Average Error: 25.9 → 18.9
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -4.5046296138063127 \cdot 10^{+74}:\\ \;\;\;\;(\left(\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{2}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_* \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{\frac{-1}{h}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}}\right)\\ \mathbf{elif}\;d \le -3.67676927850655 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{-1}{\ell}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot (\left(\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{2}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*\\ \mathbf{elif}\;d \le 1.980523357016314 \cdot 10^{+48}:\\ \;\;\;\;(\left(\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{2}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_* \cdot \left(\left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{2}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_* \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{{h}^{\frac{1}{2}}}\right)\\ \end{array}\]

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

  1. Split input into 4 regimes

  2. if d < -4.5046296138063127e+74

    1. Initial program 23.3

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

    2. Initial simplification22.6

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

    3. Taylor expanded around -inf 15.6

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

    4. Simplified11.2

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

    if -4.5046296138063127e+74 < d < -3.67676927850655e-310

    1. Initial program 27.2

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

    2. Initial simplification27.9

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

    3. Taylor expanded around -inf 26.5

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

    4. Simplified23.6

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

    if -3.67676927850655e-310 < d < 1.980523357016314e+48

    1. Initial program 27.3

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

    2. Initial simplification28.2

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

    4. Applied div-inv28.2

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

    5. Applied unpow-prod-down23.8

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

    if 1.980523357016314e+48 < d

    1. Initial program 24.7

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

    2. Initial simplification24.3

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

    3. Taylor expanded around inf 16.8

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

    4. Simplified12.4

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

  4. Final simplification18.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -4.5046296138063127 \cdot 10^{+74}:\\ \;\;\;\;(\left(\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{2}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_* \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{\frac{-1}{h}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}}\right)\\ \mathbf{elif}\;d \le -3.67676927850655 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{-1}{\ell}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot (\left(\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{2}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*\\ \mathbf{elif}\;d \le 1.980523357016314 \cdot 10^{+48}:\\ \;\;\;\;(\left(\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{2}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_* \cdot \left(\left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{2}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_* \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{{h}^{\frac{1}{2}}}\right)\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018221 +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)))))