Average Error: 59.3 → 34.2
Time: 10.0s
Precision: 64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;D \le 25187178024456.7617 \lor \neg \left(D \le 1.6551284850540221 \cdot 10^{33}\right):\\ \;\;\;\;\frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot d}{D \cdot D} \cdot \frac{d}{w \cdot h} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M} \cdot \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M}\right)\\ \end{array}\]
\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}\;D \le 25187178024456.7617 \lor \neg \left(D \le 1.6551284850540221 \cdot 10^{33}\right):\\
\;\;\;\;\frac{0}{2 \cdot w}\\

\mathbf{else}:\\
\;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot d}{D \cdot D} \cdot \frac{d}{w \cdot h} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M} \cdot \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M}\right)\\

\end{array}
double code(double c0, double w, double h, double D, double d, double M) {
	return ((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)))));
}
↓
double code(double c0, double w, double h, double D, double d, double M) {
	double VAR;
	if (((D <= 25187178024456.76) || !(D <= 1.655128485054022e+33))) {
		VAR = (0.0 / (2.0 * w));
	} else {
		VAR = ((c0 / (2.0 * w)) * ((((c0 * d) / (D * D)) * (d / (w * h))) + (sqrt((((c0 * (d * d)) / ((w * h) * (D * D))) + M)) * sqrt((((c0 * (d * d)) / ((w * h) * (D * D))) - M)))));
	}
	return VAR;
}

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 D < 25187178024456.76 or 1.655128485054022e+33 < D

    1. Initial program 59.4

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

    2. Taylor expanded around inf 35.7

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0}\]
    3. Using strategy rm

    4. Applied associate-*l/33.9

      \[\leadsto \color{blue}{\frac{c0 \cdot 0}{2 \cdot w}}\]

    5. Simplified33.9

      \[\leadsto \frac{\color{blue}{0}}{2 \cdot w}\]

    if 25187178024456.76 < D < 1.655128485054022e+33

    1. Initial program 52.4

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

    3. Applied *-commutative52.4

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

    4. Applied associate-*r*52.9

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

    5. Applied times-frac54.0

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

    7. Applied difference-of-squares54.0

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

    8. Applied sqrt-prod53.4

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot d}{D \cdot D} \cdot \frac{d}{w \cdot h} + \color{blue}{\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M} \cdot \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification34.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;D \le 25187178024456.7617 \lor \neg \left(D \le 1.6551284850540221 \cdot 10^{33}\right):\\ \;\;\;\;\frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot d}{D \cdot D} \cdot \frac{d}{w \cdot h} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M} \cdot \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ 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))))))