Average Error: 14.2 → 8.2
Time: 31.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\left(\sqrt{\sqrt{1 - \left(\left(\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}} \cdot \sqrt{\sqrt{1 - \left(\left(\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}\right) \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\left(\sqrt{\sqrt{1 - \left(\left(\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}} \cdot \sqrt{\sqrt{1 - \left(\left(\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}\right) \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r4511647 = w0;
        double r4511648 = 1.0;
        double r4511649 = M;
        double r4511650 = D;
        double r4511651 = r4511649 * r4511650;
        double r4511652 = 2.0;
        double r4511653 = d;
        double r4511654 = r4511652 * r4511653;
        double r4511655 = r4511651 / r4511654;
        double r4511656 = pow(r4511655, r4511652);
        double r4511657 = h;
        double r4511658 = l;
        double r4511659 = r4511657 / r4511658;
        double r4511660 = r4511656 * r4511659;
        double r4511661 = r4511648 - r4511660;
        double r4511662 = sqrt(r4511661);
        double r4511663 = r4511647 * r4511662;
        return r4511663;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r4511664 = 1.0;
        double r4511665 = D;
        double r4511666 = M;
        double r4511667 = r4511665 * r4511666;
        double r4511668 = 2.0;
        double r4511669 = d;
        double r4511670 = r4511668 * r4511669;
        double r4511671 = r4511667 / r4511670;
        double r4511672 = h;
        double r4511673 = cbrt(r4511672);
        double r4511674 = l;
        double r4511675 = cbrt(r4511674);
        double r4511676 = r4511673 / r4511675;
        double r4511677 = r4511671 * r4511676;
        double r4511678 = r4511677 * r4511677;
        double r4511679 = r4511678 * r4511676;
        double r4511680 = r4511664 - r4511679;
        double r4511681 = sqrt(r4511680);
        double r4511682 = sqrt(r4511681);
        double r4511683 = r4511682 * r4511682;
        double r4511684 = w0;
        double r4511685 = r4511683 * r4511684;
        return r4511685;
}

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 14.2

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

  2. Simplified14.2

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

  4. Applied add-cube-cbrt14.2

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

  5. Applied add-cube-cbrt14.3

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

  6. Applied times-frac14.3

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

  7. Applied associate-*r*11.0

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

  8. Simplified8.2

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

  10. Applied add-sqr-sqrt8.2

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

  11. Applied sqrt-prod8.2

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

  12. Final simplification8.2

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))