Henrywood and Agarwal, Equation (13)

Average Error: 59.4 → 22.9

Time: 18.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot t_1 \leq 1.3564940205041007 \cdot 10^{+168}:\\ \;\;\;\;\frac{c0 \cdot t_1}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(h \cdot M\right)\right)\right)\right)\\ \end{array} \]

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 (* d d)) (* (* w h) (* D D))))
        (t_1 (+ t_0 (sqrt (- (* t_0 t_0) (* M M))))))
   (if (<= (* (/ c0 (* 2.0 w)) t_1) 1.3564940205041007e+168)
     (/ (* c0 t_1) (* 2.0 w))
     (* 0.25 (* (/ D d) (* (/ D d) (* M (* h M))))))))

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 * (d * d)) / ((w * h) * (D * D));
	double t_1 = t_0 + sqrt(((t_0 * t_0) - (M * M)));
	double tmp;
	if (((c0 / (2.0 * w)) * t_1) <= 1.3564940205041007e+168) {
		tmp = (c0 * t_1) / (2.0 * w);
	} else {
		tmp = 0.25 * ((D / d) * ((D / d) * (M * (h * M))));
	}
	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))))) < 1.3564940205041007e168

   1. Initial program 34.7

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

   2. Applied associate-*l/_binary64 34.5

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

   if 1.3564940205041007e168 < (*.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 63.7

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

   2. Taylor expanded in c0 around -inf 34.3

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

   3. Applied unpow2_binary64 34.3

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

   4. Applied associate-*l*_binary64 31.5

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

   5. Applied add-sqr-sqrt_binary64 47.7

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

   6. Applied unpow-prod-down_binary64 47.7

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

   7. Applied times-frac_binary64 45.2

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

   8. Simplified 45.2

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

   9. Simplified 24.9

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

   10. Applied associate-*r*_binary64 20.9

       \[\leadsto 0.25 \cdot \left(\frac{D}{d} \cdot \left(\color{blue}{\left(\left(h \cdot M\right) \cdot M\right)} \cdot \frac{D}{d}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 1.3564940205041007 \cdot 10^{+168}:\\ \;\;\;\;\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(h \cdot M\right)\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022127 
(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))))))