Henrywood and Agarwal, Equation (13)

Average Error: 59.6 → 25.7

Time: 30.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 3.402992039699089 \cdot 10^{+236}:\\ \;\;\;\;\left(\frac{M}{d} \cdot \left(\frac{M}{d} \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)}{d \cdot d}\\ \end{array}\]

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) 3.402992039699089e+236)
   (* (* (/ M d) (* (/ M d) (* (* D D) h))) 0.25)
   (* 0.25 (/ (* M (* M (* D (* D 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 tmp;
	if ((D * D) <= 3.402992039699089e+236) {
		tmp = ((M / d) * ((M / d) * ((D * D) * h))) * 0.25;
	} else {
		tmp = 0.25 * ((M * (M * (D * (D * 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 D D) < 3.40299203969908883e236

   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 around -inf 38.1

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

   3. Simplified 38.1

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

   4. Taylor expanded around 0 31.7

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

   5. Simplified 31.7

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

   7. Applied associate-*l*_binary64_1042 28.5

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

   9. Applied times-frac_binary64_1107 23.2

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

   10. Simplified 22.3

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

   if 3.40299203969908883e236 < (*.f64 D D)

   1. Initial program 61.1

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

   2. Taylor expanded around -inf 59.2

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

   3. Simplified 59.2

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

   4. Taylor expanded around 0 58.0

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

   5. Simplified 58.0

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

   7. Applied associate-*l*_binary64_1042 57.6

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

   9. Applied associate-*r*_binary64_1041 46.9

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

4. Final simplification 25.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 3.402992039699089 \cdot 10^{+236}:\\ \;\;\;\;\left(\frac{M}{d} \cdot \left(\frac{M}{d} \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)}{d \cdot d}\\ \end{array}\]

Reproduce

herbie shell --seed 2021007 
(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))))))