Average Error: 25.2 → 18.6
Time: 3.8m
Precision: 64
Internal Precision: 576
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -1.281978413417261 \cdot 10^{+194}:\\ \;\;\;\;\left(\left({\left(\sqrt{\frac{d}{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt{\frac{d}{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)}\right) \cdot \left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{2 \cdot \frac{\ell}{h}}{\frac{M}{2} \cdot \frac{D}{d}}}\right)\\ \mathbf{if}\;\ell \le -5.4737872113822 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell \cdot 2}\right) \cdot \left(e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{h}\right) - \log \left(\frac{-1}{d}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;\ell \le 6.7706330310610026 \cdot 10^{-96}:\\ \;\;\;\;\left(1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell \cdot 2}\right) \cdot \left(\left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;\ell \le 5.123102837426798 \cdot 10^{+133}:\\ \;\;\;\;\left(\left({d}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot h}{\frac{\ell \cdot 2}{\frac{M}{2} \cdot \frac{D}{d}}}\right) \cdot \left(\left({\left(\sqrt{\frac{d}{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt{\frac{d}{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(e^{\frac{1}{2}}\right)}^{\left(\log d - \log \ell\right)}\right)\\ \end{array}\]

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 5 regimes

  2. if l < -1.281978413417261e+194

    1. Initial program 31.0

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

    3. Applied associate-*l/31.0

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

    4. Applied frac-times32.3

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

    6. Applied add-sqr-sqrt32.3

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

    7. Applied unpow-prod-down32.4

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

    9. Applied add-cube-cbrt32.5

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

    10. Taylor expanded around -inf 28.1

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

    11. Applied simplify24.8

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

    if -1.281978413417261e+194 < l < -5.4737872113822e-310

    1. Initial program 23.4

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

    3. Applied associate-*l/23.4

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

    4. Applied frac-times21.4

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

    5. Taylor expanded around -inf 19.4

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

    if -5.4737872113822e-310 < l < 6.7706330310610026e-96

    1. Initial program 29.0

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

    3. Applied associate-*l/29.0

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

    4. Applied frac-times22.8

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

    6. Applied div-inv22.8

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

    7. Applied unpow-prod-down10.3

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

    if 6.7706330310610026e-96 < l < 5.123102837426798e+133

    1. Initial program 18.8

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

    3. Applied associate-*l/18.8

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

    4. Applied frac-times18.3

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

    6. Applied div-inv18.3

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

    7. Applied unpow-prod-down12.8

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

    if 5.123102837426798e+133 < l

    1. Initial program 29.3

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

    3. Applied associate-*l/29.3

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

    4. Applied frac-times30.9

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

    6. Applied add-sqr-sqrt30.9

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

    7. Applied unpow-prod-down31.0

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

    9. Applied add-cube-cbrt31.1

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

    10. Taylor expanded around 0 28.2

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

    11. Applied simplify23.5

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

  4. Applied simplify18.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\ell \le -1.281978413417261 \cdot 10^{+194}:\\ \;\;\;\;\left(\left({\left(\sqrt{\frac{d}{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt{\frac{d}{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)}\right) \cdot \left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{2 \cdot \frac{\ell}{h}}{\frac{M}{2} \cdot \frac{D}{d}}}\right)\\ \mathbf{if}\;\ell \le -5.4737872113822 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell \cdot 2}\right) \cdot \left(e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{h}\right) - \log \left(\frac{-1}{d}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;\ell \le 6.7706330310610026 \cdot 10^{-96}:\\ \;\;\;\;\left(1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell \cdot 2}\right) \cdot \left(\left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;\ell \le 5.123102837426798 \cdot 10^{+133}:\\ \;\;\;\;\left(\left({d}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot h}{\frac{\ell \cdot 2}{\frac{M}{2} \cdot \frac{D}{d}}}\right) \cdot \left(\left({\left(\sqrt{\frac{d}{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt{\frac{d}{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(e^{\frac{1}{2}}\right)}^{\left(\log d - \log \ell\right)}\right)\\ \end{array}}\]

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed 2018201 
(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)))))