Henrywood and Agarwal, Equation (13)

Average Error: 59.0 → 33.1

Time: 10.2s

Precision: 64

• could not determine a ground truth for program body

  1. c0 = 2.053784194856198e+146
  2. w = 2.4943540530702483e-110
  3. h = -5.072357774180332e-263
  4. D = 2.3369147687632714e-33
  5. d = 1.862952377391352e+233
  6. M = 6031.575547282502

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \le 2476228.3530077836476266384124755859375:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;\log 1\\ \end{array}\]

C

double f(double c0, double w, double h, double D, double d, double M) {
        double r169451 = c0;
        double r169452 = 2.0;
        double r169453 = w;
        double r169454 = r169452 * r169453;
        double r169455 = r169451 / r169454;
        double r169456 = d;
        double r169457 = r169456 * r169456;
        double r169458 = r169451 * r169457;
        double r169459 = h;
        double r169460 = r169453 * r169459;
        double r169461 = D;
        double r169462 = r169461 * r169461;
        double r169463 = r169460 * r169462;
        double r169464 = r169458 / r169463;
        double r169465 = r169464 * r169464;
        double r169466 = M;
        double r169467 = r169466 * r169466;
        double r169468 = r169465 - r169467;
        double r169469 = sqrt(r169468);
        double r169470 = r169464 + r169469;
        double r169471 = r169455 * r169470;
        return r169471;
}

↓

double f(double c0, double w, double h, double D, double d, double M) {
        double r169472 = c0;
        double r169473 = 2.0;
        double r169474 = w;
        double r169475 = r169473 * r169474;
        double r169476 = r169472 / r169475;
        double r169477 = d;
        double r169478 = r169477 * r169477;
        double r169479 = r169472 * r169478;
        double r169480 = h;
        double r169481 = r169474 * r169480;
        double r169482 = D;
        double r169483 = r169482 * r169482;
        double r169484 = r169481 * r169483;
        double r169485 = r169479 / r169484;
        double r169486 = r169485 * r169485;
        double r169487 = M;
        double r169488 = r169487 * r169487;
        double r169489 = r169486 - r169488;
        double r169490 = sqrt(r169489);
        double r169491 = r169485 + r169490;
        double r169492 = r169476 * r169491;
        double r169493 = 2476228.3530077836;
        bool r169494 = r169492 <= r169493;
        double r169495 = 1.0;
        double r169496 = log(r169495);
        double r169497 = r169494 ? r169492 : r169496;
        return r169497;
}

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))))) < 2476228.3530077836

   1. Initial program 36.3

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

   if 2476228.3530077836 < (* (/ 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 63.4

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

   2. Taylor expanded around inf 34.5

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

   4. Applied add-log-exp 34.5

      \[\leadsto \color{blue}{\log \left(e^{\frac{c0}{2 \cdot w} \cdot 0}\right)}\]

   5. Simplified 32.5

      \[\leadsto \log \color{blue}{1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 33.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \le 2476228.3530077836476266384124755859375:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;\log 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ 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))))))