Average Error: 25.8 → 19.1
Time: 4.3m
Precision: 64
Internal Precision: 384
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{\ell} \le -1.2350490242485057 \cdot 10^{+46}:\\ \;\;\;\;\left(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) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell \cdot 2}\right)\\ \mathbf{if}\;\frac{1}{\ell} \le -8.445503457762819 \cdot 10^{-99}:\\ \;\;\;\;\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) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell \cdot 2}\right)\\ \mathbf{if}\;\frac{1}{\ell} \le -4.0174204410414035 \cdot 10^{-306}:\\ \;\;\;\;\left(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) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell \cdot 2}\right)\\ \mathbf{if}\;\frac{1}{\ell} \le 1.0101631959810096 \cdot 10^{-110}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\frac{1}{2}}\right)}^{\left(\log d - \log \ell\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{\frac{M}{2} \cdot \frac{D}{d}}{2}\right)\\ \mathbf{if}\;\frac{1}{\ell} \le 892010028878.1191:\\ \;\;\;\;\left(\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{1}{\ell} \cdot \left(h \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{\ell} \cdot \left(h \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right)\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)\\ \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 5 regimes

  2. if (/ 1 l) < -1.2350490242485057e+46 or -8.445503457762819e-99 < (/ 1 l) < -4.0174204410414035e-306

    1. Initial program 28.3

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

    3. Applied associate-*l/28.3

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

    4. Applied frac-times27.1

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

    5. Taylor expanded around -inf 22.9

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

    if -1.2350490242485057e+46 < (/ 1 l) < -8.445503457762819e-99

    1. Initial program 18.7

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

    3. Applied associate-*l/18.7

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

    4. Applied frac-times19.5

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

    5. Taylor expanded around -inf 15.8

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

    if -4.0174204410414035e-306 < (/ 1 l) < 1.0101631959810096e-110

    1. Initial program 27.5

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

    3. Applied div-inv27.5

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

    4. Applied associate-*r*29.5

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

    5. Taylor expanded around inf 27.6

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

    6. Applied simplify23.5

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

    if 1.0101631959810096e-110 < (/ 1 l) < 892010028878.1191

    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 div-inv18.8

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

    4. Applied associate-*r*19.3

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

    6. Applied div-inv19.3

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

    7. Applied unpow-prod-down11.9

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

    if 892010028878.1191 < (/ 1 l)

    1. Initial program 27.8

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

    3. Applied div-inv27.8

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

    4. Applied associate-*r*22.6

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

    6. Applied div-inv22.6

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

    7. Applied unpow-prod-down12.2

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

  4. Applied simplify19.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{1}{\ell} \le -1.2350490242485057 \cdot 10^{+46}:\\ \;\;\;\;\left(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) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell \cdot 2}\right)\\ \mathbf{if}\;\frac{1}{\ell} \le -8.445503457762819 \cdot 10^{-99}:\\ \;\;\;\;\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) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell \cdot 2}\right)\\ \mathbf{if}\;\frac{1}{\ell} \le -4.0174204410414035 \cdot 10^{-306}:\\ \;\;\;\;\left(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) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell \cdot 2}\right)\\ \mathbf{if}\;\frac{1}{\ell} \le 1.0101631959810096 \cdot 10^{-110}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\frac{1}{2}}\right)}^{\left(\log d - \log \ell\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{\frac{M}{2} \cdot \frac{D}{d}}{2}\right)\\ \mathbf{if}\;\frac{1}{\ell} \le 892010028878.1191:\\ \;\;\;\;\left(\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{1}{\ell} \cdot \left(h \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{\ell} \cdot \left(h \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right)\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)\\ \end{array}}\]

Runtime

Time bar (total: 4.3m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' +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)))))