Average Error: 25.2 → 16.8
Time: 1.8m
Precision: 64
Internal Precision: 128
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -4.99381406543626 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{2} \cdot \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}}{\frac{1}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\sqrt{\frac{d}{\ell}}}\right)\right)\\ \mathbf{elif}\;\ell \le 5.614652203722002 \cdot 10^{-36}:\\ \;\;\;\;\frac{d \cdot \left(1 - \left(\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot h\right) \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{2}\right) \cdot \left(\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot h\right) \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{2}\right)\right)}{\left(\sqrt{h} \cdot \sqrt{\ell}\right) \cdot \left(\frac{\frac{M \cdot D}{2 \cdot d}}{2} \cdot \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}}{\frac{1}{h}} + 1\right)}\\ \mathbf{elif}\;\ell \le 2.0042803747088486 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{2} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{d} \cdot \sqrt{d}\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{d} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{2} \cdot \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}}{\frac{1}{h}}\right)}{\sqrt{\ell}}\\ \end{array}\]

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 4 regimes

  2. if l < -4.99381406543626e-310

    1. Initial program 24.8

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

    2. Simplified24.6

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

    4. Applied times-frac22.8

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

    6. Applied div-inv22.8

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

    7. Applied associate-/r*21.4

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

    9. Applied add-sqr-sqrt21.4

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

    10. Applied sqrt-prod21.5

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

    if -4.99381406543626e-310 < l < 5.614652203722002e-36

    1. Initial program 26.9

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

    2. Simplified26.1

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

    4. Applied times-frac25.6

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

    6. Applied div-inv25.6

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

    7. Applied associate-/r*21.7

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

    9. Applied sqrt-div18.1

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

    10. Applied sqrt-div7.2

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

    11. Applied frac-times7.2

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

    12. Applied flip--11.7

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

    13. Applied frac-times12.2

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

    14. Simplified12.1

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

    if 5.614652203722002e-36 < l < 2.0042803747088486e+241

    1. Initial program 23.2

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

    2. Simplified23.1

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

    4. Applied times-frac21.1

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

    6. Applied sqrt-div14.1

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

    7. Applied sqrt-div10.8

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

    8. Applied frac-times10.8

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

    9. Applied associate-*r/10.2

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

    if 2.0042803747088486e+241 < l

    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. Simplified32.0

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

    4. Applied times-frac30.7

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

    6. Applied div-inv30.8

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

    7. Applied associate-/r*30.5

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

    9. Applied sqrt-div20.0

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

    10. Applied associate-*l/20.0

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

    11. Applied associate-*r/18.9

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

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -4.99381406543626 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{2} \cdot \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}}{\frac{1}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\sqrt{\frac{d}{\ell}}}\right)\right)\\ \mathbf{elif}\;\ell \le 5.614652203722002 \cdot 10^{-36}:\\ \;\;\;\;\frac{d \cdot \left(1 - \left(\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot h\right) \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{2}\right) \cdot \left(\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot h\right) \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{2}\right)\right)}{\left(\sqrt{h} \cdot \sqrt{\ell}\right) \cdot \left(\frac{\frac{M \cdot D}{2 \cdot d}}{2} \cdot \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}}{\frac{1}{h}} + 1\right)}\\ \mathbf{elif}\;\ell \le 2.0042803747088486 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{2} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{d} \cdot \sqrt{d}\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{d} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{2} \cdot \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}}{\frac{1}{h}}\right)}{\sqrt{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 
(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)))))