Average Error: 25.9 → 16.6
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le 3.923387526444069 \cdot 10^{-286}:\\ \;\;\;\;\left(1 - \left(h \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\left|\frac{\sqrt[3]{\frac{-1}{h}}}{\sqrt[3]{\frac{-1}{d}}}\right| \cdot \sqrt{\frac{\sqrt[3]{\frac{-1}{h}}}{\sqrt[3]{\frac{-1}{d}}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{\ell} \cdot \left(h \cdot \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\right) \cdot \left(\left(\sqrt{d} \cdot {\left(\frac{1}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \left(\left|\frac{\sqrt[3]{\frac{-1}{h}}}{\sqrt[3]{\frac{-1}{d}}}\right| \cdot \sqrt{\frac{\sqrt[3]{\frac{-1}{h}}}{\sqrt[3]{\frac{-1}{d}}}}\right)\right)\\ \end{array}\]

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 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. Using strategy rm

    3. Applied div-inv25.8

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

    4. Applied associate-*r*24.6

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

    5. Taylor expanded around -inf 22.6

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

    6. Simplified24.6

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

    8. Applied add-cube-cbrt24.9

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

    9. Applied add-cube-cbrt25.0

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

    10. Applied times-frac25.0

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

    11. Applied sqrt-prod19.7

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

    12. Simplified19.1

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

    14. Applied times-frac19.4

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

    if 3.923387526444069e-286 < l

    1. Initial program 26.0

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

    3. Applied div-inv26.0

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

    4. Applied associate-*r*25.4

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

    5. Taylor expanded around -inf 62.2

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

    6. Simplified25.4

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

    8. Applied add-cube-cbrt25.7

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

    9. Applied add-cube-cbrt25.8

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

    10. Applied times-frac25.8

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

    11. Applied sqrt-prod20.2

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

    12. Simplified19.5

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

    14. Applied div-inv19.5

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

    15. Applied unpow-prod-down13.9

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

    16. Simplified13.9

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

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le 3.923387526444069 \cdot 10^{-286}:\\ \;\;\;\;\left(1 - \left(h \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\left|\frac{\sqrt[3]{\frac{-1}{h}}}{\sqrt[3]{\frac{-1}{d}}}\right| \cdot \sqrt{\frac{\sqrt[3]{\frac{-1}{h}}}{\sqrt[3]{\frac{-1}{d}}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{\ell} \cdot \left(h \cdot \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\right) \cdot \left(\left(\sqrt{d} \cdot {\left(\frac{1}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \left(\left|\frac{\sqrt[3]{\frac{-1}{h}}}{\sqrt[3]{\frac{-1}{d}}}\right| \cdot \sqrt{\frac{\sqrt[3]{\frac{-1}{h}}}{\sqrt[3]{\frac{-1}{d}}}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 +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)))))