Average Error: 25.3 → 19.6
Time: 1.2m
Precision: 64
Internal precision: 384
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^2\right) \cdot \frac{h}{\ell}\right)\]
⬇
\[\begin{array}{l} \mathbf{if}\;d \le 1.0686035838302045 \cdot 10^{-288}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{1}{2}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^2\right)\\ \mathbf{if}\;d \le 2.155516528760191 \cdot 10^{-24}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({d}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{1}{2}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^2\right)\\ \mathbf{if}\;d \le 4.1646698243649156 \cdot 10^{+57}:\\ \;\;\;\;\left({\left(\sqrt{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right)}^2 \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{1}{2}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)}^2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({d}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^2\right) \cdot \frac{h}{\ell}\right)\\ \end{array}\]

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 < 1.0686035838302045e-288

    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. Using strategy rm

    3. Applied add-sqr-sqrt 26.2

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

    4. Applied add-sqr-sqrt 26.2

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

    5. Applied square-unprod 26.2

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

    6. Applied square-unprod 24.7

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

    if 1.0686035838302045e-288 < d < 2.155516528760191e-24

    1. Initial program 30.0

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

    3. Applied add-sqr-sqrt 30.0

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

    4. Applied add-sqr-sqrt 30.0

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

    5. Applied square-unprod 30.0

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

    6. Applied square-unprod 27.2

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

    8. Applied div-inv 27.3

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

    9. Applied unpow-prod-down 19.7

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

    if 2.155516528760191e-24 < d < 4.1646698243649156e+57

    1. Initial program 12.6

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

    3. Applied add-sqr-sqrt 12.6

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

    4. Applied add-sqr-sqrt 12.7

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

    5. Applied square-unprod 12.7

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

    6. Applied square-unprod 10.1

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

    8. Applied add-sqr-sqrt 10.3

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

    10. Applied sqrt-div 6.3

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

    if 4.1646698243649156e+57 < d

    1. Initial program 23.5

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

    3. Applied div-inv 23.5

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

    4. Applied unpow-prod-down 12.1

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

Runtime

Time bar (total: 1.2m) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064524629 4159152179 2999149171 575749698 4006532819 692958815)'
(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) (sqr (/ (* M D) (* 2 d)))) (/ h l)))))