Henrywood and Agarwal, Equation (13)

Average Error: 58.1 → 32.1

Time: 1.4m

Precision: 64

• could not determine a ground truth for program body

  1. c0 = 8.133339588411731e+237
  2. w = 4.539587674523534e-77
  3. h = -2.2562532893357764e-282
  4. D = -9.050248425741029e-70
  5. d = 1.368660464483368e+283
  6. M = 3.625702201891533e+304

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) \le 4.369340549676101 \cdot 10^{+166}:\\ \;\;\;\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double f(double c0, double w, double h, double D, double d, double M) {
        double r45493595 = c0;
        double r45493596 = 2.0;
        double r45493597 = w;
        double r45493598 = r45493596 * r45493597;
        double r45493599 = r45493595 / r45493598;
        double r45493600 = d;
        double r45493601 = r45493600 * r45493600;
        double r45493602 = r45493595 * r45493601;
        double r45493603 = h;
        double r45493604 = r45493597 * r45493603;
        double r45493605 = D;
        double r45493606 = r45493605 * r45493605;
        double r45493607 = r45493604 * r45493606;
        double r45493608 = r45493602 / r45493607;
        double r45493609 = r45493608 * r45493608;
        double r45493610 = M;
        double r45493611 = r45493610 * r45493610;
        double r45493612 = r45493609 - r45493611;
        double r45493613 = sqrt(r45493612);
        double r45493614 = r45493608 + r45493613;
        double r45493615 = r45493599 * r45493614;
        return r45493615;
}

↓

double f(double c0, double w, double h, double D, double d, double M) {
        double r45493616 = c0;
        double r45493617 = w;
        double r45493618 = 2.0;
        double r45493619 = r45493617 * r45493618;
        double r45493620 = r45493616 / r45493619;
        double r45493621 = d;
        double r45493622 = r45493621 * r45493621;
        double r45493623 = r45493616 * r45493622;
        double r45493624 = D;
        double r45493625 = r45493624 * r45493624;
        double r45493626 = h;
        double r45493627 = r45493617 * r45493626;
        double r45493628 = r45493625 * r45493627;
        double r45493629 = r45493623 / r45493628;
        double r45493630 = r45493629 * r45493629;
        double r45493631 = M;
        double r45493632 = r45493631 * r45493631;
        double r45493633 = r45493630 - r45493632;
        double r45493634 = sqrt(r45493633);
        double r45493635 = r45493634 + r45493629;
        double r45493636 = r45493620 * r45493635;
        double r45493637 = 4.369340549676101e+166;
        bool r45493638 = r45493636 <= r45493637;
        double r45493639 = 0.0;
        double r45493640 = r45493638 ? r45493636 : r45493639;
        return r45493640;
}

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 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))))) < 4.369340549676101e+166

   1. Initial program 35.3

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

   if 4.369340549676101e+166 < (* (/ 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)))))

   1. Initial program 62.5

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

   2. Simplified 55.5

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

   3. Taylor expanded around inf 33.5

      \[\leadsto \frac{c0}{w \cdot 2} \cdot \color{blue}{0}\]

   4. Taylor expanded around 0 31.5

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

4. Final simplification 32.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) \le 4.369340549676101 \cdot 10^{+166}:\\ \;\;\;\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +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))))))