Henrywood and Agarwal, Equation (13)

Average Error: 59.0 → 49.3

Time: 1.5m

Precision: 64

• could not determine a ground truth for program body

  1. c0 = 4.380842523620344e+108
  2. w = -2.3456387962029125e-288
  3. h = 1.164384249543754e-229
  4. D = 4.448914333296353e+21
  5. d = -2.0044150779320932e+149
  6. M = -4.827416740651301e-120

Math

\[\frac{c0}{2.0 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;w \le -1.4409442624034558 \cdot 10^{+204}:\\ \;\;\;\;\frac{\left(\frac{d}{D} \cdot \left(c0 \cdot \frac{\frac{2}{h}}{w}\right)\right) \cdot \frac{\frac{d}{D} \cdot c0}{w}}{2.0}\\ \mathbf{elif}\;w \le -1.9090683015159143 \cdot 10^{+115}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w} - M\right) \cdot \left(M + \frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w}\right)} + \frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w}} \cdot \left(\sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w} - M\right) \cdot \left(M + \frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w}\right)} + \frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w}} \cdot \sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w} - M\right) \cdot \left(M + \frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w}\right)} + \frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w}}\right)}{w} \cdot \frac{c0}{2.0}\\ \mathbf{elif}\;w \le -1.3308495693303094 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{1}{\frac{w}{\frac{d \cdot c0}{D} \cdot \left(\frac{\frac{2}{h}}{w} \cdot \frac{d \cdot c0}{D}\right)}}}{2.0}\\ \mathbf{elif}\;w \le 8.433285243664178 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \left(\frac{1}{w} \cdot \left(\frac{2}{h} \cdot \frac{d \cdot c0}{D}\right)\right)}{w}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(d \cdot c0\right) \cdot \left(\frac{2}{h} \cdot \left(d \cdot c0\right)\right)}{\left(w \cdot D\right) \cdot \left(w \cdot D\right)}}{2.0}\\ \end{array}\]

C

double f(double c0, double w, double h, double D, double d, double M) {
        double r9649509 = c0;
        double r9649510 = 2.0;
        double r9649511 = w;
        double r9649512 = r9649510 * r9649511;
        double r9649513 = r9649509 / r9649512;
        double r9649514 = d;
        double r9649515 = r9649514 * r9649514;
        double r9649516 = r9649509 * r9649515;
        double r9649517 = h;
        double r9649518 = r9649511 * r9649517;
        double r9649519 = D;
        double r9649520 = r9649519 * r9649519;
        double r9649521 = r9649518 * r9649520;
        double r9649522 = r9649516 / r9649521;
        double r9649523 = r9649522 * r9649522;
        double r9649524 = M;
        double r9649525 = r9649524 * r9649524;
        double r9649526 = r9649523 - r9649525;
        double r9649527 = sqrt(r9649526);
        double r9649528 = r9649522 + r9649527;
        double r9649529 = r9649513 * r9649528;
        return r9649529;
}

↓

double f(double c0, double w, double h, double D, double d, double M) {
        double r9649530 = w;
        double r9649531 = -1.4409442624034558e+204;
        bool r9649532 = r9649530 <= r9649531;
        double r9649533 = d;
        double r9649534 = D;
        double r9649535 = r9649533 / r9649534;
        double r9649536 = c0;
        double r9649537 = 2.0;
        double r9649538 = h;
        double r9649539 = r9649537 / r9649538;
        double r9649540 = r9649539 / r9649530;
        double r9649541 = r9649536 * r9649540;
        double r9649542 = r9649535 * r9649541;
        double r9649543 = r9649535 * r9649536;
        double r9649544 = r9649543 / r9649530;
        double r9649545 = r9649542 * r9649544;
        double r9649546 = 2.0;
        double r9649547 = r9649545 / r9649546;
        double r9649548 = -1.9090683015159143e+115;
        bool r9649549 = r9649530 <= r9649548;
        double r9649550 = r9649535 * r9649543;
        double r9649551 = r9649538 * r9649530;
        double r9649552 = r9649550 / r9649551;
        double r9649553 = M;
        double r9649554 = r9649552 - r9649553;
        double r9649555 = r9649553 + r9649552;
        double r9649556 = r9649554 * r9649555;
        double r9649557 = sqrt(r9649556);
        double r9649558 = r9649557 + r9649552;
        double r9649559 = cbrt(r9649558);
        double r9649560 = r9649559 * r9649559;
        double r9649561 = r9649559 * r9649560;
        double r9649562 = r9649561 / r9649530;
        double r9649563 = r9649536 / r9649546;
        double r9649564 = r9649562 * r9649563;
        double r9649565 = -1.3308495693303094e-88;
        bool r9649566 = r9649530 <= r9649565;
        double r9649567 = 1.0;
        double r9649568 = r9649533 * r9649536;
        double r9649569 = r9649568 / r9649534;
        double r9649570 = r9649540 * r9649569;
        double r9649571 = r9649569 * r9649570;
        double r9649572 = r9649530 / r9649571;
        double r9649573 = r9649567 / r9649572;
        double r9649574 = r9649573 / r9649546;
        double r9649575 = 8.433285243664178e+80;
        bool r9649576 = r9649530 <= r9649575;
        double r9649577 = r9649567 / r9649530;
        double r9649578 = r9649539 * r9649569;
        double r9649579 = r9649577 * r9649578;
        double r9649580 = r9649543 * r9649579;
        double r9649581 = r9649580 / r9649530;
        double r9649582 = r9649581 / r9649546;
        double r9649583 = r9649539 * r9649568;
        double r9649584 = r9649568 * r9649583;
        double r9649585 = r9649530 * r9649534;
        double r9649586 = r9649585 * r9649585;
        double r9649587 = r9649584 / r9649586;
        double r9649588 = r9649587 / r9649546;
        double r9649589 = r9649576 ? r9649582 : r9649588;
        double r9649590 = r9649566 ? r9649574 : r9649589;
        double r9649591 = r9649549 ? r9649564 : r9649590;
        double r9649592 = r9649532 ? r9649547 : r9649591;
        return r9649592;
}

Error

Bits error versus c0

Bits error versus w

Bits error versus h

Bits error versus D

Bits error versus d

Bits error versus M

Derivation

1. Split input into 5 regimes

2. if w < -1.4409442624034558e+204

   1. Initial program 57.1

      \[\frac{c0}{2.0 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

   2. Simplified 52.0

      \[\leadsto \color{blue}{\frac{c0}{2.0} \cdot \frac{\sqrt{\left(M + \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h}}{w}}\]

   3. Taylor expanded around 0 57.1

      \[\leadsto \frac{c0}{2.0} \cdot \frac{\color{blue}{2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{w}\]

   4. Simplified 46.7

      \[\leadsto \frac{c0}{2.0} \cdot \frac{\color{blue}{\frac{2}{\frac{w \cdot h}{\frac{d}{D}}} \cdot \left(c0 \cdot \frac{d}{D}\right)}}{w}\]
   5. Using strategy rm

   6. Applied associate-*l/ 46.7

      \[\leadsto \color{blue}{\frac{c0 \cdot \frac{\frac{2}{\frac{w \cdot h}{\frac{d}{D}}} \cdot \left(c0 \cdot \frac{d}{D}\right)}{w}}{2.0}}\]

   7. Simplified 47.8

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{2}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}}{2.0}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 47.8

      \[\leadsto \frac{\frac{\left(\frac{\color{blue}{1 \cdot 2}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}{2.0}\]

   10. Applied times-frac 47.7

       \[\leadsto \frac{\frac{\left(\color{blue}{\left(\frac{1}{w} \cdot \frac{2}{h}\right)} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}{2.0}\]

   11. Applied associate-*l* 50.0

       \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{w} \cdot \left(\frac{2}{h} \cdot \left(\frac{d}{D} \cdot c0\right)\right)\right)} \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}{2.0}\]
   12. Using strategy rm

   13. Applied *-un-lft-identity 50.0

       \[\leadsto \frac{\frac{\left(\frac{1}{w} \cdot \left(\frac{2}{h} \cdot \left(\frac{d}{D} \cdot c0\right)\right)\right) \cdot \left(\frac{d}{D} \cdot c0\right)}{\color{blue}{1 \cdot w}}}{2.0}\]

   14. Applied times-frac 49.7

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{w} \cdot \left(\frac{2}{h} \cdot \left(\frac{d}{D} \cdot c0\right)\right)}{1} \cdot \frac{\frac{d}{D} \cdot c0}{w}}}{2.0}\]

   15. Simplified 44.2

       \[\leadsto \frac{\color{blue}{\left(\left(\frac{\frac{2}{h}}{w} \cdot c0\right) \cdot \frac{d}{D}\right)} \cdot \frac{\frac{d}{D} \cdot c0}{w}}{2.0}\]

   if -1.4409442624034558e+204 < w < -1.9090683015159143e+115

   1. Initial program 57.5

      \[\frac{c0}{2.0 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

   2. Simplified 50.0

      \[\leadsto \color{blue}{\frac{c0}{2.0} \cdot \frac{\sqrt{\left(M + \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h}}{w}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 50.1

      \[\leadsto \frac{c0}{2.0} \cdot \frac{\color{blue}{\left(\sqrt[3]{\sqrt{\left(M + \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h}} \cdot \sqrt[3]{\sqrt{\left(M + \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h}}\right) \cdot \sqrt[3]{\sqrt{\left(M + \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h}}}}{w}\]

   if -1.9090683015159143e+115 < w < -1.3308495693303094e-88

   1. Initial program 59.1

      \[\frac{c0}{2.0 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

   2. Simplified 50.0

      \[\leadsto \color{blue}{\frac{c0}{2.0} \cdot \frac{\sqrt{\left(M + \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h}}{w}}\]

   3. Taylor expanded around 0 59.0

      \[\leadsto \frac{c0}{2.0} \cdot \frac{\color{blue}{2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{w}\]

   4. Simplified 49.2

      \[\leadsto \frac{c0}{2.0} \cdot \frac{\color{blue}{\frac{2}{\frac{w \cdot h}{\frac{d}{D}}} \cdot \left(c0 \cdot \frac{d}{D}\right)}}{w}\]
   5. Using strategy rm

   6. Applied associate-*l/ 49.2

      \[\leadsto \color{blue}{\frac{c0 \cdot \frac{\frac{2}{\frac{w \cdot h}{\frac{d}{D}}} \cdot \left(c0 \cdot \frac{d}{D}\right)}{w}}{2.0}}\]

   7. Simplified 47.5

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{2}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}}{2.0}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 47.5

      \[\leadsto \frac{\frac{\left(\frac{\color{blue}{1 \cdot 2}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}{2.0}\]

   10. Applied times-frac 47.5

       \[\leadsto \frac{\frac{\left(\color{blue}{\left(\frac{1}{w} \cdot \frac{2}{h}\right)} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}{2.0}\]

   11. Applied associate-*l* 47.1

       \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{w} \cdot \left(\frac{2}{h} \cdot \left(\frac{d}{D} \cdot c0\right)\right)\right)} \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}{2.0}\]
   12. Using strategy rm

   13. Applied clear-num 47.1

       \[\leadsto \frac{\color{blue}{\frac{1}{\frac{w}{\left(\frac{1}{w} \cdot \left(\frac{2}{h} \cdot \left(\frac{d}{D} \cdot c0\right)\right)\right) \cdot \left(\frac{d}{D} \cdot c0\right)}}}}{2.0}\]

   14. Simplified 46.7

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{w}{\left(\frac{c0 \cdot d}{D} \cdot \frac{\frac{2}{h}}{w}\right) \cdot \frac{c0 \cdot d}{D}}}}}{2.0}\]

   if -1.3308495693303094e-88 < w < 8.433285243664178e+80

   1. Initial program 60.1

      \[\frac{c0}{2.0 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

   2. Simplified 55.2

      \[\leadsto \color{blue}{\frac{c0}{2.0} \cdot \frac{\sqrt{\left(M + \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h}}{w}}\]

   3. Taylor expanded around 0 61.0

      \[\leadsto \frac{c0}{2.0} \cdot \frac{\color{blue}{2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{w}\]

   4. Simplified 56.1

      \[\leadsto \frac{c0}{2.0} \cdot \frac{\color{blue}{\frac{2}{\frac{w \cdot h}{\frac{d}{D}}} \cdot \left(c0 \cdot \frac{d}{D}\right)}}{w}\]
   5. Using strategy rm

   6. Applied associate-*l/ 56.0

      \[\leadsto \color{blue}{\frac{c0 \cdot \frac{\frac{2}{\frac{w \cdot h}{\frac{d}{D}}} \cdot \left(c0 \cdot \frac{d}{D}\right)}{w}}{2.0}}\]

   7. Simplified 53.7

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{2}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}}{2.0}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 53.7

      \[\leadsto \frac{\frac{\left(\frac{\color{blue}{1 \cdot 2}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}{2.0}\]

   10. Applied times-frac 53.7

       \[\leadsto \frac{\frac{\left(\color{blue}{\left(\frac{1}{w} \cdot \frac{2}{h}\right)} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}{2.0}\]

   11. Applied associate-*l* 51.5

       \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{w} \cdot \left(\frac{2}{h} \cdot \left(\frac{d}{D} \cdot c0\right)\right)\right)} \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}{2.0}\]
   12. Using strategy rm

   13. Applied associate-*l/ 52.1

       \[\leadsto \frac{\frac{\left(\frac{1}{w} \cdot \left(\frac{2}{h} \cdot \color{blue}{\frac{d \cdot c0}{D}}\right)\right) \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}{2.0}\]

   if 8.433285243664178e+80 < w

   1. Initial program 57.6

      \[\frac{c0}{2.0 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

   2. Simplified 49.9

      \[\leadsto \color{blue}{\frac{c0}{2.0} \cdot \frac{\sqrt{\left(M + \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{w \cdot h}}{w}}\]

   3. Taylor expanded around 0 57.3

      \[\leadsto \frac{c0}{2.0} \cdot \frac{\color{blue}{2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{w}\]

   4. Simplified 46.9

      \[\leadsto \frac{c0}{2.0} \cdot \frac{\color{blue}{\frac{2}{\frac{w \cdot h}{\frac{d}{D}}} \cdot \left(c0 \cdot \frac{d}{D}\right)}}{w}\]
   5. Using strategy rm

   6. Applied associate-*l/ 46.9

      \[\leadsto \color{blue}{\frac{c0 \cdot \frac{\frac{2}{\frac{w \cdot h}{\frac{d}{D}}} \cdot \left(c0 \cdot \frac{d}{D}\right)}{w}}{2.0}}\]

   7. Simplified 48.3

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{2}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}}{2.0}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 48.3

      \[\leadsto \frac{\frac{\left(\frac{\color{blue}{1 \cdot 2}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}{2.0}\]

   10. Applied times-frac 48.3

       \[\leadsto \frac{\frac{\left(\color{blue}{\left(\frac{1}{w} \cdot \frac{2}{h}\right)} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}{2.0}\]

   11. Applied associate-*l* 49.6

       \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{w} \cdot \left(\frac{2}{h} \cdot \left(\frac{d}{D} \cdot c0\right)\right)\right)} \cdot \left(\frac{d}{D} \cdot c0\right)}{w}}{2.0}\]
   12. Using strategy rm

   13. Applied associate-*l/ 50.2

       \[\leadsto \frac{\frac{\left(\frac{1}{w} \cdot \left(\frac{2}{h} \cdot \left(\frac{d}{D} \cdot c0\right)\right)\right) \cdot \color{blue}{\frac{d \cdot c0}{D}}}{w}}{2.0}\]

   14. Applied associate-*l/ 47.6

       \[\leadsto \frac{\frac{\left(\frac{1}{w} \cdot \left(\frac{2}{h} \cdot \color{blue}{\frac{d \cdot c0}{D}}\right)\right) \cdot \frac{d \cdot c0}{D}}{w}}{2.0}\]

   15. Applied associate-*r/ 48.7

       \[\leadsto \frac{\frac{\left(\frac{1}{w} \cdot \color{blue}{\frac{\frac{2}{h} \cdot \left(d \cdot c0\right)}{D}}\right) \cdot \frac{d \cdot c0}{D}}{w}}{2.0}\]

   16. Applied frac-times 47.2

       \[\leadsto \frac{\frac{\color{blue}{\frac{1 \cdot \left(\frac{2}{h} \cdot \left(d \cdot c0\right)\right)}{w \cdot D}} \cdot \frac{d \cdot c0}{D}}{w}}{2.0}\]

   17. Applied frac-times 50.3

       \[\leadsto \frac{\frac{\color{blue}{\frac{\left(1 \cdot \left(\frac{2}{h} \cdot \left(d \cdot c0\right)\right)\right) \cdot \left(d \cdot c0\right)}{\left(w \cdot D\right) \cdot D}}}{w}}{2.0}\]

   18. Applied associate-/l/ 48.6

       \[\leadsto \frac{\color{blue}{\frac{\left(1 \cdot \left(\frac{2}{h} \cdot \left(d \cdot c0\right)\right)\right) \cdot \left(d \cdot c0\right)}{w \cdot \left(\left(w \cdot D\right) \cdot D\right)}}}{2.0}\]

   19. Simplified 45.9

       \[\leadsto \frac{\frac{\left(1 \cdot \left(\frac{2}{h} \cdot \left(d \cdot c0\right)\right)\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(w \cdot D\right) \cdot \left(w \cdot D\right)}}}{2.0}\]
3. Recombined 5 regimes into one program.

4. Final simplification 49.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \le -1.4409442624034558 \cdot 10^{+204}:\\ \;\;\;\;\frac{\left(\frac{d}{D} \cdot \left(c0 \cdot \frac{\frac{2}{h}}{w}\right)\right) \cdot \frac{\frac{d}{D} \cdot c0}{w}}{2.0}\\ \mathbf{elif}\;w \le -1.9090683015159143 \cdot 10^{+115}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w} - M\right) \cdot \left(M + \frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w}\right)} + \frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w}} \cdot \left(\sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w} - M\right) \cdot \left(M + \frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w}\right)} + \frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w}} \cdot \sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w} - M\right) \cdot \left(M + \frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w}\right)} + \frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w}}\right)}{w} \cdot \frac{c0}{2.0}\\ \mathbf{elif}\;w \le -1.3308495693303094 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{1}{\frac{w}{\frac{d \cdot c0}{D} \cdot \left(\frac{\frac{2}{h}}{w} \cdot \frac{d \cdot c0}{D}\right)}}}{2.0}\\ \mathbf{elif}\;w \le 8.433285243664178 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \left(\frac{1}{w} \cdot \left(\frac{2}{h} \cdot \frac{d \cdot c0}{D}\right)\right)}{w}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(d \cdot c0\right) \cdot \left(\frac{2}{h} \cdot \left(d \cdot c0\right)\right)}{\left(w \cdot D\right) \cdot \left(w \cdot D\right)}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  (* (/ c0 (* 2.0 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))