Henrywood and Agarwal, Equation (13)

Average Error: 57.9 → 50.7

Time: 55.3s

Precision: 64

• could not determine a ground truth for program body

  1. c0 = 2.1349606945297384e+240
  2. w = 1.2254588308927825e-216
  3. h = -5.879678403871345e-165
  4. D = -4.041865292113949e+35
  5. d = -2.1293592977956366e+226
  6. M = 15.597049771107491

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \cdot M \le 7.865697586031107 \cdot 10^{-80}:\\ \;\;\;\;\frac{c0}{2} \cdot \frac{\sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} + M\right) \cdot \left(\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} - M\right)} + \frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w}} \cdot \left(\sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} + M\right) \cdot \left(\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} - M\right)} + \frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w}} \cdot \sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} + M\right) \cdot \left(\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} - M\right)} + \frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w}}\right)}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} + \sqrt{\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} + M} \cdot \sqrt{\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} - M}}{w} \cdot \frac{c0}{2}\\ \end{array}\]

C

double f(double c0, double w, double h, double D, double d, double M) {
        double r7852121 = c0;
        double r7852122 = 2.0;
        double r7852123 = w;
        double r7852124 = r7852122 * r7852123;
        double r7852125 = r7852121 / r7852124;
        double r7852126 = d;
        double r7852127 = r7852126 * r7852126;
        double r7852128 = r7852121 * r7852127;
        double r7852129 = h;
        double r7852130 = r7852123 * r7852129;
        double r7852131 = D;
        double r7852132 = r7852131 * r7852131;
        double r7852133 = r7852130 * r7852132;
        double r7852134 = r7852128 / r7852133;
        double r7852135 = r7852134 * r7852134;
        double r7852136 = M;
        double r7852137 = r7852136 * r7852136;
        double r7852138 = r7852135 - r7852137;
        double r7852139 = sqrt(r7852138);
        double r7852140 = r7852134 + r7852139;
        double r7852141 = r7852125 * r7852140;
        return r7852141;
}

↓

double f(double c0, double w, double h, double D, double d, double M) {
        double r7852142 = M;
        double r7852143 = r7852142 * r7852142;
        double r7852144 = 7.865697586031107e-80;
        bool r7852145 = r7852143 <= r7852144;
        double r7852146 = c0;
        double r7852147 = 2.0;
        double r7852148 = r7852146 / r7852147;
        double r7852149 = d;
        double r7852150 = D;
        double r7852151 = r7852149 / r7852150;
        double r7852152 = r7852146 * r7852151;
        double r7852153 = r7852151 * r7852152;
        double r7852154 = h;
        double r7852155 = w;
        double r7852156 = r7852154 * r7852155;
        double r7852157 = r7852153 / r7852156;
        double r7852158 = r7852157 + r7852142;
        double r7852159 = r7852157 - r7852142;
        double r7852160 = r7852158 * r7852159;
        double r7852161 = sqrt(r7852160);
        double r7852162 = r7852161 + r7852157;
        double r7852163 = cbrt(r7852162);
        double r7852164 = r7852163 * r7852163;
        double r7852165 = r7852163 * r7852164;
        double r7852166 = r7852165 / r7852155;
        double r7852167 = r7852148 * r7852166;
        double r7852168 = sqrt(r7852158);
        double r7852169 = sqrt(r7852159);
        double r7852170 = r7852168 * r7852169;
        double r7852171 = r7852157 + r7852170;
        double r7852172 = r7852171 / r7852155;
        double r7852173 = r7852172 * r7852148;
        double r7852174 = r7852145 ? r7852167 : r7852173;
        return r7852174;
}

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 (* M M) < 7.865697586031107e-80

   1. Initial program 55.6

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

   2. Simplified 45.7

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

   4. Applied add-cube-cbrt 45.8

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

   if 7.865697586031107e-80 < (* M M)

   1. Initial program 61.3

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

   2. Simplified 59.3

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

   4. Applied times-frac 59.8

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

   6. Applied sqrt-prod 58.8

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

   7. Simplified 58.0

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

4. Final simplification 50.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \le 7.865697586031107 \cdot 10^{-80}:\\ \;\;\;\;\frac{c0}{2} \cdot \frac{\sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} + M\right) \cdot \left(\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} - M\right)} + \frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w}} \cdot \left(\sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} + M\right) \cdot \left(\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} - M\right)} + \frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w}} \cdot \sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} + M\right) \cdot \left(\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} - M\right)} + \frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w}}\right)}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} + \sqrt{\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} + M} \cdot \sqrt{\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{h \cdot w} - M}}{w} \cdot \frac{c0}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(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))))))