Average Error: 13.8 → 11.5
Time: 1.1m
Precision: 64
Internal Precision: 576
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;D \le -6.774046224605452 \cdot 10^{-126}:\\ \;\;\;\;\sqrt{1 - \frac{\left(\frac{M}{d} \cdot \frac{D}{\frac{2}{h}}\right) \cdot \frac{M \cdot D}{d \cdot 2}}{\ell}} \cdot w0\\ \mathbf{elif}\;D \le 0.07296854825418927:\\ \;\;\;\;\sqrt{1 - \frac{\left(M \cdot \frac{D}{\frac{2}{h}}\right) \cdot \frac{\frac{M}{\frac{d \cdot 2}{D}}}{\ell}}{d}} \cdot w0\\ \mathbf{elif}\;D \le 6.15391543601957 \cdot 10^{+244}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\left(\sqrt[3]{\frac{M}{d}} \cdot \frac{D}{\frac{2}{h}}\right) \cdot \left(\sqrt[3]{\frac{M}{d}} \cdot \sqrt[3]{\frac{M}{d}}\right)\right)} \cdot w0\\ \end{array}\]

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

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 < -6.774046224605452e-126

    1. Initial program 17.6

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

    2. Initial simplification17.1

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}} \cdot w0\]
    3. Using strategy rm

    4. Applied div-inv17.0

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\color{blue}{\ell \cdot \frac{1}{h}}}} \cdot w0\]

    5. Applied times-frac12.3

      \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}}} \cdot w0\]

    6. Simplified16.0

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{D}{\frac{2}{h}}\right)}} \cdot w0\]
    7. Using strategy rm

    8. Applied associate-*l/16.7

      \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M}{d} \cdot \frac{D}{\frac{2}{h}}\right)}{\ell}}} \cdot w0\]

    if -6.774046224605452e-126 < D < 0.07296854825418927

    1. Initial program 9.4

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

    2. Initial simplification8.9

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}} \cdot w0\]
    3. Using strategy rm

    4. Applied div-inv8.9

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\color{blue}{\ell \cdot \frac{1}{h}}}} \cdot w0\]

    5. Applied times-frac3.2

      \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}}} \cdot w0\]

    6. Simplified4.9

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{D}{\frac{2}{h}}\right)}} \cdot w0\]
    7. Using strategy rm

    8. Applied associate-*l/4.2

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \color{blue}{\frac{M \cdot \frac{D}{\frac{2}{h}}}{d}}} \cdot w0\]

    9. Applied associate-*r/4.3

      \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \left(M \cdot \frac{D}{\frac{2}{h}}\right)}{d}}} \cdot w0\]
    10. Using strategy rm

    11. Applied associate-/l*4.2

      \[\leadsto \sqrt{1 - \frac{\frac{\color{blue}{\frac{M}{\frac{2 \cdot d}{D}}}}{\ell} \cdot \left(M \cdot \frac{D}{\frac{2}{h}}\right)}{d}} \cdot w0\]

    if 0.07296854825418927 < D < 6.15391543601957e+244

    1. Initial program 18.0

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

    2. Initial simplification17.7

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}} \cdot w0\]
    3. Using strategy rm

    4. Applied div-inv17.7

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\color{blue}{\ell \cdot \frac{1}{h}}}} \cdot w0\]

    5. Applied times-frac13.8

      \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}}} \cdot w0\]

    6. Simplified18.6

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{D}{\frac{2}{h}}\right)}} \cdot w0\]
    7. Using strategy rm

    8. Applied associate-*l/18.9

      \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M}{d} \cdot \frac{D}{\frac{2}{h}}\right)}{\ell}}} \cdot w0\]

    9. Taylor expanded around 0 20.3

      \[\leadsto \color{blue}{1} \cdot w0\]

    if 6.15391543601957e+244 < D

    1. Initial program 29.8

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

    2. Initial simplification30.0

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}} \cdot w0\]
    3. Using strategy rm

    4. Applied div-inv30.0

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\color{blue}{\ell \cdot \frac{1}{h}}}} \cdot w0\]

    5. Applied times-frac26.3

      \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}}} \cdot w0\]

    6. Simplified36.1

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{D}{\frac{2}{h}}\right)}} \cdot w0\]
    7. Using strategy rm

    8. Applied add-cube-cbrt36.1

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\frac{M}{d}} \cdot \sqrt[3]{\frac{M}{d}}\right) \cdot \sqrt[3]{\frac{M}{d}}\right)} \cdot \frac{D}{\frac{2}{h}}\right)} \cdot w0\]

    9. Applied associate-*l*36.1

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{M}{d}} \cdot \sqrt[3]{\frac{M}{d}}\right) \cdot \left(\sqrt[3]{\frac{M}{d}} \cdot \frac{D}{\frac{2}{h}}\right)\right)}} \cdot w0\]
  3. Recombined 4 regimes into one program.

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;D \le -6.774046224605452 \cdot 10^{-126}:\\ \;\;\;\;\sqrt{1 - \frac{\left(\frac{M}{d} \cdot \frac{D}{\frac{2}{h}}\right) \cdot \frac{M \cdot D}{d \cdot 2}}{\ell}} \cdot w0\\ \mathbf{elif}\;D \le 0.07296854825418927:\\ \;\;\;\;\sqrt{1 - \frac{\left(M \cdot \frac{D}{\frac{2}{h}}\right) \cdot \frac{\frac{M}{\frac{d \cdot 2}{D}}}{\ell}}{d}} \cdot w0\\ \mathbf{elif}\;D \le 6.15391543601957 \cdot 10^{+244}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\left(\sqrt[3]{\frac{M}{d}} \cdot \frac{D}{\frac{2}{h}}\right) \cdot \left(\sqrt[3]{\frac{M}{d}} \cdot \sqrt[3]{\frac{M}{d}}\right)\right)} \cdot w0\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018255 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))