Henrywood and Agarwal, Equation (13)

Average Error: 59.2 → 32.3

Time: 38.8s

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 r7544598 = c0;
        double r7544599 = 2.0;
        double r7544600 = w;
        double r7544601 = r7544599 * r7544600;
        double r7544602 = r7544598 / r7544601;
        double r7544603 = d;
        double r7544604 = r7544603 * r7544603;
        double r7544605 = r7544598 * r7544604;
        double r7544606 = h;
        double r7544607 = r7544600 * r7544606;
        double r7544608 = D;
        double r7544609 = r7544608 * r7544608;
        double r7544610 = r7544607 * r7544609;
        double r7544611 = r7544605 / r7544610;
        double r7544612 = r7544611 * r7544611;
        double r7544613 = M;
        double r7544614 = r7544613 * r7544613;
        double r7544615 = r7544612 - r7544614;
        double r7544616 = sqrt(r7544615);
        double r7544617 = r7544611 + r7544616;
        double r7544618 = r7544602 * r7544617;
        return r7544618;
}

↓

double f(double c0, double w, double h, double D, double d, double M) {
        double r7544619 = c0;
        double r7544620 = w;
        double r7544621 = 2.0;
        double r7544622 = r7544620 * r7544621;
        double r7544623 = r7544619 / r7544622;
        double r7544624 = d;
        double r7544625 = r7544624 * r7544624;
        double r7544626 = r7544619 * r7544625;
        double r7544627 = D;
        double r7544628 = r7544627 * r7544627;
        double r7544629 = h;
        double r7544630 = r7544620 * r7544629;
        double r7544631 = r7544628 * r7544630;
        double r7544632 = r7544626 / r7544631;
        double r7544633 = r7544632 * r7544632;
        double r7544634 = M;
        double r7544635 = r7544634 * r7544634;
        double r7544636 = r7544633 - r7544635;
        double r7544637 = sqrt(r7544636);
        double r7544638 = r7544637 + r7544632;
        double r7544639 = r7544623 * r7544638;
        double r7544640 = 3.593395567930998e+241;
        bool r7544641 = r7544639 <= r7544640;
        double r7544642 = 0.0;
        double r7544643 = r7544641 ? r7544639 : r7544642;
        return r7544643;
}

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 56.9

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

   3. Taylor expanded around inf 34.3

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

   5. Applied mul0 31.9

      \[\leadsto \frac{\color{blue}{0}}{2}\]
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))))))