Average Error: 13.1 → 7.9
Time: 1.6m
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\left(\sqrt{\sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \frac{\sqrt[3]{\frac{M \cdot D}{2 \cdot d} \cdot h} \cdot \left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d} \cdot h} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d} \cdot h}\right)}{\ell}}} \cdot \sqrt{\sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}}}\right) \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\left(\sqrt{\sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \frac{\sqrt[3]{\frac{M \cdot D}{2 \cdot d} \cdot h} \cdot \left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d} \cdot h} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d} \cdot h}\right)}{\ell}}} \cdot \sqrt{\sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}}}\right) \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r25160526 = w0;
        double r25160527 = 1.0;
        double r25160528 = M;
        double r25160529 = D;
        double r25160530 = r25160528 * r25160529;
        double r25160531 = 2.0;
        double r25160532 = d;
        double r25160533 = r25160531 * r25160532;
        double r25160534 = r25160530 / r25160533;
        double r25160535 = pow(r25160534, r25160531);
        double r25160536 = h;
        double r25160537 = l;
        double r25160538 = r25160536 / r25160537;
        double r25160539 = r25160535 * r25160538;
        double r25160540 = r25160527 - r25160539;
        double r25160541 = sqrt(r25160540);
        double r25160542 = r25160526 * r25160541;
        return r25160542;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r25160543 = 1.0;
        double r25160544 = M;
        double r25160545 = D;
        double r25160546 = r25160544 * r25160545;
        double r25160547 = 2.0;
        double r25160548 = d;
        double r25160549 = r25160547 * r25160548;
        double r25160550 = r25160546 / r25160549;
        double r25160551 = h;
        double r25160552 = r25160550 * r25160551;
        double r25160553 = cbrt(r25160552);
        double r25160554 = r25160553 * r25160553;
        double r25160555 = r25160553 * r25160554;
        double r25160556 = l;
        double r25160557 = r25160555 / r25160556;
        double r25160558 = r25160550 * r25160557;
        double r25160559 = r25160543 - r25160558;
        double r25160560 = sqrt(r25160559);
        double r25160561 = sqrt(r25160560);
        double r25160562 = r25160552 / r25160556;
        double r25160563 = r25160562 * r25160550;
        double r25160564 = r25160543 - r25160563;
        double r25160565 = sqrt(r25160564);
        double r25160566 = sqrt(r25160565);
        double r25160567 = r25160561 * r25160566;
        double r25160568 = w0;
        double r25160569 = r25160567 * r25160568;
        return r25160569;
}

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 13.1

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

  2. Simplified11.8

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

  4. Applied associate-*l/7.9

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

  6. Applied add-sqr-sqrt7.9

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

  7. Applied sqrt-prod7.9

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

  9. Applied add-cube-cbrt7.9

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

  10. Final simplification7.9

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

Reproduce

herbie shell --seed 2019124 +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))))))