Henrywood and Agarwal, Equation (13)

Average Error: 58.1 → 54.8

Time: 59.0s

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 r2882111 = c0;
        double r2882112 = 2.0;
        double r2882113 = w;
        double r2882114 = r2882112 * r2882113;
        double r2882115 = r2882111 / r2882114;
        double r2882116 = d;
        double r2882117 = r2882116 * r2882116;
        double r2882118 = r2882111 * r2882117;
        double r2882119 = h;
        double r2882120 = r2882113 * r2882119;
        double r2882121 = D;
        double r2882122 = r2882121 * r2882121;
        double r2882123 = r2882120 * r2882122;
        double r2882124 = r2882118 / r2882123;
        double r2882125 = r2882124 * r2882124;
        double r2882126 = M;
        double r2882127 = r2882126 * r2882126;
        double r2882128 = r2882125 - r2882127;
        double r2882129 = sqrt(r2882128);
        double r2882130 = r2882124 + r2882129;
        double r2882131 = r2882115 * r2882130;
        return r2882131;
}

↓

double f(double c0, double w, double h, double D, double d, double M) {
        double r2882132 = c0;
        double r2882133 = 5.43376491402612e-292;
        bool r2882134 = r2882132 <= r2882133;
        double r2882135 = w;
        double r2882136 = r2882132 / r2882135;
        double r2882137 = d;
        double r2882138 = D;
        double r2882139 = r2882137 / r2882138;
        double r2882140 = r2882132 * r2882139;
        double r2882141 = r2882140 / r2882135;
        double r2882142 = h;
        double r2882143 = r2882139 / r2882142;
        double r2882144 = r2882141 * r2882143;
        double r2882145 = r2882139 * r2882139;
        double r2882146 = r2882145 / r2882142;
        double r2882147 = r2882146 * r2882136;
        double r2882148 = M;
        double r2882149 = r2882147 - r2882148;
        double r2882150 = r2882148 + r2882147;
        double r2882151 = r2882149 * r2882150;
        double r2882152 = sqrt(r2882151);
        double r2882153 = r2882144 + r2882152;
        double r2882154 = cbrt(r2882153);
        double r2882155 = r2882154 * r2882154;
        double r2882156 = r2882155 * r2882154;
        double r2882157 = 2.0;
        double r2882158 = r2882156 / r2882157;
        double r2882159 = r2882136 * r2882158;
        double r2882160 = sqrt(r2882149);
        double r2882161 = sqrt(r2882150);
        double r2882162 = r2882160 * r2882161;
        double r2882163 = r2882144 + r2882162;
        double r2882164 = r2882163 / r2882157;
        double r2882165 = r2882164 * r2882136;
        double r2882166 = r2882134 ? r2882159 : r2882165;
        return r2882166;
}

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))))))