Average Error: 25.8 → 17.5
Time: 1.7m
Precision: 64
Internal Precision: 576
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;h \le -8.814693978161449 \cdot 10^{-72}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{\frac{-1}{\ell}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{h \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right)\\ \mathbf{elif}\;h \le -2.68609340941986 \cdot 10^{-309}:\\ \;\;\;\;\left(1 - \frac{h \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right) \cdot \left(\frac{\sqrt{\frac{-1}{h}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;h \le 3.2401082412735674 \cdot 10^{-257}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{{h}^{\frac{1}{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right) \cdot \left(\frac{\sqrt{d}}{{\ell}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array}\]

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 < -8.814693978161449e-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 associate-*l/23.7

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

    4. Applied frac-times22.1

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

    5. Taylor expanded around -inf 19.9

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

    6. Simplified16.3

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

    if -8.814693978161449e-72 < h < -2.68609340941986e-309

    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 associate-*l/28.8

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

    4. Applied frac-times28.8

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

    5. Taylor expanded around -inf 21.7

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

    6. Simplified17.8

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

    if -2.68609340941986e-309 < 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. Taylor expanded around inf 19.9

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

    3. Simplified14.8

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

    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 associate-*l/25.2

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

    4. Applied frac-times24.2

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

    5. Taylor expanded around inf 21.9

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

    6. Simplified18.5

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;h \le -8.814693978161449 \cdot 10^{-72}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{\frac{-1}{\ell}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{h \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right)\\ \mathbf{elif}\;h \le -2.68609340941986 \cdot 10^{-309}:\\ \;\;\;\;\left(1 - \frac{h \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right) \cdot \left(\frac{\sqrt{\frac{-1}{h}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;h \le 3.2401082412735674 \cdot 10^{-257}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{{h}^{\frac{1}{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right) \cdot \left(\frac{\sqrt{d}}{{\ell}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018225 +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)))))