Henrywood and Agarwal, Equation (13)

Average Error: 58.2 → 34.9

Time: 4.6m

Precision: 64

• could not determine a ground truth for program body

  1. c0 = -1.747008936416313e+285
  2. w = -1.4550244133772744e+126
  3. h = -3.2409033373260667e+72
  4. D = -3.0504287297335623e-229
  5. d = -3.8412183007530713e+270
  6. M = -3.0900647107111323e+158

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \le -9.149099578267195 \cdot 10^{+66}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \le -3.0984117268558066 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{c0 \cdot \left(\sqrt{\left(\frac{\frac{\frac{d}{D}}{\frac{h}{c0}}}{\frac{w}{\frac{d}{D}}} - M\right) \cdot \left(M + \frac{\frac{\frac{d}{D}}{\frac{h}{c0}}}{\frac{w}{\frac{d}{D}}}\right)} + \frac{\frac{\frac{d}{D}}{\frac{h}{c0}}}{\frac{w}{\frac{d}{D}}}\right)}{w}}{2}\\ \mathbf{elif}\;d \le -4.661221006329722 \cdot 10^{-176}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \le 4.6238459383082094 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{w \cdot h} + \sqrt{\left(\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{w \cdot h} + M\right) \cdot \left(\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{w \cdot h} - M\right)}}{\frac{w}{c0}}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double f(double c0, double w, double h, double D, double d, double M) {
        double r81896121 = c0;
        double r81896122 = 2.0;
        double r81896123 = w;
        double r81896124 = r81896122 * r81896123;
        double r81896125 = r81896121 / r81896124;
        double r81896126 = d;
        double r81896127 = r81896126 * r81896126;
        double r81896128 = r81896121 * r81896127;
        double r81896129 = h;
        double r81896130 = r81896123 * r81896129;
        double r81896131 = D;
        double r81896132 = r81896131 * r81896131;
        double r81896133 = r81896130 * r81896132;
        double r81896134 = r81896128 / r81896133;
        double r81896135 = r81896134 * r81896134;
        double r81896136 = M;
        double r81896137 = r81896136 * r81896136;
        double r81896138 = r81896135 - r81896137;
        double r81896139 = sqrt(r81896138);
        double r81896140 = r81896134 + r81896139;
        double r81896141 = r81896125 * r81896140;
        return r81896141;
}

↓

double f(double c0, double w, double h, double D, double d, double M) {
        double r81896142 = d;
        double r81896143 = -9.149099578267195e+66;
        bool r81896144 = r81896142 <= r81896143;
        double r81896145 = 0.0;
        double r81896146 = -3.0984117268558066e-57;
        bool r81896147 = r81896142 <= r81896146;
        double r81896148 = c0;
        double r81896149 = D;
        double r81896150 = r81896142 / r81896149;
        double r81896151 = h;
        double r81896152 = r81896151 / r81896148;
        double r81896153 = r81896150 / r81896152;
        double r81896154 = w;
        double r81896155 = r81896154 / r81896150;
        double r81896156 = r81896153 / r81896155;
        double r81896157 = M;
        double r81896158 = r81896156 - r81896157;
        double r81896159 = r81896157 + r81896156;
        double r81896160 = r81896158 * r81896159;
        double r81896161 = sqrt(r81896160);
        double r81896162 = r81896161 + r81896156;
        double r81896163 = r81896148 * r81896162;
        double r81896164 = r81896163 / r81896154;
        double r81896165 = 2.0;
        double r81896166 = r81896164 / r81896165;
        double r81896167 = -4.661221006329722e-176;
        bool r81896168 = r81896142 <= r81896167;
        double r81896169 = 4.6238459383082094e-126;
        bool r81896170 = r81896142 <= r81896169;
        double r81896171 = r81896148 * r81896150;
        double r81896172 = r81896150 * r81896171;
        double r81896173 = r81896154 * r81896151;
        double r81896174 = r81896172 / r81896173;
        double r81896175 = r81896174 + r81896157;
        double r81896176 = r81896174 - r81896157;
        double r81896177 = r81896175 * r81896176;
        double r81896178 = sqrt(r81896177);
        double r81896179 = r81896174 + r81896178;
        double r81896180 = r81896154 / r81896148;
        double r81896181 = r81896179 / r81896180;
        double r81896182 = r81896181 / r81896165;
        double r81896183 = r81896170 ? r81896182 : r81896145;
        double r81896184 = r81896168 ? r81896145 : r81896183;
        double r81896185 = r81896147 ? r81896166 : r81896184;
        double r81896186 = r81896144 ? r81896145 : r81896185;
        return r81896186;
}

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 3 regimes

2. if d < -9.149099578267195e+66 or -3.0984117268558066e-57 < d < -4.661221006329722e-176 or 4.6238459383082094e-126 < d

   1. Initial program 58.6

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

   2. Simplified 53.9

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

   4. Applied associate-*r* 54.8

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

   6. Applied *-un-lft-identity 54.8

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

   7. Applied *-un-lft-identity 54.8

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

   8. Applied distribute-lft-out 54.8

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

   9. Simplified 51.6

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

   10. Taylor expanded around -inf 31.6

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

   if -9.149099578267195e+66 < d < -3.0984117268558066e-57

   1. Initial program 52.6

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

   2. Simplified 51.7

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

   4. Applied associate-*r* 52.6

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

   6. Applied *-un-lft-identity 52.6

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

   7. Applied *-un-lft-identity 52.6

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

   8. Applied distribute-lft-out 52.6

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

   9. Simplified 49.0

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

   11. Applied associate-*r/ 48.9

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

   12. Simplified 49.3

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

   if -4.661221006329722e-176 < d < 4.6238459383082094e-126

   1. Initial program 60.9

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

   2. Simplified 41.7

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

   4. Applied associate-*r* 43.7

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

   6. Applied *-un-lft-identity 43.7

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

   7. Applied *-un-lft-identity 43.7

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

   8. Applied distribute-lft-out 43.7

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

   9. Simplified 40.9

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

   11. Applied pow1 40.9

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

   12. Applied pow1 40.9

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

   13. Applied pow-prod-down 40.9

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

   14. Simplified 43.0

       \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt{\left(\frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w} - M\right) \cdot \left(M + \frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w}\right)} + \frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot c0\right)}{h \cdot w}}{\frac{w}{c0}}\right)}}^{1}}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 34.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -9.149099578267195 \cdot 10^{+66}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \le -3.0984117268558066 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{c0 \cdot \left(\sqrt{\left(\frac{\frac{\frac{d}{D}}{\frac{h}{c0}}}{\frac{w}{\frac{d}{D}}} - M\right) \cdot \left(M + \frac{\frac{\frac{d}{D}}{\frac{h}{c0}}}{\frac{w}{\frac{d}{D}}}\right)} + \frac{\frac{\frac{d}{D}}{\frac{h}{c0}}}{\frac{w}{\frac{d}{D}}}\right)}{w}}{2}\\ \mathbf{elif}\;d \le -4.661221006329722 \cdot 10^{-176}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \le 4.6238459383082094 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{w \cdot h} + \sqrt{\left(\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{w \cdot h} + M\right) \cdot \left(\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{w \cdot h} - M\right)}}{\frac{w}{c0}}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(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))))))