Average Error: 13.3 → 7.3
Time: 2.7m
Precision: 64
Internal Precision: 320
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\frac{\ell}{h}}{\frac{M}{2} \cdot \frac{D}{d}}}} \le 1.1688363266013715 \cdot 10^{+146}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\frac{\ell}{h}}{\frac{M}{2} \cdot \frac{D}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{\frac{M \cdot D}{2 \cdot d}}}}{\frac{1}{h}}}\\ \end{array}\]

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 2 regimes

  2. if (sqrt (- 1 (* 1 (/ (* (/ M 2) (/ D d)) (/ (/ l h) (* (/ M 2) (/ D d))))))) < 1.1688363266013715e+146

    1. Initial program 3.4

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

    2. Initial simplification2.9

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

    4. Applied div-inv2.9

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

    5. Applied associate-/r*3.4

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

    7. Applied *-un-lft-identity3.4

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

    8. Applied *-un-lft-identity3.4

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

    9. Applied times-frac3.4

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

    10. Simplified3.4

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

    11. Simplified0.7

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

    if 1.1688363266013715e+146 < (sqrt (- 1 (* 1 (/ (* (/ M 2) (/ D d)) (/ (/ l h) (* (/ M 2) (/ D d)))))))

    1. Initial program 57.4

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

    2. Initial simplification57.4

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

    4. Applied div-inv57.4

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

    5. Applied associate-/r*39.4

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

    7. Applied associate-/l*36.4

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

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\frac{\ell}{h}}{\frac{M}{2} \cdot \frac{D}{d}}}} \le 1.1688363266013715 \cdot 10^{+146}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\frac{\ell}{h}}{\frac{M}{2} \cdot \frac{D}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{\frac{M \cdot D}{2 \cdot d}}}}{\frac{1}{h}}}\\ \end{array}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018217 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))