Henrywood and Agarwal, Equation (13)

Average Error: 59.2 → 30.2

Time: 19.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := c0 \cdot \left(d \cdot d\right)\\ t_2 := \frac{t_1}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_3 := \sqrt{t_2 \cdot t_2 - M \cdot M}\\ \mathbf{if}\;\begin{array}{l} t_4 := t_0 \cdot \left(t_2 + t_3\right)\\ t_4 \leq 0 \lor \neg \left(t_4 \leq 1.608586741825546 \cdot 10^{+295}\right) \end{array}:\\ \;\;\;\;0.25 \cdot \frac{\frac{{D}^{2} \cdot \left(M \cdot \left(h \cdot M\right)\right)}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_3 + \frac{\frac{t_1}{w \cdot h}}{D \cdot D}\right)\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :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))))))

↓

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (* c0 (* d d)))
        (t_2 (/ t_1 (* (* w h) (* D D))))
        (t_3 (sqrt (- (* t_2 t_2) (* M M)))))
   (if (let* ((t_4 (* t_0 (+ t_2 t_3))))
         (or (<= t_4 0.0) (not (<= t_4 1.608586741825546e+295))))
     (* 0.25 (/ (/ (* (pow D 2.0) (* M (* h M))) d) d))
     (* t_0 (+ t_3 (/ (/ t_1 (* w h)) (* D D)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (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)));
}

↓

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = c0 * (d * d);
	double t_2 = t_1 / ((w * h) * (D * D));
	double t_3 = sqrt((t_2 * t_2) - (M * M));
	double t_4 = t_0 * (t_2 + t_3);
	double tmp;
	if ((t_4 <= 0.0) || !(t_4 <= 1.608586741825546e+295)) {
		tmp = 0.25 * (((pow(D, 2.0) * (M * (h * M))) / d) / d);
	} else {
		tmp = t_0 * (t_3 + ((t_1 / (w * h)) / (D * D)));
	}
	return tmp;
}

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 (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -0.0 or 1.60858674182554608e295 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 60.1

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

   2. Taylor expanded in c0 around -inf 41.9

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot c0}\right)} \]

   3. Taylor expanded in c0 around 0 35.1

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]

   4. Applied unpow2_binary64 35.1

      \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

   5. Applied associate-/r*_binary64 32.5

      \[\leadsto 0.25 \cdot \color{blue}{\frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}}{d}} \]

   6. Applied unpow2_binary64 32.5

      \[\leadsto 0.25 \cdot \frac{\frac{{D}^{2} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)}{d}}{d} \]

   7. Applied associate-*l*_binary64 30.6

      \[\leadsto 0.25 \cdot \frac{\frac{{D}^{2} \cdot \color{blue}{\left(M \cdot \left(M \cdot h\right)\right)}}{d}}{d} \]

   8. Simplified 30.6

      \[\leadsto 0.25 \cdot \frac{\frac{{D}^{2} \cdot \left(M \cdot \color{blue}{\left(h \cdot M\right)}\right)}{d}}{d} \]

   if -0.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < 1.60858674182554608e295

   1. Initial program 8.0

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

   2. Applied associate-/r*_binary64 10.5

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0 \lor \neg \left(\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 1.608586741825546 \cdot 10^{+295}\right):\\ \;\;\;\;0.25 \cdot \frac{\frac{{D}^{2} \cdot \left(M \cdot \left(h \cdot M\right)\right)}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M} + \frac{\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}}{D \cdot D}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022039 
(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))))))