Average Error: 13.4 → 8.0
Time: 26.7s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \left(\frac{\frac{D \cdot M}{2 \cdot d} \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{D \cdot M}{2 \cdot d}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\sqrt{1 - \left(\frac{\frac{D \cdot M}{2 \cdot d} \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{D \cdot M}{2 \cdot d}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r2791380 = w0;
        double r2791381 = 1.0;
        double r2791382 = M;
        double r2791383 = D;
        double r2791384 = r2791382 * r2791383;
        double r2791385 = 2.0;
        double r2791386 = d;
        double r2791387 = r2791385 * r2791386;
        double r2791388 = r2791384 / r2791387;
        double r2791389 = pow(r2791388, r2791385);
        double r2791390 = h;
        double r2791391 = l;
        double r2791392 = r2791390 / r2791391;
        double r2791393 = r2791389 * r2791392;
        double r2791394 = r2791381 - r2791393;
        double r2791395 = sqrt(r2791394);
        double r2791396 = r2791380 * r2791395;
        return r2791396;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r2791397 = 1.0;
        double r2791398 = D;
        double r2791399 = M;
        double r2791400 = r2791398 * r2791399;
        double r2791401 = 2.0;
        double r2791402 = d;
        double r2791403 = r2791401 * r2791402;
        double r2791404 = r2791400 / r2791403;
        double r2791405 = h;
        double r2791406 = cbrt(r2791405);
        double r2791407 = r2791406 * r2791406;
        double r2791408 = r2791404 * r2791407;
        double r2791409 = l;
        double r2791410 = cbrt(r2791409);
        double r2791411 = r2791410 * r2791410;
        double r2791412 = r2791408 / r2791411;
        double r2791413 = r2791412 * r2791404;
        double r2791414 = r2791406 / r2791410;
        double r2791415 = r2791413 * r2791414;
        double r2791416 = r2791397 - r2791415;
        double r2791417 = sqrt(r2791416);
        double r2791418 = w0;
        double r2791419 = r2791417 * r2791418;
        return r2791419;
}

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 13.4

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

  2. Simplified13.4

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

  4. Applied add-cube-cbrt13.5

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

  5. Applied add-cube-cbrt13.5

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

  6. Applied times-frac13.5

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

  7. Applied associate-*r*10.5

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

  8. Simplified8.9

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

  10. Applied frac-times8.9

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

  11. Applied associate-*r/8.0

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

  12. Final simplification8.0

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

Reproduce

herbie shell --seed 2019128 +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))))))