Average Error: 14.4 → 8.8
Time: 24.5s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - {\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}\right)}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - {\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}\right)}
double f(double w0, double M, double D, double h, double l, double d) {
        double r233104 = w0;
        double r233105 = 1.0;
        double r233106 = M;
        double r233107 = D;
        double r233108 = r233106 * r233107;
        double r233109 = 2.0;
        double r233110 = d;
        double r233111 = r233109 * r233110;
        double r233112 = r233108 / r233111;
        double r233113 = pow(r233112, r233109);
        double r233114 = h;
        double r233115 = l;
        double r233116 = r233114 / r233115;
        double r233117 = r233113 * r233116;
        double r233118 = r233105 - r233117;
        double r233119 = sqrt(r233118);
        double r233120 = r233104 * r233119;
        return r233120;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r233121 = w0;
        double r233122 = 1.0;
        double r233123 = 1.0;
        double r233124 = 2.0;
        double r233125 = d;
        double r233126 = r233124 * r233125;
        double r233127 = M;
        double r233128 = D;
        double r233129 = r233127 * r233128;
        double r233130 = r233126 / r233129;
        double r233131 = r233123 / r233130;
        double r233132 = 2.0;
        double r233133 = r233124 / r233132;
        double r233134 = pow(r233131, r233133);
        double r233135 = r233123 / r233126;
        double r233136 = r233129 * r233135;
        double r233137 = pow(r233136, r233133);
        double r233138 = h;
        double r233139 = r233137 * r233138;
        double r233140 = l;
        double r233141 = r233123 / r233140;
        double r233142 = r233139 * r233141;
        double r233143 = r233134 * r233142;
        double r233144 = r233122 - r233143;
        double r233145 = sqrt(r233144);
        double r233146 = r233121 * r233145;
        return r233146;
}

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. Initial program 14.4

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

  3. Applied sqr-pow14.4

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

  4. Applied associate-*l*12.6

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

  6. Applied div-inv12.6

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

  7. Applied associate-*r*8.8

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

  9. Applied clear-num8.8

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

  11. Applied div-inv8.8

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

  12. Final simplification8.8

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))