Error

Derivation

1. Split input into 5 regimes.

2. if h < -1.759225207600335e+58

   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. Applied simplify 26.1

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

   4. Applied div-inv 26.1

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

   5. Applied *-un-lft-identity 26.1

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

   6. Applied square-prod 26.1

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

   7. Applied times-frac 24.0

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

   8. Applied simplify 24.0

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

   9. Applied simplify 24.0

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

   if -1.759225207600335e+58 < h < -1.3193602092139957e-197

   1. Initial program 21.0

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

   2. Applied simplify 21.0

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

   4. Applied add-sqr-sqrt 21.0

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

   5. Applied square-undiv 18.6

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

   7. Applied *-un-lft-identity 18.6

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

   8. Applied times-frac 19.5

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

   10. Applied add-sqr-sqrt 19.6

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

   if -1.3193602092139957e-197 < h < 7.060529603865913e-283

   1. Initial program 35.1

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

   2. Applied simplify 35.1

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

   4. Applied div-inv 35.1

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

   5. Applied square-mult 35.1

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

   6. Applied times-frac 32.2

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

   7. Applied simplify 34.5

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

   if 7.060529603865913e-283 < h < 8.953263541046556e-147

   1. Initial program 29.5

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

   2. Applied simplify 29.5

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

   4. Applied add-sqr-sqrt 29.5

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

   5. Applied square-undiv 27.7

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

   7. Applied *-un-lft-identity 27.7

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

   8. Applied times-frac 29.4

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

   10. Applied div-inv 29.4

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

   11. Applied unpow-prod-down 14.3

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

   if 8.953263541046556e-147 < h

   1. Initial program 23.6

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

   2. Applied simplify 23.4

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

   4. Applied add-sqr-sqrt 23.4

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

   5. Applied square-undiv 22.0

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

   7. Applied div-inv 22.0

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

   8. Applied unpow-prod-down 17.5

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

Runtime

Time bar (total: 3.7m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(3735396131 803941924 2811322739 4215829258 3945640984 889435658)'
(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)))))