Henrywood and Agarwal, Equation (13)

Average Error: 59.7 → 28.6

Time: 17.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;D \leq -1.3535396070238666 \cdot 10^{+154}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \frac{D \cdot \left(D \cdot h\right)}{d}}{d}\\ \mathbf{elif}\;D \leq 3.601076346426854 \cdot 10^{-175}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right)}{d}\\ \mathbf{elif}\;D \leq 2.921876147211044 \cdot 10^{-162}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \frac{D \cdot \left(D \cdot h\right)}{d}}{d}\\ \mathbf{elif}\;D \leq 3.8477647255578 \cdot 10^{-32}:\\ \;\;\;\;0.25 \cdot \frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot D\right)\right) \cdot \frac{h}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot \left(D \cdot D\right)}{d} \cdot \frac{M \cdot M}{d}\right)\\ \end{array}\]

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 -1.3535396070238666e+154)
   (* 0.25 (/ (* (* M M) (/ (* D (* D h)) d)) d))
   (if (<= D 3.601076346426854e-175)
     (* 0.25 (/ (* M (* M (/ (* h (* D D)) d))) d))
     (if (<= D 2.921876147211044e-162)
       (* 0.25 (/ (* (* M M) (/ (* D (* D h)) d)) d))
       (if (<= D 3.8477647255578e-32)
         (* 0.25 (/ (* (* (* M M) (* D D)) (/ h d)) d))
         (* 0.25 (* (/ (* h (* D D)) d) (/ (* M M) 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 <= -1.3535396070238666e+154) {
		tmp = 0.25 * (((M * M) * ((D * (D * h)) / d)) / d);
	} else if (D <= 3.601076346426854e-175) {
		tmp = 0.25 * ((M * (M * ((h * (D * D)) / d))) / d);
	} else if (D <= 2.921876147211044e-162) {
		tmp = 0.25 * (((M * M) * ((D * (D * h)) / d)) / d);
	} else if (D <= 3.8477647255578e-32) {
		tmp = 0.25 * ((((M * M) * (D * D)) * (h / d)) / d);
	} else {
		tmp = 0.25 * (((h * (D * D)) / d) * ((M * M) / 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 4 regimes

2. if D < -1.3535396070238666e154 or 3.6010763464268542e-175 < D < 2.92187614721104395e-162

   1. Initial program 63.0

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

   2. Taylor expanded around -inf 58.2

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

   3. Simplified 58.2

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

   4. Taylor expanded around 0 57.4

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

   5. Simplified 57.4

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

   7. Applied associate-/r*_binary64 57.1

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

   8. Simplified 57.0

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

   10. Applied associate-*l*_binary64 43.6

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

   if -1.3535396070238666e154 < D < 3.6010763464268542e-175

   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 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 39.0

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

   4. Taylor expanded around 0 31.8

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

   5. Simplified 31.8

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

   7. Applied associate-/r*_binary64 28.6

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

   8. Simplified 28.2

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

   10. Applied associate-*l*_binary64 24.3

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

   if 2.92187614721104395e-162 < D < 3.84776472555779985e-32

   1. Initial program 57.2

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

   2. Taylor expanded around -inf 37.1

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

   3. Simplified 37.2

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

   4. Taylor expanded around 0 30.0

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

   5. Simplified 30.0

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

   7. Applied associate-/r*_binary64 28.1

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

   8. Simplified 27.8

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

   10. Applied *-un-lft-identity_binary64 27.8

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

   11. Applied times-frac_binary64 27.6

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

   12. Applied associate-*r*_binary64 27.2

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

   if 3.84776472555779985e-32 < D

   1. Initial program 57.8

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

   2. Taylor expanded around -inf 47.8

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

   3. Simplified 49.0

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

   4. Taylor expanded around 0 42.3

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

   5. Simplified 42.3

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

   7. Applied times-frac_binary64 39.7

      \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{\left(D \cdot D\right) \cdot h}{d}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 28.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq -1.3535396070238666 \cdot 10^{+154}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \frac{D \cdot \left(D \cdot h\right)}{d}}{d}\\ \mathbf{elif}\;D \leq 3.601076346426854 \cdot 10^{-175}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right)}{d}\\ \mathbf{elif}\;D \leq 2.921876147211044 \cdot 10^{-162}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \frac{D \cdot \left(D \cdot h\right)}{d}}{d}\\ \mathbf{elif}\;D \leq 3.8477647255578 \cdot 10^{-32}:\\ \;\;\;\;0.25 \cdot \frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot D\right)\right) \cdot \frac{h}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot \left(D \cdot D\right)}{d} \cdot \frac{M \cdot M}{d}\right)\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(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))))))