Henrywood and Agarwal, Equation (13)

Average Error: 59.4 → 26.4

Time: 22.2s

Precision: binary64

Cost: 21314

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 8.99373257184573 \cdot 10^{-165}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 3.749449414383238 \cdot 10^{+248}:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\frac{h \cdot \left(D \cdot D\right)}{d} \cdot \frac{M}{\sqrt[3]{d}}\right)\right)\\ \end{array}\]

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 (<= (* M M) 8.99373257184573e-165)
   0.0
   (if (<= (* M M) 3.749449414383238e+248)
     (* 0.25 (* (/ (* M M) d) (/ (* D (* D h)) d)))
     (*
      0.25
      (*
       (/ M (* (cbrt d) (cbrt d)))
       (* (/ (* h (* D D)) d) (/ M (cbrt 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 ((M * M) <= 8.99373257184573e-165) {
		tmp = 0.0;
	} else if ((M * M) <= 3.749449414383238e+248) {
		tmp = 0.25 * (((M * M) / d) * ((D * (D * h)) / d));
	} else {
		tmp = 0.25 * ((M / (cbrt(d) * cbrt(d))) * (((h * (D * D)) / d) * (M / cbrt(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

Alternatives

Alternative 1
Error	26.6
Cost	1858

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 8.99373257184573 \cdot 10^{-165}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 3.749449414383238 \cdot 10^{+248}:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(M \cdot \left(\frac{h \cdot \left(D \cdot D\right)}{d} \cdot \frac{M}{d}\right)\right)\\ \end{array}\]

Alternative 2
Error	28.0
Cost	1602

\[\begin{array}{l} \mathbf{if}\;d \leq -1.5789612593083392 \cdot 10^{+240}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq 1.1472860962098607 \cdot 10^{+184}:\\ \;\;\;\;0.25 \cdot \left(M \cdot \left(\frac{h \cdot \left(D \cdot D\right)}{d} \cdot \frac{M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 3
Error	28.3
Cost	1858

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 1.805254340274376 \cdot 10^{-214}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 6.495598949195897 \cdot 10^{+288}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{M \cdot M}{d} \cdot h\right) \cdot \frac{D \cdot D}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)}{d \cdot d}\\ \end{array}\]

Alternative 4
Error	28.7
Cost	1858

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 8.99373257184573 \cdot 10^{-165}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 6.495598949195897 \cdot 10^{+288}:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)}{d \cdot d}\\ \end{array}\]

Alternative 5
Error	29.1
Cost	1858

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 6.787167257433162 \cdot 10^{-164}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 2.6923044536263433 \cdot 10^{+289}:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 6
Error	29.0
Cost	1858

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 3.295220516778699 \cdot 10^{-178}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 2.6923044536263433 \cdot 10^{+289}:\\ \;\;\;\;0.25 \cdot \frac{\frac{M \cdot M}{d} \cdot \left(h \cdot \left(D \cdot D\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 7
Error	29.0
Cost	1858

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 1.0273504708314887 \cdot 10^{-164}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 2.6923044536263433 \cdot 10^{+289}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \frac{h \cdot \left(D \cdot D\right)}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 8
Error	29.6
Cost	1858

\[\begin{array}{l} \mathbf{if}\;d \cdot d \leq 3.6829237027202115 \cdot 10^{-308}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \cdot d \leq 4.044613264486354 \cdot 10^{+300}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \left(D \cdot \left(D \cdot h\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 9
Error	30.6
Cost	1858

\[\begin{array}{l} \mathbf{if}\;d \cdot d \leq 6.11052838450271 \cdot 10^{-307}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \cdot d \leq 5.394462067156222 \cdot 10^{+300}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 10
Error	32.1
Cost	64

\[0\]

Alternative 11
Error	61.9
Cost	64

\[1\]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M M) < 8.9937325718457304e-165

   1. Initial program 24.3

      \[0\]

   if 8.9937325718457304e-165 < (*.f64 M M) < 3.7494494143832381e248

   1. Initial program 61.7

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

   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 38.5

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

   4. Taylor expanded around 0 31.1

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

   5. Simplified 31.1

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

   7. Applied times-frac_binary64_1107 27.3

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

   9. Applied associate-*r*_binary64_1041 23.9

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

   10. Simplified 23.9

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

   if 3.7494494143832381e248 < (*.f64 M M)

   1. Initial program 64.0

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

   2. Taylor expanded around -inf 59.7

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

   3. Simplified 59.7

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

   4. Taylor expanded around 0 58.8

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

   5. Simplified 58.8

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

   7. Applied times-frac_binary64_1107 57.4

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

   9. Applied add-cube-cbrt_binary64_1136 57.5

      \[\leadsto \left(\frac{M \cdot M}{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}} \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right) \cdot 0.25\]

   10. Applied times-frac_binary64_1107 43.9

       \[\leadsto \left(\color{blue}{\left(\frac{M}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \frac{M}{\sqrt[3]{d}}\right)} \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right) \cdot 0.25\]

   11. Applied associate-*l*_binary64_1042 36.7

       \[\leadsto \color{blue}{\left(\frac{M}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\frac{M}{\sqrt[3]{d}} \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right)\right)} \cdot 0.25\]

   12. Simplified 36.7

       \[\leadsto \left(\frac{M}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \color{blue}{\left(\frac{h \cdot \left(D \cdot D\right)}{d} \cdot \frac{M}{\sqrt[3]{d}}\right)}\right) \cdot 0.25\]

   13. Simplified 36.7

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

4. Final simplification 26.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 8.99373257184573 \cdot 10^{-165}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 3.749449414383238 \cdot 10^{+248}:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\frac{h \cdot \left(D \cdot D\right)}{d} \cdot \frac{M}{\sqrt[3]{d}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))))