Average Error: 25.8 → 17.6
Time: 4.0m
Precision: 64
Internal Precision: 576
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;h \le -3.764381787251415 \cdot 10^{-72}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{\frac{-1}{\ell}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{h \cdot \left({\left(\frac{1}{d \cdot 2} \cdot \left(M \cdot D\right)\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right)\\ \mathbf{elif}\;h \le 2.25962810701194 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{1}{d \cdot 2} \cdot \left(M \cdot D\right)\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left(\frac{\sqrt{\frac{-1}{h}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;h \le 3.2401082412735674 \cdot 10^{-257}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{1}{d \cdot 2} \cdot \left(M \cdot D\right)\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{{h}^{\frac{1}{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{1}{d \cdot 2} \cdot \left(M \cdot D\right)\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left(\frac{\sqrt{d}}{{\ell}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array}\]

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 4 regimes

  2. if h < -3.764381787251415e-72

    1. Initial program 23.7

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

    3. Applied div-inv23.7

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

    5. Applied associate-*r/22.1

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

    6. Taylor expanded around -inf 19.9

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

    7. Simplified16.3

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

    if -3.764381787251415e-72 < h < 2.25962810701194e-310

    1. Initial program 28.8

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

    3. Applied div-inv28.8

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

    5. Applied associate-*r/28.8

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

    6. Taylor expanded around -inf 21.7

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

    7. Simplified17.8

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

    if 2.25962810701194e-310 < h < 3.2401082412735674e-257

    1. Initial program 35.5

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

    3. Applied div-inv35.5

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

    5. Applied associate-*r/35.5

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

    6. Taylor expanded around inf 19.9

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

    7. Simplified14.8

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

    if 3.2401082412735674e-257 < h

    1. Initial program 25.2

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

    3. Applied div-inv25.2

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

    5. Applied associate-*r/24.2

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

    6. Taylor expanded around inf 22.0

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

    7. Simplified18.5

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

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;h \le -3.764381787251415 \cdot 10^{-72}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{\frac{-1}{\ell}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{h \cdot \left({\left(\frac{1}{d \cdot 2} \cdot \left(M \cdot D\right)\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right)\\ \mathbf{elif}\;h \le 2.25962810701194 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{1}{d \cdot 2} \cdot \left(M \cdot D\right)\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left(\frac{\sqrt{\frac{-1}{h}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;h \le 3.2401082412735674 \cdot 10^{-257}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{1}{d \cdot 2} \cdot \left(M \cdot D\right)\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{{h}^{\frac{1}{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{1}{d \cdot 2} \cdot \left(M \cdot D\right)\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left(\frac{\sqrt{d}}{{\ell}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array}\]

Runtime

Time bar (total: 4.0m)Debug logProfile

herbie shell --seed 2018225 
(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)))))