Henrywood and Agarwal, Equation (13)

Average Error: 59.6 → 25.4

Time: 22.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 8.433590193383779 \cdot 10^{+281}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(\frac{M}{d} \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \log \left({\left(e^{\frac{M \cdot M}{d}}\right)}^{\left(\frac{\left(D \cdot D\right) \cdot h}{d}\right)}\right)\\ \end{array}\]

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
 (if (<= (* D D) 8.433590193383779e+281)
   (* 0.25 (/ (* M (* (/ M d) (* (* D D) h))) d))
   (* 0.25 (log (pow (exp (/ (* M M) d)) (/ (* (* D D) h) 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) <= 8.433590193383779e+281) {
		tmp = 0.25 * ((M * ((M / d) * ((D * D) * h))) / d);
	} else {
		tmp = 0.25 * log(pow(exp((M * M) / d), (((D * D) * h) / 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) < 8.4335901933837792e281

   1. Initial program 59.3

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

   2. Taylor expanded around -inf 38.5

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

   3. Simplified 39.5

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

   4. Taylor expanded around 0 32.3

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

   5. Simplified 32.3

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

   7. Applied associate-/r*_binary64_1045 29.4

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

   8. Simplified 28.9

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

   10. Applied *-un-lft-identity_binary64_1101 28.9

       \[\leadsto 0.25 \cdot \frac{\frac{M \cdot M}{\color{blue}{1 \cdot d}} \cdot \left(\left(D \cdot D\right) \cdot h\right)}{d}\]

   11. Applied times-frac_binary64_1107 24.9

       \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(\frac{M}{1} \cdot \frac{M}{d}\right)} \cdot \left(\left(D \cdot D\right) \cdot h\right)}{d}\]

   12. Applied associate-*l*_binary64_1042 23.1

       \[\leadsto 0.25 \cdot \frac{\color{blue}{\frac{M}{1} \cdot \left(\frac{M}{d} \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)}}{d}\]

   if 8.4335901933837792e281 < (*.f64 D D)

   1. Initial program 62.2

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

   2. Taylor expanded around -inf 62.1

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

   3. Simplified 62.3

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

   4. Taylor expanded around 0 61.6

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

   5. Simplified 61.6

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

   7. Applied associate-/r*_binary64_1045 61.5

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

   8. Simplified 61.3

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

   10. Applied add-log-exp_binary64_1140 62.2

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

   11. Simplified 43.5

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

4. Final simplification 25.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 8.433590193383779 \cdot 10^{+281}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(\frac{M}{d} \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \log \left({\left(e^{\frac{M \cdot M}{d}}\right)}^{\left(\frac{\left(D \cdot D\right) \cdot h}{d}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021047 
(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))))))