Henrywood and Agarwal, Equation (13)

Average Error: 58.1 → 34.8

Time: 53.3s

Precision: 64

• could not determine a ground truth for program body

  1. c0 = -8.067133419157703e+291
  2. w = 8.283822717227434e-223
  3. h = 1.2925874203003307e-304
  4. D = 4.799562527043637e-22
  5. d = 3.2126910245618796e+48
  6. M = 6.777537671086472e-301

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;D \le -2.4768410751898854 \cdot 10^{+36}:\\ \;\;\;\;\left(\frac{\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{w}}{h} + \sqrt{\frac{\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{w}}{h} + M} \cdot \sqrt{\frac{\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{w}}{h} - M}\right) \cdot \frac{c0}{w \cdot 2}\\ \mathbf{elif}\;D \le 2.7797976866862625 \cdot 10^{+42}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \le 4.721751036118839 \cdot 10^{+112}:\\ \;\;\;\;\left(\frac{\frac{1}{2}}{w} \cdot \frac{\mathsf{fma}\left(\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} - M \cdot M}\right), \left(\frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} - M \cdot M\right), \left(\left(\frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}}\right) \cdot \frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}}\right)\right)}{\mathsf{fma}\left(\left(\frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}}\right), \left(\frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} - \sqrt{\frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} - M \cdot M}\right), \left(\frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} - M \cdot M\right)\right)}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double f(double c0, double w, double h, double D, double d, double M) {
        double r5735955 = c0;
        double r5735956 = 2.0;
        double r5735957 = w;
        double r5735958 = r5735956 * r5735957;
        double r5735959 = r5735955 / r5735958;
        double r5735960 = d;
        double r5735961 = r5735960 * r5735960;
        double r5735962 = r5735955 * r5735961;
        double r5735963 = h;
        double r5735964 = r5735957 * r5735963;
        double r5735965 = D;
        double r5735966 = r5735965 * r5735965;
        double r5735967 = r5735964 * r5735966;
        double r5735968 = r5735962 / r5735967;
        double r5735969 = r5735968 * r5735968;
        double r5735970 = M;
        double r5735971 = r5735970 * r5735970;
        double r5735972 = r5735969 - r5735971;
        double r5735973 = sqrt(r5735972);
        double r5735974 = r5735968 + r5735973;
        double r5735975 = r5735959 * r5735974;
        return r5735975;
}

↓

double f(double c0, double w, double h, double D, double d, double M) {
        double r5735976 = D;
        double r5735977 = -2.4768410751898854e+36;
        bool r5735978 = r5735976 <= r5735977;
        double r5735979 = d;
        double r5735980 = r5735979 / r5735976;
        double r5735981 = c0;
        double r5735982 = r5735981 * r5735980;
        double r5735983 = r5735980 * r5735982;
        double r5735984 = w;
        double r5735985 = r5735983 / r5735984;
        double r5735986 = h;
        double r5735987 = r5735985 / r5735986;
        double r5735988 = M;
        double r5735989 = r5735987 + r5735988;
        double r5735990 = sqrt(r5735989);
        double r5735991 = r5735987 - r5735988;
        double r5735992 = sqrt(r5735991);
        double r5735993 = r5735990 * r5735992;
        double r5735994 = r5735987 + r5735993;
        double r5735995 = 2.0;
        double r5735996 = r5735984 * r5735995;
        double r5735997 = r5735981 / r5735996;
        double r5735998 = r5735994 * r5735997;
        double r5735999 = 2.7797976866862625e+42;
        bool r5736000 = r5735976 <= r5735999;
        double r5736001 = 0.0;
        double r5736002 = 4.721751036118839e+112;
        bool r5736003 = r5735976 <= r5736002;
        double r5736004 = 0.5;
        double r5736005 = r5736004 / r5735984;
        double r5736006 = r5735980 / r5735986;
        double r5736007 = r5735980 * r5736006;
        double r5736008 = r5735984 / r5735981;
        double r5736009 = r5736007 / r5736008;
        double r5736010 = r5736009 * r5736009;
        double r5736011 = r5735988 * r5735988;
        double r5736012 = r5736010 - r5736011;
        double r5736013 = sqrt(r5736012);
        double r5736014 = r5736010 * r5736009;
        double r5736015 = fma(r5736013, r5736012, r5736014);
        double r5736016 = r5736009 - r5736013;
        double r5736017 = fma(r5736009, r5736016, r5736012);
        double r5736018 = r5736015 / r5736017;
        double r5736019 = r5736005 * r5736018;
        double r5736020 = r5736019 * r5735981;
        double r5736021 = r5736003 ? r5736020 : r5736001;
        double r5736022 = r5736000 ? r5736001 : r5736021;
        double r5736023 = r5735978 ? r5735998 : r5736022;
        return r5736023;
}

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 < -2.4768410751898854e+36

   1. Initial program 56.2

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

   2. Simplified 43.9

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

   4. Applied difference-of-squares 43.9

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

   5. Applied sqrt-prod 48.3

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

   if -2.4768410751898854e+36 < D < 2.7797976866862625e+42 or 4.721751036118839e+112 < 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 52.2

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

   3. Taylor expanded around -inf 31.6

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

   if 2.7797976866862625e+42 < D < 4.721751036118839e+112

   1. Initial program 54.6

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

   2. Simplified 47.3

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

   4. Applied div-inv 47.4

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

   5. Applied associate-*l* 47.2

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

   6. Simplified 46.2

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

   8. Applied flip3-+ 51.0

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

   9. Simplified 54.6

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

   10. Simplified 54.1

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

4. Final simplification 34.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \le -2.4768410751898854 \cdot 10^{+36}:\\ \;\;\;\;\left(\frac{\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{w}}{h} + \sqrt{\frac{\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{w}}{h} + M} \cdot \sqrt{\frac{\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{d}{D}\right)}{w}}{h} - M}\right) \cdot \frac{c0}{w \cdot 2}\\ \mathbf{elif}\;D \le 2.7797976866862625 \cdot 10^{+42}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \le 4.721751036118839 \cdot 10^{+112}:\\ \;\;\;\;\left(\frac{\frac{1}{2}}{w} \cdot \frac{\mathsf{fma}\left(\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} - M \cdot M}\right), \left(\frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} - M \cdot M\right), \left(\left(\frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}}\right) \cdot \frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}}\right)\right)}{\mathsf{fma}\left(\left(\frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}}\right), \left(\frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} - \sqrt{\frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} - M \cdot M}\right), \left(\frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}}{\frac{w}{c0}} - M \cdot M\right)\right)}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(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))))))