Average Error: 25.1 → 18.4
Time: 1.8m
Precision: 64
Internal Precision: 128
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;h \le 1.1119893539492268 \cdot 10^{-297}:\\ \;\;\;\;\left(1 - h \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell \cdot 2}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\sqrt{\frac{d}{\ell}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sqrt{\sqrt{d}} \cdot \sqrt{\sqrt{d}}\right) \cdot \sqrt{d}\right) \cdot \left(1 - h \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell \cdot 2}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array}\]

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 h < 1.1119893539492268e-297

    1. Initial program 25.1

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

    2. Simplified24.7

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

    4. Applied associate-*l/24.7

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

    5. Applied associate-/r/23.9

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

    7. Applied add-sqr-sqrt24.0

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

    if 1.1119893539492268e-297 < h

    1. Initial program 25.1

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

    2. Simplified24.9

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

    4. Applied associate-*l/24.9

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

    5. Applied associate-/r/24.0

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

    7. Applied add-sqr-sqrt24.2

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

    9. Applied sqrt-div18.3

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

    10. Applied sqrt-div18.3

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

    11. Applied sqrt-div18.3

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

    12. Applied sqrt-div12.4

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

    13. Applied sqrt-div12.5

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

    14. Applied frac-times12.5

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

    15. Applied frac-times12.5

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

    16. Applied associate-*r/12.6

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

    17. Simplified12.6

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

  4. Final simplification18.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;h \le 1.1119893539492268 \cdot 10^{-297}:\\ \;\;\;\;\left(1 - h \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell \cdot 2}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\sqrt{\frac{d}{\ell}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sqrt{\sqrt{d}} \cdot \sqrt{\sqrt{d}}\right) \cdot \sqrt{d}\right) \cdot \left(1 - h \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell \cdot 2}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 
(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)))))