Average Error: 58.0 → 35.4
Time: 1.7m
Precision: 64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;w \le 1.938412383063818 \cdot 10^{-184}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \le 3.2348425285823535 \cdot 10^{-79}:\\ \;\;\;\;(\left(\sqrt{\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{h}}{w} + M}\right) \cdot \left(\sqrt{\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{h}}{w} - M}\right) + \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{h}}{w}\right))_* \cdot \frac{c0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}\;w \le 1.938412383063818 \cdot 10^{-184}:\\
\;\;\;\;0\\

\mathbf{elif}\;w \le 3.2348425285823535 \cdot 10^{-79}:\\
\;\;\;\;(\left(\sqrt{\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{h}}{w} + M}\right) \cdot \left(\sqrt{\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{h}}{w} - M}\right) + \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{h}}{w}\right))_* \cdot \frac{c0}{2 \cdot w}\\

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

\end{array}
double f(double c0, double w, double h, double D, double d, double M) {
        double r43224188 = c0;
        double r43224189 = 2.0;
        double r43224190 = w;
        double r43224191 = r43224189 * r43224190;
        double r43224192 = r43224188 / r43224191;
        double r43224193 = d;
        double r43224194 = r43224193 * r43224193;
        double r43224195 = r43224188 * r43224194;
        double r43224196 = h;
        double r43224197 = r43224190 * r43224196;
        double r43224198 = D;
        double r43224199 = r43224198 * r43224198;
        double r43224200 = r43224197 * r43224199;
        double r43224201 = r43224195 / r43224200;
        double r43224202 = r43224201 * r43224201;
        double r43224203 = M;
        double r43224204 = r43224203 * r43224203;
        double r43224205 = r43224202 - r43224204;
        double r43224206 = sqrt(r43224205);
        double r43224207 = r43224201 + r43224206;
        double r43224208 = r43224192 * r43224207;
        return r43224208;
}
↓
double f(double c0, double w, double h, double D, double d, double M) {
        double r43224209 = w;
        double r43224210 = 1.938412383063818e-184;
        bool r43224211 = r43224209 <= r43224210;
        double r43224212 = 0.0;
        double r43224213 = 3.2348425285823535e-79;
        bool r43224214 = r43224209 <= r43224213;
        double r43224215 = d;
        double r43224216 = D;
        double r43224217 = r43224215 / r43224216;
        double r43224218 = r43224217 * r43224217;
        double r43224219 = c0;
        double r43224220 = h;
        double r43224221 = r43224219 / r43224220;
        double r43224222 = r43224218 * r43224221;
        double r43224223 = r43224222 / r43224209;
        double r43224224 = M;
        double r43224225 = r43224223 + r43224224;
        double r43224226 = sqrt(r43224225);
        double r43224227 = r43224223 - r43224224;
        double r43224228 = sqrt(r43224227);
        double r43224229 = fma(r43224226, r43224228, r43224223);
        double r43224230 = 2.0;
        double r43224231 = r43224230 * r43224209;
        double r43224232 = r43224219 / r43224231;
        double r43224233 = r43224229 * r43224232;
        double r43224234 = r43224214 ? r43224233 : r43224212;
        double r43224235 = r43224211 ? r43224212 : r43224234;
        return r43224235;
}

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 2 regimes

  2. if w < 1.938412383063818e-184 or 3.2348425285823535e-79 < w

    1. Initial program 57.9

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

    2. Simplified53.2

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

    3. Taylor expanded around -inf 35.2

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

    4. Taylor expanded around 0 33.6

      \[\leadsto \color{blue}{0}\]

    if 1.938412383063818e-184 < w < 3.2348425285823535e-79

    1. Initial program 59.1

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

    2. Simplified53.4

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

    4. Applied difference-of-squares53.4

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

    5. Applied sqrt-prod54.0

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

    6. Applied fma-def54.0

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

  4. Final simplification35.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;w \le 1.938412383063818 \cdot 10^{-184}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \le 3.2348425285823535 \cdot 10^{-79}:\\ \;\;\;\;(\left(\sqrt{\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{h}}{w} + M}\right) \cdot \left(\sqrt{\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{h}}{w} - M}\right) + \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{h}}{w}\right))_* \cdot \frac{c0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(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))))))