Henrywood and Agarwal, Equation (13)

Average Error: 58.1 → 54.8

Time: 47.5s

Precision: 64

• could not determine a ground truth for program body

  1. c0 = -9.06417075773683e+295
  2. w = -1.1185926496885313e+79
  3. h = -1.3378188464547656e+264
  4. D = 2.40245628775927e-284
  5. d = 1.1163916175002776e+249
  6. M = 3.1891419124655103e+174

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c0 \le 5.43376491402612 \cdot 10^{-292}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\left(\sqrt[3]{\frac{c0 \cdot \frac{d}{D}}{w} \cdot \frac{\frac{d}{D}}{h} + \sqrt{\left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} - M\right) \cdot \left(M + \frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w}\right)}} \cdot \sqrt[3]{\frac{c0 \cdot \frac{d}{D}}{w} \cdot \frac{\frac{d}{D}}{h} + \sqrt{\left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} - M\right) \cdot \left(M + \frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w}\right)}}\right) \cdot \sqrt[3]{\frac{c0 \cdot \frac{d}{D}}{w} \cdot \frac{\frac{d}{D}}{h} + \sqrt{\left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} - M\right) \cdot \left(M + \frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w}\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0 \cdot \frac{d}{D}}{w} \cdot \frac{\frac{d}{D}}{h} + \sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} - M} \cdot \sqrt{M + \frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w}}}{2} \cdot \frac{c0}{w}\\ \end{array}\]

C

double f(double c0, double w, double h, double D, double d, double M) {
        double r2801911 = c0;
        double r2801912 = 2.0;
        double r2801913 = w;
        double r2801914 = r2801912 * r2801913;
        double r2801915 = r2801911 / r2801914;
        double r2801916 = d;
        double r2801917 = r2801916 * r2801916;
        double r2801918 = r2801911 * r2801917;
        double r2801919 = h;
        double r2801920 = r2801913 * r2801919;
        double r2801921 = D;
        double r2801922 = r2801921 * r2801921;
        double r2801923 = r2801920 * r2801922;
        double r2801924 = r2801918 / r2801923;
        double r2801925 = r2801924 * r2801924;
        double r2801926 = M;
        double r2801927 = r2801926 * r2801926;
        double r2801928 = r2801925 - r2801927;
        double r2801929 = sqrt(r2801928);
        double r2801930 = r2801924 + r2801929;
        double r2801931 = r2801915 * r2801930;
        return r2801931;
}

↓

double f(double c0, double w, double h, double D, double d, double M) {
        double r2801932 = c0;
        double r2801933 = 5.43376491402612e-292;
        bool r2801934 = r2801932 <= r2801933;
        double r2801935 = w;
        double r2801936 = r2801932 / r2801935;
        double r2801937 = d;
        double r2801938 = D;
        double r2801939 = r2801937 / r2801938;
        double r2801940 = r2801932 * r2801939;
        double r2801941 = r2801940 / r2801935;
        double r2801942 = h;
        double r2801943 = r2801939 / r2801942;
        double r2801944 = r2801941 * r2801943;
        double r2801945 = r2801939 * r2801939;
        double r2801946 = r2801945 / r2801942;
        double r2801947 = r2801946 * r2801936;
        double r2801948 = M;
        double r2801949 = r2801947 - r2801948;
        double r2801950 = r2801948 + r2801947;
        double r2801951 = r2801949 * r2801950;
        double r2801952 = sqrt(r2801951);
        double r2801953 = r2801944 + r2801952;
        double r2801954 = cbrt(r2801953);
        double r2801955 = r2801954 * r2801954;
        double r2801956 = r2801955 * r2801954;
        double r2801957 = 2.0;
        double r2801958 = r2801956 / r2801957;
        double r2801959 = r2801936 * r2801958;
        double r2801960 = sqrt(r2801949);
        double r2801961 = sqrt(r2801950);
        double r2801962 = r2801960 * r2801961;
        double r2801963 = r2801944 + r2801962;
        double r2801964 = r2801963 / r2801957;
        double r2801965 = r2801964 * r2801936;
        double r2801966 = r2801934 ? r2801959 : r2801965;
        return r2801966;
}

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 < 5.43376491402612e-292

   1. Initial program 58.2

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

   2. Simplified 52.7

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

   4. Applied *-un-lft-identity 52.7

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

   5. Applied times-frac 53.5

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

   6. Applied associate-*r* 53.9

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

   7. Simplified 54.3

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

   9. Applied add-cube-cbrt 54.4

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

   if 5.43376491402612e-292 < c0

   1. Initial program 58.0

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

   2. Simplified 52.6

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

   4. Applied *-un-lft-identity 52.6

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

   5. Applied times-frac 53.8

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

   6. Applied associate-*r* 54.2

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

   7. Simplified 54.5

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

   9. Applied sqrt-prod 55.2

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

4. Final simplification 54.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \le 5.43376491402612 \cdot 10^{-292}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\left(\sqrt[3]{\frac{c0 \cdot \frac{d}{D}}{w} \cdot \frac{\frac{d}{D}}{h} + \sqrt{\left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} - M\right) \cdot \left(M + \frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w}\right)}} \cdot \sqrt[3]{\frac{c0 \cdot \frac{d}{D}}{w} \cdot \frac{\frac{d}{D}}{h} + \sqrt{\left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} - M\right) \cdot \left(M + \frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w}\right)}}\right) \cdot \sqrt[3]{\frac{c0 \cdot \frac{d}{D}}{w} \cdot \frac{\frac{d}{D}}{h} + \sqrt{\left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} - M\right) \cdot \left(M + \frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w}\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0 \cdot \frac{d}{D}}{w} \cdot \frac{\frac{d}{D}}{h} + \sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} - M} \cdot \sqrt{M + \frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w}}}{2} \cdot \frac{c0}{w}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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))))))