Henrywood and Agarwal, Equation (13)

Average Error: 59.5 → 32.5

Time: 14.1s

Precision: binary64

• could not determine a ground truth for program body

  1. c0 = -6.436876158065135e+222
  2. w = 7.579122439236971e+112
  3. h = 3.446777546335459e-102
  4. D = 1.5826036261607966e-301
  5. d = 8.110632023437778e+267
  6. M = 6.511727365055e-183

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \le 8.55333432252809391 \cdot 10^{70}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \le +inf.0:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(d \cdot \frac{d}{w}\right) \cdot \frac{c0}{h \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double code(double c0, double w, double h, double D, double d, double M) {
	return ((double) ((c0 / ((double) (2.0 * w))) * ((double) ((((double) (c0 * ((double) (d * d)))) / ((double) (((double) (w * h)) * ((double) (D * D))))) + ((double) sqrt(((double) (((double) ((((double) (c0 * ((double) (d * d)))) / ((double) (((double) (w * h)) * ((double) (D * D))))) * (((double) (c0 * ((double) (d * d)))) / ((double) (((double) (w * h)) * ((double) (D * D))))))) - ((double) (M * M))))))))));
}

↓

double code(double c0, double w, double h, double D, double d, double M) {
	double VAR;
	if ((((double) ((c0 / ((double) (2.0 * w))) * ((double) ((((double) (c0 * ((double) (d * d)))) / ((double) (((double) (w * h)) * ((double) (D * D))))) + ((double) sqrt(((double) (((double) ((((double) (c0 * ((double) (d * d)))) / ((double) (((double) (w * h)) * ((double) (D * D))))) * (((double) (c0 * ((double) (d * d)))) / ((double) (((double) (w * h)) * ((double) (D * D))))))) - ((double) (M * M)))))))))) <= 8.553334322528094e+70)) {
		VAR = ((double) ((c0 / ((double) (2.0 * w))) * ((double) ((((double) (c0 * ((double) (d * d)))) / ((double) (((double) (w * h)) * ((double) (D * D))))) + ((double) sqrt(((double) (((double) ((((double) (c0 * ((double) (d * d)))) / ((double) (((double) (w * h)) * ((double) (D * D))))) * (((double) (c0 * ((double) (d * d)))) / ((double) (((double) (w * h)) * ((double) (D * D))))))) - ((double) (M * M))))))))));
	} else {
		double VAR_1;
		if ((((double) ((c0 / ((double) (2.0 * w))) * ((double) ((((double) (c0 * ((double) (d * d)))) / ((double) (((double) (w * h)) * ((double) (D * D))))) + ((double) sqrt(((double) (((double) ((((double) (c0 * ((double) (d * d)))) / ((double) (((double) (w * h)) * ((double) (D * D))))) * (((double) (c0 * ((double) (d * d)))) / ((double) (((double) (w * h)) * ((double) (D * D))))))) - ((double) (M * M)))))))))) <= +inf.0)) {
			VAR_1 = ((double) ((c0 / ((double) (2.0 * w))) * ((double) (2.0 * ((double) (((double) (d * (d / w))) * (c0 / ((double) (h * ((double) (D * D)))))))))));
		} else {
			VAR_1 = 0.0;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 (* (/ 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))))) < 8.55333432252809391e70

   1. Initial program 37.1

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

   if 8.55333432252809391e70 < (* (/ 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))))) < +inf.0

   1. Initial program 55.8

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

   2. Simplified 56.5

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} + \sqrt{\frac{c0}{w \cdot h} \cdot \left({\left(\frac{d}{D}\right)}^{4} \cdot \frac{c0}{w \cdot h}\right) - M \cdot M}\right)}\]
   3. Using strategy rm

   4. Applied flip-+ 63.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right) - \sqrt{\frac{c0}{w \cdot h} \cdot \left({\left(\frac{d}{D}\right)}^{4} \cdot \frac{c0}{w \cdot h}\right) - M \cdot M} \cdot \sqrt{\frac{c0}{w \cdot h} \cdot \left({\left(\frac{d}{D}\right)}^{4} \cdot \frac{c0}{w \cdot h}\right) - M \cdot M}}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - \sqrt{\frac{c0}{w \cdot h} \cdot \left({\left(\frac{d}{D}\right)}^{4} \cdot \frac{c0}{w \cdot h}\right) - M \cdot M}}}\]

   5. Simplified 57.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{0 + M \cdot M}}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - \sqrt{\frac{c0}{w \cdot h} \cdot \left({\left(\frac{d}{D}\right)}^{4} \cdot \frac{c0}{w \cdot h}\right) - M \cdot M}}\]

   6. Simplified 57.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\color{blue}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - \sqrt{\frac{c0}{w \cdot h} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{4}\right) - M \cdot M}}}\]

   7. Taylor expanded around 0 49.2

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

   8. Simplified 49.1

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

   if +inf.0 < (* (/ 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 64.0

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

   2. Simplified 59.9

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} + \sqrt{\frac{c0}{w \cdot h} \cdot \left({\left(\frac{d}{D}\right)}^{4} \cdot \frac{c0}{w \cdot h}\right) - M \cdot M}\right)}\]

   3. Taylor expanded around inf 32.8

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

   5. Applied mul0-r 30.4

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

4. Final simplification 32.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \le 8.55333432252809391 \cdot 10^{70}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \le +inf.0:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(d \cdot \frac{d}{w}\right) \cdot \frac{c0}{h \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ 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))))))