Average Error: 58.5 → 32.2
Time: 1.0m
Precision: 64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) \le 9.221058218589873 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(\sqrt{\left(\frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{h \cdot \frac{D}{d}} - M\right) \cdot \left(M + \frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{h \cdot \frac{D}{d}}\right)} + \frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{h \cdot \frac{D}{d}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\frac{c0}{2 \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}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) \le 9.221058218589873 \cdot 10^{+88}:\\
\;\;\;\;\frac{\frac{c0}{w} \cdot \left(\sqrt{\left(\frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{h \cdot \frac{D}{d}} - M\right) \cdot \left(M + \frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{h \cdot \frac{D}{d}}\right)} + \frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{h \cdot \frac{D}{d}}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double c0, double w, double h, double D, double d, double M) {
        double r5709287 = c0;
        double r5709288 = 2.0;
        double r5709289 = w;
        double r5709290 = r5709288 * r5709289;
        double r5709291 = r5709287 / r5709290;
        double r5709292 = d;
        double r5709293 = r5709292 * r5709292;
        double r5709294 = r5709287 * r5709293;
        double r5709295 = h;
        double r5709296 = r5709289 * r5709295;
        double r5709297 = D;
        double r5709298 = r5709297 * r5709297;
        double r5709299 = r5709296 * r5709298;
        double r5709300 = r5709294 / r5709299;
        double r5709301 = r5709300 * r5709300;
        double r5709302 = M;
        double r5709303 = r5709302 * r5709302;
        double r5709304 = r5709301 - r5709303;
        double r5709305 = sqrt(r5709304);
        double r5709306 = r5709300 + r5709305;
        double r5709307 = r5709291 * r5709306;
        return r5709307;
}
↓
double f(double c0, double w, double h, double D, double d, double M) {
        double r5709308 = c0;
        double r5709309 = w;
        double r5709310 = 2.0;
        double r5709311 = r5709309 * r5709310;
        double r5709312 = r5709308 / r5709311;
        double r5709313 = d;
        double r5709314 = r5709313 * r5709313;
        double r5709315 = r5709308 * r5709314;
        double r5709316 = D;
        double r5709317 = r5709316 * r5709316;
        double r5709318 = h;
        double r5709319 = r5709309 * r5709318;
        double r5709320 = r5709317 * r5709319;
        double r5709321 = r5709315 / r5709320;
        double r5709322 = r5709321 * r5709321;
        double r5709323 = M;
        double r5709324 = r5709323 * r5709323;
        double r5709325 = r5709322 - r5709324;
        double r5709326 = sqrt(r5709325);
        double r5709327 = r5709326 + r5709321;
        double r5709328 = r5709312 * r5709327;
        double r5709329 = 9.221058218589873e+88;
        bool r5709330 = r5709328 <= r5709329;
        double r5709331 = r5709308 / r5709309;
        double r5709332 = r5709316 / r5709313;
        double r5709333 = r5709331 / r5709332;
        double r5709334 = r5709318 * r5709332;
        double r5709335 = r5709333 / r5709334;
        double r5709336 = r5709335 - r5709323;
        double r5709337 = r5709323 + r5709335;
        double r5709338 = r5709336 * r5709337;
        double r5709339 = sqrt(r5709338);
        double r5709340 = r5709339 + r5709335;
        double r5709341 = r5709331 * r5709340;
        double r5709342 = r5709341 / r5709310;
        double r5709343 = 0.0;
        double r5709344 = r5709330 ? r5709342 : r5709343;
        return r5709344;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (* (/ c0 (* 2 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))))) < 9.221058218589873e+88

    1. Initial program 37.0

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

    2. Simplified38.4

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(M + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}\right) \cdot \left(\frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h} - M\right)} + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}\right) \cdot \frac{c0}{w}}{2}}\]
    3. Using strategy rm

    4. Applied associate-*r*40.6

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

    6. Applied *-un-lft-identity40.6

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

    7. Applied associate-*r*40.6

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

    8. Simplified34.6

      \[\leadsto \frac{\color{blue}{\left(\frac{\frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{\frac{D}{d}}}{h} + \sqrt{\frac{\frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{\frac{D}{d}}}{h} \cdot \frac{\frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{\frac{D}{d}}}{h} - M \cdot M}\right)} \cdot \frac{c0}{w}}{2}\]
    9. Using strategy rm

    10. Applied *-un-lft-identity34.6

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

    11. Applied associate-*r*34.6

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

    12. Simplified33.0

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

    if 9.221058218589873e+88 < (* (/ c0 (* 2 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)))))

    1. Initial program 62.4

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

    2. Simplified55.0

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(M + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}\right) \cdot \left(\frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h} - M\right)} + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}\right) \cdot \frac{c0}{w}}{2}}\]
    3. Using strategy rm

    4. Applied associate-*r*55.5

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

    5. Taylor expanded around -inf 32.1

      \[\leadsto \frac{\color{blue}{0}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification32.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) \le 9.221058218589873 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(\sqrt{\left(\frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{h \cdot \frac{D}{d}} - M\right) \cdot \left(M + \frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{h \cdot \frac{D}{d}}\right)} + \frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{h \cdot \frac{D}{d}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  (* (/ c0 (* 2 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))))))