Average Error: 25.5 → 15.3
Time: 1.8m
Precision: 64
Internal Precision: 576
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -2.086334155827768 \cdot 10^{+25}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{-1}{h}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot \left(\sqrt{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \sqrt{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right)\right) \cdot \left(1 - \frac{\frac{D}{2} \cdot \frac{M}{d}}{2} \cdot \left(\frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell} \cdot h\right)\right)\\ \mathbf{elif}\;d \le -1.3518961194486003 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{-1}{\ell}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{D}{2} \cdot \frac{M}{d}}{2} \cdot \left(\frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell} \cdot h\right)\right)\\ \mathbf{elif}\;d \le -3.5980871304962084 \cdot 10^{-306}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{-1}{h}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{D}{2} \cdot \frac{M}{d}}{\frac{\ell}{h}} \cdot \frac{\frac{D}{2} \cdot \frac{M}{d}}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\frac{D}{2} \cdot \frac{M}{d}}{2} \cdot \left(\frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell} \cdot h\right)\right) \cdot \left(\left({d}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array}\]

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 d < -2.086334155827768e+25

    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. Initial simplification23.1

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

    4. Applied times-frac22.4

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

    6. Applied associate-/r/20.8

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

    8. Applied add-sqr-sqrt21.0

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

    9. Taylor expanded around -inf 14.0

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

    10. Simplified9.7

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

    if -2.086334155827768e+25 < d < -1.3518961194486003e-241

    1. Initial program 25.3

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

    2. Initial simplification25.5

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

    4. Applied times-frac23.6

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

    6. Applied associate-/r/21.6

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

    7. Taylor expanded around -inf 19.1

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

    8. Simplified15.5

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

    if -1.3518961194486003e-241 < d < -3.5980871304962084e-306

    1. Initial program 43.1

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

    2. Initial simplification43.5

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

    4. Applied times-frac41.5

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

    5. Taylor expanded around -inf 40.9

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

    6. Simplified39.7

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

    if -3.5980871304962084e-306 < d

    1. Initial program 25.0

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

    2. Initial simplification25.0

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

    4. Applied times-frac23.7

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

    6. Applied associate-/r/21.7

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

    8. Applied div-inv21.7

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

    9. Applied unpow-prod-down15.7

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

  4. Final simplification15.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -2.086334155827768 \cdot 10^{+25}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{-1}{h}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot \left(\sqrt{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \sqrt{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right)\right) \cdot \left(1 - \frac{\frac{D}{2} \cdot \frac{M}{d}}{2} \cdot \left(\frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell} \cdot h\right)\right)\\ \mathbf{elif}\;d \le -1.3518961194486003 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{-1}{\ell}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{D}{2} \cdot \frac{M}{d}}{2} \cdot \left(\frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell} \cdot h\right)\right)\\ \mathbf{elif}\;d \le -3.5980871304962084 \cdot 10^{-306}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{-1}{h}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{D}{2} \cdot \frac{M}{d}}{\frac{\ell}{h}} \cdot \frac{\frac{D}{2} \cdot \frac{M}{d}}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\frac{D}{2} \cdot \frac{M}{d}}{2} \cdot \left(\frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell} \cdot h\right)\right) \cdot \left(\left({d}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2018234 
(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)))))