Henrywood and Agarwal, Equation (13)

Average Error: 59.1 → 31.8

Time: 1.3m

Precision: 64

• could not determine a ground truth for program body

  1. c0 = -5.592015412387592e+58
  2. w = 7.831553493153687e-85
  3. h = 1.1529852324293761e-245
  4. D = -2.0201089279346698e-209
  5. d = 1.6265452175469944e+222
  6. M = 6.881733804517132e-260

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) \le 2.250378991519621914334036295758019115197 \cdot 10^{263}:\\ \;\;\;\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double f(double c0, double w, double h, double D, double d, double M) {
        double r7719965 = c0;
        double r7719966 = 2.0;
        double r7719967 = w;
        double r7719968 = r7719966 * r7719967;
        double r7719969 = r7719965 / r7719968;
        double r7719970 = d;
        double r7719971 = r7719970 * r7719970;
        double r7719972 = r7719965 * r7719971;
        double r7719973 = h;
        double r7719974 = r7719967 * r7719973;
        double r7719975 = D;
        double r7719976 = r7719975 * r7719975;
        double r7719977 = r7719974 * r7719976;
        double r7719978 = r7719972 / r7719977;
        double r7719979 = r7719978 * r7719978;
        double r7719980 = M;
        double r7719981 = r7719980 * r7719980;
        double r7719982 = r7719979 - r7719981;
        double r7719983 = sqrt(r7719982);
        double r7719984 = r7719978 + r7719983;
        double r7719985 = r7719969 * r7719984;
        return r7719985;
}

↓

double f(double c0, double w, double h, double D, double d, double M) {
        double r7719986 = c0;
        double r7719987 = w;
        double r7719988 = 2.0;
        double r7719989 = r7719987 * r7719988;
        double r7719990 = r7719986 / r7719989;
        double r7719991 = d;
        double r7719992 = r7719991 * r7719991;
        double r7719993 = r7719986 * r7719992;
        double r7719994 = D;
        double r7719995 = r7719994 * r7719994;
        double r7719996 = h;
        double r7719997 = r7719987 * r7719996;
        double r7719998 = r7719995 * r7719997;
        double r7719999 = r7719993 / r7719998;
        double r7720000 = r7719999 * r7719999;
        double r7720001 = M;
        double r7720002 = r7720001 * r7720001;
        double r7720003 = r7720000 - r7720002;
        double r7720004 = sqrt(r7720003);
        double r7720005 = r7720004 + r7719999;
        double r7720006 = r7719990 * r7720005;
        double r7720007 = 2.250378991519622e+263;
        bool r7720008 = r7720006 <= r7720007;
        double r7720009 = 0.0;
        double r7720010 = r7720008 ? r7720006 : r7720009;
        return r7720010;
}

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 (* 2.0 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))))) < 2.250378991519622e+263

   1. Initial program 33.8

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

   if 2.250378991519622e+263 < (* (/ c0 (* 2.0 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)))))

   1. Initial program 64.0

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

   2. Simplified 56.5

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

   3. Taylor expanded around inf 31.4

      \[\leadsto \color{blue}{0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 31.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) \le 2.250378991519621914334036295758019115197 \cdot 10^{263}:\\ \;\;\;\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  (* (/ c0 (* 2.0 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))))))