Average Error: 25.9 → 17.2
Time: 3.1m
Precision: 64
Internal Precision: 576
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -5.2248836320409264 \cdot 10^{+97}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{\frac{-1}{h}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\ell}{h}} \cdot \frac{\frac{M}{2} \cdot \frac{D}{d}}{2}\right)\\ \mathbf{elif}\;d \le -3.90065203139097 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\ell}{h}} \cdot \frac{\frac{M}{2} \cdot \frac{D}{d}}{2}\right) \cdot \left(\frac{\sqrt{\frac{-1}{\ell}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;d \le 8.104720669399042 \cdot 10^{+100}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\ell}{h}} \cdot \frac{\frac{M}{2} \cdot \frac{D}{d}}{2}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({d}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)\\ \mathbf{elif}\;d \le 8.410593098276458 \cdot 10^{+275}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{{\ell}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot 2}\right)\\ \end{array}\]

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 d < -5.2248836320409264e+97

    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. Initial simplification24.5

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

    4. Applied times-frac24.2

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

    5. Taylor expanded around -inf 15.6

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

    6. Simplified11.1

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

    if -5.2248836320409264e+97 < d < -3.90065203139097e-310

    1. Initial program 25.1

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

    2. Initial simplification25.9

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

    4. Applied times-frac24.0

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

    5. Taylor expanded around -inf 22.6

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

    6. Simplified19.4

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

    if -3.90065203139097e-310 < d < 8.104720669399042e+100

    1. Initial program 26.2

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

    2. Initial simplification26.7

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

    4. Applied times-frac24.7

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

    6. Applied div-inv24.7

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

    7. Applied unpow-prod-down19.8

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

    if 8.104720669399042e+100 < d < 8.410593098276458e+275

    1. Initial program 26.6

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

    2. Initial simplification25.9

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

    3. Taylor expanded around inf 16.9

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

    4. Simplified12.5

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

    if 8.410593098276458e+275 < d

    1. Initial program 35.3

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

    2. Initial simplification34.5

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

    3. Taylor expanded around inf 28.9

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

    4. Simplified25.4

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -5.2248836320409264 \cdot 10^{+97}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{\frac{-1}{h}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\ell}{h}} \cdot \frac{\frac{M}{2} \cdot \frac{D}{d}}{2}\right)\\ \mathbf{elif}\;d \le -3.90065203139097 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\ell}{h}} \cdot \frac{\frac{M}{2} \cdot \frac{D}{d}}{2}\right) \cdot \left(\frac{\sqrt{\frac{-1}{\ell}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;d \le 8.104720669399042 \cdot 10^{+100}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\ell}{h}} \cdot \frac{\frac{M}{2} \cdot \frac{D}{d}}{2}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({d}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)\\ \mathbf{elif}\;d \le 8.410593098276458 \cdot 10^{+275}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{{\ell}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot 2}\right)\\ \end{array}\]

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed 2018215 +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)))))