Henrywood and Agarwal, Equation (13)

Average Error: 58.1 → 52.7

Time: 52.3s

Precision: 64

• could not determine a ground truth for program body

  1. c0 = 2.365951752076539e+256
  2. w = -1.597050900171122e-98
  3. h = 2.7421650851285317e-178
  4. D = 1.951146446221059e-304
  5. d = -4.1946076574544436e-132
  6. M = -3.7762789558399307e+40

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;D \le 2.8151256814079048 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} + \sqrt{\left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} - M\right) \cdot \left(M + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}\right)}\right)}{2}\\ \mathbf{elif}\;D \le 1.3218565804590423 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{\left(\left(d \cdot c0\right) \cdot d\right) \cdot 2}{w \cdot \left(\left(D \cdot D\right) \cdot h\right)} \cdot \frac{c0}{w}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} - M\right)} + \frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}} \cdot \left(\sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} - M\right)} + \frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}} \cdot \sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} - M\right)} + \frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}}\right)\right) \cdot \frac{c0}{w}}{2}\\ \end{array}\]

C

double f(double c0, double w, double h, double D, double d, double M) {
        double r2939293 = c0;
        double r2939294 = 2.0;
        double r2939295 = w;
        double r2939296 = r2939294 * r2939295;
        double r2939297 = r2939293 / r2939296;
        double r2939298 = d;
        double r2939299 = r2939298 * r2939298;
        double r2939300 = r2939293 * r2939299;
        double r2939301 = h;
        double r2939302 = r2939295 * r2939301;
        double r2939303 = D;
        double r2939304 = r2939303 * r2939303;
        double r2939305 = r2939302 * r2939304;
        double r2939306 = r2939300 / r2939305;
        double r2939307 = r2939306 * r2939306;
        double r2939308 = M;
        double r2939309 = r2939308 * r2939308;
        double r2939310 = r2939307 - r2939309;
        double r2939311 = sqrt(r2939310);
        double r2939312 = r2939306 + r2939311;
        double r2939313 = r2939297 * r2939312;
        return r2939313;
}

↓

double f(double c0, double w, double h, double D, double d, double M) {
        double r2939314 = D;
        double r2939315 = 2.8151256814079048e-161;
        bool r2939316 = r2939314 <= r2939315;
        double r2939317 = c0;
        double r2939318 = w;
        double r2939319 = r2939317 / r2939318;
        double r2939320 = d;
        double r2939321 = r2939320 / r2939314;
        double r2939322 = r2939319 * r2939321;
        double r2939323 = r2939322 * r2939321;
        double r2939324 = h;
        double r2939325 = r2939323 / r2939324;
        double r2939326 = M;
        double r2939327 = r2939325 - r2939326;
        double r2939328 = r2939321 * r2939321;
        double r2939329 = r2939328 * r2939319;
        double r2939330 = r2939329 / r2939324;
        double r2939331 = r2939326 + r2939330;
        double r2939332 = r2939327 * r2939331;
        double r2939333 = sqrt(r2939332);
        double r2939334 = r2939325 + r2939333;
        double r2939335 = r2939319 * r2939334;
        double r2939336 = 2.0;
        double r2939337 = r2939335 / r2939336;
        double r2939338 = 1.3218565804590423e+114;
        bool r2939339 = r2939314 <= r2939338;
        double r2939340 = r2939320 * r2939317;
        double r2939341 = r2939340 * r2939320;
        double r2939342 = r2939341 * r2939336;
        double r2939343 = r2939314 * r2939314;
        double r2939344 = r2939343 * r2939324;
        double r2939345 = r2939318 * r2939344;
        double r2939346 = r2939342 / r2939345;
        double r2939347 = r2939346 * r2939319;
        double r2939348 = r2939347 / r2939336;
        double r2939349 = r2939330 - r2939326;
        double r2939350 = r2939331 * r2939349;
        double r2939351 = sqrt(r2939350);
        double r2939352 = r2939351 + r2939325;
        double r2939353 = cbrt(r2939352);
        double r2939354 = r2939353 * r2939353;
        double r2939355 = r2939353 * r2939354;
        double r2939356 = r2939355 * r2939319;
        double r2939357 = r2939356 / r2939336;
        double r2939358 = r2939339 ? r2939348 : r2939357;
        double r2939359 = r2939316 ? r2939337 : r2939358;
        return r2939359;
}

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 < 2.8151256814079048e-161

   1. Initial program 59.3

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

   2. Simplified 53.4

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

   4. Applied associate-*r* 54.3

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

   6. Applied associate-*r* 53.9

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

   if 2.8151256814079048e-161 < D < 1.3218565804590423e+114

   1. Initial program 54.1

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

   2. Simplified 52.4

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

   4. Applied associate-*r* 53.1

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

   5. Taylor expanded around 0 54.6

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

   6. Simplified 52.2

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

   if 1.3218565804590423e+114 < D

   1. Initial program 59.9

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

   2. Simplified 42.6

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

   4. Applied associate-*r* 43.4

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

   6. Applied add-cube-cbrt 43.5

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

4. Final simplification 52.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \le 2.8151256814079048 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} + \sqrt{\left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} - M\right) \cdot \left(M + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}\right)}\right)}{2}\\ \mathbf{elif}\;D \le 1.3218565804590423 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{\left(\left(d \cdot c0\right) \cdot d\right) \cdot 2}{w \cdot \left(\left(D \cdot D\right) \cdot h\right)} \cdot \frac{c0}{w}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} - M\right)} + \frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}} \cdot \left(\sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} - M\right)} + \frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}} \cdot \sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} - M\right)} + \frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}}\right)\right) \cdot \frac{c0}{w}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(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))))))