Average Error: 25.4 → 17.2
Time: 4.1m
Precision: 64
Internal Precision: 128
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;h \le -1.77030403846294 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell \cdot 2}{h \cdot \frac{M \cdot D}{2 \cdot d}}}\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \le 3.823537858472912 \cdot 10^{+214}:\\ \;\;\;\;\frac{\left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\frac{\ell}{h} \cdot 2}{\frac{M \cdot D}{2 \cdot d}}}\right) \cdot \left(\sqrt{d} \cdot \sqrt{d}\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell \cdot 2}{h \cdot \frac{M \cdot D}{2 \cdot d}}} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell \cdot 2}{h \cdot \frac{M \cdot D}{2 \cdot d}}}\right) \cdot \left(\sqrt{d} \cdot \sqrt{d}\right)}{\left(\sqrt{h} \cdot \sqrt{\ell}\right) \cdot \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell \cdot 2}{h \cdot \frac{M \cdot D}{2 \cdot d}}} + 1\right)}\\ \end{array}\]

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 3 regimes

  2. if h < -1.77030403846294e-310

    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. Simplified25.2

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

    4. Applied associate-/l*23.7

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

    6. Applied associate-*l/23.7

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

    7. Applied associate-/l/22.2

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

    9. Applied associate-*r*22.4

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

    if -1.77030403846294e-310 < h < 3.823537858472912e+214

    1. Initial program 24.5

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

    2. Simplified24.4

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

    4. Applied associate-/l*22.7

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

    6. Applied sqrt-div16.6

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

    7. Applied sqrt-div11.0

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

    8. Applied frac-times11.0

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

    9. Applied associate-*r/10.4

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

    if 3.823537858472912e+214 < h

    1. Initial program 32.0

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

    2. Simplified31.5

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

    4. Applied associate-/l*31.5

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

    6. Applied associate-*l/31.5

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

    7. Applied associate-/l/27.5

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

    9. Applied sqrt-div23.1

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

    10. Applied sqrt-div17.3

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

    11. Applied frac-times17.3

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

    12. Applied flip--20.1

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell \cdot 2}{\frac{M \cdot D}{2 \cdot d} \cdot h}} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell \cdot 2}{\frac{M \cdot D}{2 \cdot d} \cdot h}}}{1 + \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell \cdot 2}{\frac{M \cdot D}{2 \cdot d} \cdot h}}}} \cdot \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}\]

    13. Applied frac-times22.3

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;h \le -1.77030403846294 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell \cdot 2}{h \cdot \frac{M \cdot D}{2 \cdot d}}}\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \le 3.823537858472912 \cdot 10^{+214}:\\ \;\;\;\;\frac{\left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\frac{\ell}{h} \cdot 2}{\frac{M \cdot D}{2 \cdot d}}}\right) \cdot \left(\sqrt{d} \cdot \sqrt{d}\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell \cdot 2}{h \cdot \frac{M \cdot D}{2 \cdot d}}} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell \cdot 2}{h \cdot \frac{M \cdot D}{2 \cdot d}}}\right) \cdot \left(\sqrt{d} \cdot \sqrt{d}\right)}{\left(\sqrt{h} \cdot \sqrt{\ell}\right) \cdot \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell \cdot 2}{h \cdot \frac{M \cdot D}{2 \cdot d}}} + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019051 
(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)))))