Average Error: 58.0 → 32.6
Time: 46.2s
Precision: 64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) \le -3.3911979403647276 \cdot 10^{-210}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\mathsf{fma}\left(\sqrt{M + \frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}}, \sqrt{\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h} - M}, \frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(0\right)\\ \end{array}\]
\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}\;\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 -3.3911979403647276 \cdot 10^{-210}:\\
\;\;\;\;\frac{c0}{w} \cdot \frac{\mathsf{fma}\left(\sqrt{M + \frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}}, \sqrt{\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h} - M}, \frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\left(0\right)\\

\end{array}
double f(double c0, double w, double h, double D, double d, double M) {
        double r2757955 = c0;
        double r2757956 = 2.0;
        double r2757957 = w;
        double r2757958 = r2757956 * r2757957;
        double r2757959 = r2757955 / r2757958;
        double r2757960 = d;
        double r2757961 = r2757960 * r2757960;
        double r2757962 = r2757955 * r2757961;
        double r2757963 = h;
        double r2757964 = r2757957 * r2757963;
        double r2757965 = D;
        double r2757966 = r2757965 * r2757965;
        double r2757967 = r2757964 * r2757966;
        double r2757968 = r2757962 / r2757967;
        double r2757969 = r2757968 * r2757968;
        double r2757970 = M;
        double r2757971 = r2757970 * r2757970;
        double r2757972 = r2757969 - r2757971;
        double r2757973 = sqrt(r2757972);
        double r2757974 = r2757968 + r2757973;
        double r2757975 = r2757959 * r2757974;
        return r2757975;
}
↓
double f(double c0, double w, double h, double D, double d, double M) {
        double r2757976 = c0;
        double r2757977 = w;
        double r2757978 = 2.0;
        double r2757979 = r2757977 * r2757978;
        double r2757980 = r2757976 / r2757979;
        double r2757981 = d;
        double r2757982 = r2757981 * r2757981;
        double r2757983 = r2757976 * r2757982;
        double r2757984 = D;
        double r2757985 = r2757984 * r2757984;
        double r2757986 = h;
        double r2757987 = r2757977 * r2757986;
        double r2757988 = r2757985 * r2757987;
        double r2757989 = r2757983 / r2757988;
        double r2757990 = r2757989 * r2757989;
        double r2757991 = M;
        double r2757992 = r2757991 * r2757991;
        double r2757993 = r2757990 - r2757992;
        double r2757994 = sqrt(r2757993);
        double r2757995 = r2757994 + r2757989;
        double r2757996 = r2757980 * r2757995;
        double r2757997 = -3.3911979403647276e-210;
        bool r2757998 = r2757996 <= r2757997;
        double r2757999 = r2757976 / r2757977;
        double r2758000 = r2757981 / r2757984;
        double r2758001 = r2758000 * r2758000;
        double r2758002 = r2758001 / r2757986;
        double r2758003 = r2757999 * r2758002;
        double r2758004 = r2757991 + r2758003;
        double r2758005 = sqrt(r2758004);
        double r2758006 = r2758003 - r2757991;
        double r2758007 = sqrt(r2758006);
        double r2758008 = fma(r2758005, r2758007, r2758003);
        double r2758009 = r2758008 / r2757978;
        double r2758010 = r2757999 * r2758009;
        double r2758011 = 0.0;
        double r2758012 = /* ERROR: no posit support in C */;
        double r2758013 = /* ERROR: no posit support in C */;
        double r2758014 = r2757998 ? r2758010 : r2758013;
        return r2758014;
}

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 (* (/ 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))))) < -3.3911979403647276e-210

    1. Initial program 48.1

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

    2. Simplified44.4

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

    4. Applied difference-of-squares44.4

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

    5. Applied sqrt-prod40.6

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

    6. Applied fma-def40.6

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

    if -3.3911979403647276e-210 < (* (/ 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 58.7

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

    2. Simplified53.4

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

    3. Taylor expanded around inf 33.7

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

    5. Applied insert-posit1633.7

      \[\leadsto \color{blue}{\left(\left(\frac{c0}{w} \cdot \frac{0}{2}\right)\right)}\]

    6. Simplified32.0

      \[\leadsto \color{blue}{\left(0\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification32.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) \le -3.3911979403647276 \cdot 10^{-210}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\mathsf{fma}\left(\sqrt{M + \frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}}, \sqrt{\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h} - M}, \frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(0\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +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))))))