Henrywood and Agarwal, Equation (13)

Average Error: 59.6 → 27.2

Time: 24.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 5.214799871769385 \cdot 10^{+210}:\\ \;\;\;\;0.25 \cdot \frac{\left(D \cdot D\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}{d}\\ \end{array} \]

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
 (if (<= (* D D) 5.214799871769385e+210)
   (* 0.25 (/ (* (* D D) (/ (* M (* h M)) d)) d))
   (* 0.25 (/ (* D (* D (/ (* h (* M M)) 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 tmp;
	if ((D * D) <= 5.214799871769385e+210) {
		tmp = 0.25 * (((D * D) * ((M * (h * M)) / d)) / d);
	} else {
		tmp = 0.25 * ((D * (D * ((h * (M * M)) / 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 D D) < 5.2147998717693848e210

   1. Initial program 59.4

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

   2. Taylor expanded in c0 around -inf 39.4

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

   3. Simplified 39.4

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

   4. Taylor expanded in c0 around 0 32.1

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

   5. Simplified 32.1

      \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]
   6. Using strategy rm

   7. Applied associate-/r*_binary64 29.1

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

   8. Simplified 28.5

      \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d}}}{d} \]
   9. Using strategy rm

   10. Applied associate-*r*_binary64 25.6

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

   if 5.2147998717693848e210 < (*.f64 D D)

   1. Initial program 60.5

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

   2. Taylor expanded in c0 around -inf 56.9

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

   3. Simplified 56.9

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

   4. Taylor expanded in c0 around 0 54.4

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

   5. Simplified 54.4

      \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]
   6. Using strategy rm

   7. Applied associate-/r*_binary64 53.6

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

   8. Simplified 53.1

      \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d}}}{d} \]
   9. Using strategy rm

   10. Applied associate-*l*_binary64 35.8

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

4. Final simplification 27.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 5.214799871769385 \cdot 10^{+210}:\\ \;\;\;\;0.25 \cdot \frac{\left(D \cdot D\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}{d}\\ \end{array} \]

Reproduce

herbie shell --seed 2021206 
(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))))))