    1. Initial program 7.3

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

    3. Applied associate-*l/7.3

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

    4. Applied frac-times9.0

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

    5. Simplified9.8

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

    6. Taylor expanded around -inf 36.6

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

    7. Simplified9.8

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

    9. Applied add-cube-cbrt10.2

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

    10. Applied add-cube-cbrt10.4

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

    11. Applied times-frac10.3

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

    12. Applied sqrt-prod6.7

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

    13. Simplified6.5

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

    15. Applied pow16.5

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

    16. Applied pow16.5

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

    17. Applied pow-prod-down6.5

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

    18. Simplified3.6

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

  4. Final simplification16.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) = -\infty:\\ \;\;\;\;\left(1 - \frac{\frac{\left(M \cdot \frac{D}{2}\right) \cdot h}{\frac{d}{M \cdot \frac{D}{2}}}}{d \cdot \left(\ell \cdot 2\right)}\right) \cdot \left(\left(\left|\frac{\sqrt[3]{\frac{-1}{\ell}}}{\sqrt[3]{\frac{-1}{d}}}\right| \cdot \sqrt{\frac{\sqrt[3]{\frac{-1}{\ell}}}{\sqrt[3]{\frac{-1}{d}}}}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le 2.2174029682308495 \cdot 10^{+182}:\\ \;\;\;\;\left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{2}{\frac{h}{\ell}}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{\frac{-1}{\ell}}}{\sqrt[3]{\frac{-1}{d}}}} \cdot \left(\left|\frac{\sqrt[3]{\frac{-1}{\ell}}}{\sqrt[3]{\frac{-1}{d}}}\right| \cdot \sqrt{\frac{d}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\frac{\left(M \cdot \frac{D}{2}\right) \cdot h}{\frac{d}{M \cdot \frac{D}{2}}}}{d \cdot \left(\ell \cdot 2\right)}\right) \cdot \left(\left(\left|\frac{\sqrt[3]{\frac{-1}{\ell}}}{\sqrt[3]{\frac{-1}{d}}}\right| \cdot \sqrt{\frac{\sqrt[3]{\frac{-1}{\ell}}}{\sqrt[3]{\frac{-1}{d}}}}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +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)))))