Average Error: 13.8 → 8.4
Time: 1.5m
Precision: 64
Internal Precision: 576
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{D \cdot h}{d} \cdot \frac{M}{2}\right) \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \le -1.2180960052631341 \cdot 10^{+268}:\\ \;\;\;\;\left(\sqrt{\sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\frac{\ell}{h}}{\frac{M \cdot D}{2 \cdot d}}}}} \cdot \sqrt{\sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\frac{\ell}{h}}{\frac{M \cdot D}{2 \cdot d}}}}}\right) \cdot w0\\ \mathbf{elif}\;\left(\frac{D \cdot h}{d} \cdot \frac{M}{2}\right) \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \le -34009522142315936.0:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D \cdot h}{d} \cdot \frac{M}{2}\right) \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{D}{d} \cdot \frac{M}{2}}{\frac{\ell}{\frac{\frac{D}{d} \cdot \frac{M}{2}}{\frac{1}{h}}}}}\\ \end{array}\]

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 3 regimes

  2. if (* (/ (/ (* M D) (* 2 d)) l) (* (/ (* h D) d) (/ M 2))) < -1.2180960052631341e+268

    1. Initial program 51.4

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

    2. Initial simplification51.2

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

    4. Applied associate-/l*49.1

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

    6. Applied add-sqr-sqrt49.1

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

    7. Applied sqrt-prod49.2

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

    if -1.2180960052631341e+268 < (* (/ (/ (* M D) (* 2 d)) l) (* (/ (* h D) d) (/ M 2))) < -34009522142315936.0

    1. Initial program 21.1

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

    2. Initial simplification20.8

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

    4. Applied div-inv20.8

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

    5. Applied times-frac0.6

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

    6. Simplified1.1

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

    if -34009522142315936.0 < (* (/ (/ (* M D) (* 2 d)) l) (* (/ (* h D) d) (/ M 2)))

    1. Initial program 5.8

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

    2. Initial simplification5.3

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

    4. Applied associate-/l*4.5

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

    6. Applied div-inv4.5

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

    7. Applied associate-/l*0.9

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

    9. Applied times-frac1.4

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

    11. Applied times-frac0.9

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

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{D \cdot h}{d} \cdot \frac{M}{2}\right) \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \le -1.2180960052631341 \cdot 10^{+268}:\\ \;\;\;\;\left(\sqrt{\sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\frac{\ell}{h}}{\frac{M \cdot D}{2 \cdot d}}}}} \cdot \sqrt{\sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\frac{\ell}{h}}{\frac{M \cdot D}{2 \cdot d}}}}}\right) \cdot w0\\ \mathbf{elif}\;\left(\frac{D \cdot h}{d} \cdot \frac{M}{2}\right) \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \le -34009522142315936.0:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D \cdot h}{d} \cdot \frac{M}{2}\right) \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{D}{d} \cdot \frac{M}{2}}{\frac{\ell}{\frac{\frac{D}{d} \cdot \frac{M}{2}}{\frac{1}{h}}}}}\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018225 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))