Henrywood and Agarwal, Equation (13)

Average Error: 59.5 → 27.3

Time: 28.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;D \leq -1.7964762795118962 \cdot 10^{+146}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \sqrt{-M \cdot M}\\ \mathbf{elif}\;D \leq 2.3663530675469865 \cdot 10^{+125}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \frac{\left(D \cdot D\right) \cdot h}{d}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \frac{D \cdot \left(D \cdot h\right)}{d}}{d}\\ \end{array}\]

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 -1.7964762795118962e+146)
   (* (/ c0 (* 2.0 w)) (sqrt (- (* M M))))
   (if (<= D 2.3663530675469865e+125)
     (* 0.25 (/ (* M (* M (/ (* (* D D) h) d))) d))
     (* 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 <= -1.7964762795118962e+146) {
		tmp = (c0 / (2.0 * w)) * sqrt(-(M * M));
	} else if (D <= 2.3663530675469865e+125) {
		tmp = 0.25 * ((M * (M * (((D * D) * h) / d))) / d);
	} 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 3 regimes

2. if D < -1.7964762795118962e146

   1. Initial program 61.8

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

   2. Taylor expanded around 0 50.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\sqrt{-{M}^{2}}}\]

   3. Simplified 50.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\sqrt{-M \cdot M}}\]

   if -1.7964762795118962e146 < D < 2.36635306754698647e125

   1. Initial program 59.1

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

   2. Taylor expanded around -inf 38.2

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

   3. Simplified 38.9

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

   4. Taylor expanded around 0 31.6

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

   5. Simplified 31.6

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

   7. Applied associate-/r*_binary64_1045 29.1

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

   8. Simplified 28.5

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

   10. Applied associate-*l*_binary64_1042 24.4

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

   if 2.36635306754698647e125 < D

   1. Initial program 62.0

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

   2. Taylor expanded around -inf 59.6

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

   3. Simplified 60.5

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

   4. Taylor expanded around 0 59.2

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

   5. Simplified 59.2

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

   7. Applied associate-/r*_binary64_1045 59.0

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

   8. Simplified 58.6

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

   10. Applied associate-*l*_binary64_1042 47.1

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

4. Final simplification 27.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq -1.7964762795118962 \cdot 10^{+146}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \sqrt{-M \cdot M}\\ \mathbf{elif}\;D \leq 2.3663530675469865 \cdot 10^{+125}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \frac{\left(D \cdot D\right) \cdot h}{d}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \frac{D \cdot \left(D \cdot h\right)}{d}}{d}\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(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))))))