Average Error: 25.8 → 18.2
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;h \le -1.3311461438088012 \cdot 10^{-102}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{\ell}}\right)\right)\\ \mathbf{elif}\;h \le -6.71204672900335 \cdot 10^{-294}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left(\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right)\\ \mathbf{elif}\;h \le 2.206278862898223 \cdot 10^{-82}:\\ \;\;\;\;\left(\left({\left(\frac{1}{h}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{elif}\;h \le 5.154774397990406 \cdot 10^{+84}:\\ \;\;\;\;\frac{\left(\frac{\frac{\frac{M}{2}}{\frac{d}{D}}}{\frac{\ell}{h}} \cdot \frac{\frac{\frac{M}{2}}{\frac{d}{D}}}{2}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(-\sqrt{d}\right)\right)}{\sqrt{\ell}} + {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}} + \left(\left(\left(\sqrt{\sqrt{\frac{d}{h}}} \cdot \sqrt{\sqrt{\frac{d}{h}}}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-\frac{\frac{\frac{M}{2}}{\frac{d}{D}}}{\frac{\ell}{h}}\right)\right) \cdot \frac{\frac{D}{d}}{\frac{4}{M}}\\ \end{array}\]

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 5 regimes

  2. if h < -1.3311461438088012e-102

    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. Using strategy rm

    3. Applied associate-*r/21.9

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

    4. Taylor expanded around -inf 19.2

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

    5. Simplified15.6

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

    if -1.3311461438088012e-102 < h < -6.71204672900335e-294

    1. Initial program 30.5

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

    3. Applied associate-*r/30.4

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

    4. Taylor expanded around -inf 20.7

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

    5. Simplified16.7

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

    if -6.71204672900335e-294 < h < 2.206278862898223e-82

    1. Initial program 30.5

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

    3. Applied div-inv30.5

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

    4. Applied unpow-prod-down19.1

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

    5. Simplified19.1

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

    if 2.206278862898223e-82 < h < 5.154774397990406e+84

    1. Initial program 18.2

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

    3. Applied associate-*r/18.3

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

    5. Applied sub-neg18.3

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

    6. Applied distribute-rgt-in18.3

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

    7. Simplified16.7

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

    9. Applied sqrt-div16.1

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

    10. Applied associate-*l/16.1

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

    11. Applied associate-*l/15.9

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

    if 5.154774397990406e+84 < h

    1. Initial program 27.0

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

    3. Applied associate-*r/24.7

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

    5. Applied sub-neg24.7

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

    6. Applied distribute-rgt-in24.8

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

    7. Simplified26.1

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

    9. Applied associate-*r*24.8

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

    10. Simplified24.9

      \[\leadsto 1 \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) + \left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(-\sqrt{\frac{d}{h}}\right)\right) \cdot \frac{\frac{\frac{M}{2}}{\frac{d}{D}}}{\frac{\ell}{h}}\right) \cdot \color{blue}{\frac{\frac{D}{d}}{\frac{4}{M}}}\]
    11. Using strategy rm

    12. Applied add-sqr-sqrt24.9

      \[\leadsto 1 \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) + \left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(-\sqrt{\color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{h}}}}\right)\right) \cdot \frac{\frac{\frac{M}{2}}{\frac{d}{D}}}{\frac{\ell}{h}}\right) \cdot \frac{\frac{D}{d}}{\frac{4}{M}}\]

    13. Applied sqrt-prod24.9

      \[\leadsto 1 \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) + \left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(-\color{blue}{\sqrt{\sqrt{\frac{d}{h}}} \cdot \sqrt{\sqrt{\frac{d}{h}}}}\right)\right) \cdot \frac{\frac{\frac{M}{2}}{\frac{d}{D}}}{\frac{\ell}{h}}\right) \cdot \frac{\frac{D}{d}}{\frac{4}{M}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;h \le -1.3311461438088012 \cdot 10^{-102}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{\ell}}\right)\right)\\ \mathbf{elif}\;h \le -6.71204672900335 \cdot 10^{-294}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left(\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right)\\ \mathbf{elif}\;h \le 2.206278862898223 \cdot 10^{-82}:\\ \;\;\;\;\left(\left({\left(\frac{1}{h}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{elif}\;h \le 5.154774397990406 \cdot 10^{+84}:\\ \;\;\;\;\frac{\left(\frac{\frac{\frac{M}{2}}{\frac{d}{D}}}{\frac{\ell}{h}} \cdot \frac{\frac{\frac{M}{2}}{\frac{d}{D}}}{2}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(-\sqrt{d}\right)\right)}{\sqrt{\ell}} + {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}} + \left(\left(\left(\sqrt{\sqrt{\frac{d}{h}}} \cdot \sqrt{\sqrt{\frac{d}{h}}}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-\frac{\frac{\frac{M}{2}}{\frac{d}{D}}}{\frac{\ell}{h}}\right)\right) \cdot \frac{\frac{D}{d}}{\frac{4}{M}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 +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)))))