Henrywood and Agarwal, Equation (13)

Average Error: 58.0 → 50.1

Time: 57.3s

Precision: 64

• could not determine a ground truth for program body

  1. c0 = -5.727971624273011e+226
  2. w = -4.7512274716228236e+176
  3. h = -1.4894749161561686e-112
  4. D = 5.152094513058003e-306
  5. d = -4.918120335736088e+289
  6. M = -2.607418302614673e+239

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \cdot M \le 5.939879892392706 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\sqrt{\left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} + M\right) \cdot \left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} - M\right)} + \frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}\right) \cdot \left(\left(\sqrt{\left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} + M\right) \cdot \left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} - M\right)} + \frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}\right) \cdot \left(\sqrt{\left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} + M\right) \cdot \left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} - M\right)} + \frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}\right)\right)} \cdot \frac{c0}{w}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} + \sqrt{\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} + M} \cdot \sqrt{\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} - M}\right) \cdot \frac{c0}{w}}{2}\\ \end{array}\]

C

double f(double c0, double w, double h, double D, double d, double M) {
        double r4409366 = c0;
        double r4409367 = 2.0;
        double r4409368 = w;
        double r4409369 = r4409367 * r4409368;
        double r4409370 = r4409366 / r4409369;
        double r4409371 = d;
        double r4409372 = r4409371 * r4409371;
        double r4409373 = r4409366 * r4409372;
        double r4409374 = h;
        double r4409375 = r4409368 * r4409374;
        double r4409376 = D;
        double r4409377 = r4409376 * r4409376;
        double r4409378 = r4409375 * r4409377;
        double r4409379 = r4409373 / r4409378;
        double r4409380 = r4409379 * r4409379;
        double r4409381 = M;
        double r4409382 = r4409381 * r4409381;
        double r4409383 = r4409380 - r4409382;
        double r4409384 = sqrt(r4409383);
        double r4409385 = r4409379 + r4409384;
        double r4409386 = r4409370 * r4409385;
        return r4409386;
}

↓

double f(double c0, double w, double h, double D, double d, double M) {
        double r4409387 = M;
        double r4409388 = r4409387 * r4409387;
        double r4409389 = 5.939879892392706e-39;
        bool r4409390 = r4409388 <= r4409389;
        double r4409391 = c0;
        double r4409392 = w;
        double r4409393 = r4409391 / r4409392;
        double r4409394 = d;
        double r4409395 = D;
        double r4409396 = r4409394 / r4409395;
        double r4409397 = r4409393 * r4409396;
        double r4409398 = r4409397 * r4409396;
        double r4409399 = h;
        double r4409400 = r4409398 / r4409399;
        double r4409401 = r4409400 + r4409387;
        double r4409402 = r4409400 - r4409387;
        double r4409403 = r4409401 * r4409402;
        double r4409404 = sqrt(r4409403);
        double r4409405 = r4409404 + r4409400;
        double r4409406 = r4409405 * r4409405;
        double r4409407 = r4409405 * r4409406;
        double r4409408 = cbrt(r4409407);
        double r4409409 = r4409408 * r4409393;
        double r4409410 = 2.0;
        double r4409411 = r4409409 / r4409410;
        double r4409412 = sqrt(r4409401);
        double r4409413 = sqrt(r4409402);
        double r4409414 = r4409412 * r4409413;
        double r4409415 = r4409400 + r4409414;
        double r4409416 = r4409415 * r4409393;
        double r4409417 = r4409416 / r4409410;
        double r4409418 = r4409390 ? r4409411 : r4409417;
        return r4409418;
}

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) < 5.939879892392706e-39

   1. Initial program 55.8

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

   2. Simplified 47.3

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

   4. Applied associate-*r* 48.9

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

   6. Applied associate-*r* 48.3

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

   8. Applied add-cbrt-cube 50.0

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

   9. Simplified 45.8

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

   if 5.939879892392706e-39 < (* M M)

   1. Initial program 61.7

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

   2. Simplified 60.4

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

   4. Applied associate-*r* 60.4

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

   6. Applied associate-*r* 60.4

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

   8. Applied sqrt-prod 58.2

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

   9. Simplified 57.3

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

4. Final simplification 50.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \le 5.939879892392706 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\sqrt{\left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} + M\right) \cdot \left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} - M\right)} + \frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}\right) \cdot \left(\left(\sqrt{\left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} + M\right) \cdot \left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} - M\right)} + \frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}\right) \cdot \left(\sqrt{\left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} + M\right) \cdot \left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} - M\right)} + \frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}\right)\right)} \cdot \frac{c0}{w}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} + \sqrt{\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} + M} \cdot \sqrt{\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} - M}\right) \cdot \frac{c0}{w}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(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))))))