Henrywood and Agarwal, Equation (13)

Average Error: 59.2 → 32.3

Time: 34.4s

Precision: 64

• could not determine a ground truth for program body

  1. c0 = 4.046401737432886e+231
  2. w = -1.3554673749956964e-176
  3. h = 9.954080561998736e-55
  4. D = 4.360008815267399e-238
  5. d = 2.536808514302376e+298
  6. M = -1.9185588755758526e-34

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) \le 3.59339556793099782905505395424773047473 \cdot 10^{241}:\\ \;\;\;\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double f(double c0, double w, double h, double D, double d, double M) {
        double r151585 = c0;
        double r151586 = 2.0;
        double r151587 = w;
        double r151588 = r151586 * r151587;
        double r151589 = r151585 / r151588;
        double r151590 = d;
        double r151591 = r151590 * r151590;
        double r151592 = r151585 * r151591;
        double r151593 = h;
        double r151594 = r151587 * r151593;
        double r151595 = D;
        double r151596 = r151595 * r151595;
        double r151597 = r151594 * r151596;
        double r151598 = r151592 / r151597;
        double r151599 = r151598 * r151598;
        double r151600 = M;
        double r151601 = r151600 * r151600;
        double r151602 = r151599 - r151601;
        double r151603 = sqrt(r151602);
        double r151604 = r151598 + r151603;
        double r151605 = r151589 * r151604;
        return r151605;
}

↓

double f(double c0, double w, double h, double D, double d, double M) {
        double r151606 = c0;
        double r151607 = w;
        double r151608 = 2.0;
        double r151609 = r151607 * r151608;
        double r151610 = r151606 / r151609;
        double r151611 = d;
        double r151612 = r151611 * r151611;
        double r151613 = r151606 * r151612;
        double r151614 = D;
        double r151615 = r151614 * r151614;
        double r151616 = h;
        double r151617 = r151607 * r151616;
        double r151618 = r151615 * r151617;
        double r151619 = r151613 / r151618;
        double r151620 = r151619 * r151619;
        double r151621 = M;
        double r151622 = r151621 * r151621;
        double r151623 = r151620 - r151622;
        double r151624 = sqrt(r151623);
        double r151625 = r151624 + r151619;
        double r151626 = r151610 * r151625;
        double r151627 = 3.593395567930998e+241;
        bool r151628 = r151626 <= r151627;
        double r151629 = 0.0;
        double r151630 = r151628 ? r151626 : r151629;
        return r151630;
}

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))))) < 3.593395567930998e+241

   1. Initial program 34.1

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

   if 3.593395567930998e+241 < (* (/ 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 63.9

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

   2. Simplified 61.9

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

   3. Taylor expanded around inf 34.3

      \[\leadsto \frac{c0}{w \cdot 2} \cdot \color{blue}{0}\]
   4. Using strategy rm

   5. Applied add-cbrt-cube 34.3

      \[\leadsto \frac{c0}{w \cdot 2} \cdot \color{blue}{\sqrt[3]{\left(0 \cdot 0\right) \cdot 0}}\]

   6. Applied add-cbrt-cube 34.3

      \[\leadsto \frac{c0}{w \cdot \color{blue}{\sqrt[3]{\left(2 \cdot 2\right) \cdot 2}}} \cdot \sqrt[3]{\left(0 \cdot 0\right) \cdot 0}\]

   7. Applied add-cbrt-cube 41.6

      \[\leadsto \frac{c0}{\color{blue}{\sqrt[3]{\left(w \cdot w\right) \cdot w}} \cdot \sqrt[3]{\left(2 \cdot 2\right) \cdot 2}} \cdot \sqrt[3]{\left(0 \cdot 0\right) \cdot 0}\]

   8. Applied cbrt-unprod 41.6

      \[\leadsto \frac{c0}{\color{blue}{\sqrt[3]{\left(\left(w \cdot w\right) \cdot w\right) \cdot \left(\left(2 \cdot 2\right) \cdot 2\right)}}} \cdot \sqrt[3]{\left(0 \cdot 0\right) \cdot 0}\]

   9. Applied add-cbrt-cube 48.8

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(c0 \cdot c0\right) \cdot c0}}}{\sqrt[3]{\left(\left(w \cdot w\right) \cdot w\right) \cdot \left(\left(2 \cdot 2\right) \cdot 2\right)}} \cdot \sqrt[3]{\left(0 \cdot 0\right) \cdot 0}\]

   10. Applied cbrt-undiv 49.3

       \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(c0 \cdot c0\right) \cdot c0}{\left(\left(w \cdot w\right) \cdot w\right) \cdot \left(\left(2 \cdot 2\right) \cdot 2\right)}}} \cdot \sqrt[3]{\left(0 \cdot 0\right) \cdot 0}\]

   11. Applied cbrt-unprod 49.3

       \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(c0 \cdot c0\right) \cdot c0}{\left(\left(w \cdot w\right) \cdot w\right) \cdot \left(\left(2 \cdot 2\right) \cdot 2\right)} \cdot \left(\left(0 \cdot 0\right) \cdot 0\right)}}\]

   12. Simplified 31.9

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

4. Final simplification 32.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) \le 3.59339556793099782905505395424773047473 \cdot 10^{241}:\\ \;\;\;\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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))))))