Average Error: 26.2 → 21.0
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;h \le 6.303505144207272 \cdot 10^{-299}:\\ \;\;\;\;\left(1 - \frac{h}{\frac{\ell}{\frac{\frac{D \cdot M}{2 \cdot d} \cdot \frac{D \cdot M}{2 \cdot d}}{2}}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{\frac{\frac{-1}{h}}{\frac{-1}{d}}}\right)\\ \mathbf{elif}\;h \le 8.543904292155608 \cdot 10^{-153}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left({\left(\frac{1}{h}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \frac{h}{\frac{\ell}{\frac{\frac{D \cdot M}{2 \cdot d} \cdot \frac{D \cdot M}{2 \cdot d}}{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h}{\frac{\frac{\ell}{\sqrt{\frac{\frac{D \cdot M}{2 \cdot d} \cdot \frac{D \cdot M}{2 \cdot d}}{2}}}}{\sqrt{\frac{\frac{D \cdot M}{2 \cdot d} \cdot \frac{D \cdot M}{2 \cdot d}}{2}}}}\right) \cdot \left(\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \end{array}\]

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 3 regimes

  2. if h < 6.303505144207272e-299

    1. Initial program 26.4

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

    3. Applied associate-*r/25.8

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

    5. Applied *-un-lft-identity25.8

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

    6. Applied associate-*r*25.8

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

    7. Simplified25.1

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

    8. Taylor expanded around -inf 22.8

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

    9. Simplified25.2

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

    if 6.303505144207272e-299 < h < 8.543904292155608e-153

    1. Initial program 32.8

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

    3. Applied associate-*r/32.7

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

    5. Applied *-un-lft-identity32.7

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

    6. Applied associate-*r*32.7

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

    7. Simplified34.4

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

    9. Applied div-inv34.4

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

    10. Applied unpow-prod-down19.7

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

    11. Simplified19.7

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

    if 8.543904292155608e-153 < h

    1. Initial program 23.9

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

    3. Applied associate-*r/22.7

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

    5. Applied *-un-lft-identity22.7

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

    6. Applied associate-*r*22.7

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

    7. Simplified22.0

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

    9. Applied add-sqr-sqrt22.0

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

    10. Applied associate-/r*22.0

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

    12. Applied div-inv22.0

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

    13. Applied unpow-prod-down15.9

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

    14. Simplified15.9

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

  4. Final simplification21.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;h \le 6.303505144207272 \cdot 10^{-299}:\\ \;\;\;\;\left(1 - \frac{h}{\frac{\ell}{\frac{\frac{D \cdot M}{2 \cdot d} \cdot \frac{D \cdot M}{2 \cdot d}}{2}}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{\frac{\frac{-1}{h}}{\frac{-1}{d}}}\right)\\ \mathbf{elif}\;h \le 8.543904292155608 \cdot 10^{-153}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left({\left(\frac{1}{h}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \frac{h}{\frac{\ell}{\frac{\frac{D \cdot M}{2 \cdot d} \cdot \frac{D \cdot M}{2 \cdot d}}{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h}{\frac{\frac{\ell}{\sqrt{\frac{\frac{D \cdot M}{2 \cdot d} \cdot \frac{D \cdot M}{2 \cdot d}}{2}}}}{\sqrt{\frac{\frac{D \cdot M}{2 \cdot d} \cdot \frac{D \cdot M}{2 \cdot d}}{2}}}}\right) \cdot \left(\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 +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)))))