Henrywood and Agarwal, Equation (13)

Average Error: 59.5 → 25.8

Time: 26.0s

Precision: binary64

Cost: 1602

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;D \leq -2.9238707485049054 \cdot 10^{+63}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 1.6115875034803501 \cdot 10^{+146}:\\ \;\;\;\;0.25 \cdot \left(\left(M \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right) \cdot \frac{M}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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 -2.9238707485049054e+63)
   0.0
   (if (<= D 1.6115875034803501e+146)
     (* 0.25 (* (* M (/ (* h (* D D)) d)) (/ M d)))
     0.0)))

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 <= -2.9238707485049054e+63) {
		tmp = 0.0;
	} else if (D <= 1.6115875034803501e+146) {
		tmp = 0.25 * ((M * ((h * (D * D)) / d)) * (M / d));
	} else {
		tmp = 0.0;
	}
	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	27.0
Cost	21058

\[\begin{array}{l} \mathbf{if}\;M \leq 1.0089094772484812 \cdot 10^{-187}:\\ \;\;\;\;0.25 \cdot \left(\left(M \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right) \cdot \frac{M}{d}\right)\\ \mathbf{elif}\;M \leq 6.647082908848975 \cdot 10^{+32}:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{h \cdot \left(D \cdot D\right)}{d} \cdot \frac{M}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{M}{\sqrt[3]{d}}\right)\\ \end{array}\]

Alternative 2
Error	26.0
Cost	1409

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 2.3824290065840326 \cdot 10^{+299}:\\ \;\;\;\;0.25 \cdot \left(\left(M \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right) \cdot \frac{M}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \left(D \cdot \left(D \cdot h\right)\right)}{d \cdot d}\\ \end{array}\]

Alternative 3
Error	27.0
Cost	21314

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 8.9040510693509 \cdot 10^{-320}:\\ \;\;\;\;0.25 \cdot \left(\left(M \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right) \cdot \frac{M}{d}\right)\\ \mathbf{elif}\;M \cdot M \leq 1.4195040828507912 \cdot 10^{+55}:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{\frac{d}{D \cdot D}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{h \cdot \left(D \cdot D\right)}{d} \cdot \frac{M}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{M}{\sqrt[3]{d}}\right)\\ \end{array}\]

Alternative 4
Error	27.0
Cost	21000

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 8.9040510693509 \cdot 10^{-320} \lor \neg \left(M \cdot M \leq 1.4195040828507912 \cdot 10^{+55}\right):\\ \;\;\;\;0.25 \cdot \left(\left(\frac{h \cdot \left(D \cdot D\right)}{d} \cdot \frac{M}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{M}{\sqrt[3]{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{\frac{d}{D \cdot D}}\right)\\ \end{array}\]

Alternative 5
Error	27.0
Cost	21000

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 0 \lor \neg \left(M \cdot M \leq 2.62195929256777 \cdot 10^{+105}\right):\\ \;\;\;\;0.25 \cdot \left(\left(\frac{h \cdot \left(D \cdot D\right)}{d} \cdot \frac{M}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{M}{\sqrt[3]{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot M}{d} \cdot \left(\frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \frac{D \cdot D}{\sqrt[3]{d}}\right)\right)\\ \end{array}\]

Alternative 6
Error	27.1
Cost	21000

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 0 \lor \neg \left(M \cdot M \leq 6.89936055196706 \cdot 10^{+117}\right):\\ \;\;\;\;0.25 \cdot \left(\left(\frac{h \cdot \left(D \cdot D\right)}{d} \cdot \frac{M}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{M}{\sqrt[3]{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot D}{\sqrt[3]{d}}\right)\right)\\ \end{array}\]

Alternative 7
Error	27.4
Cost	20416

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

Alternative 8
Error	30.7
Cost	27528

\[\begin{array}{l} \mathbf{if}\;M \leq -1.3365260190186299 \cdot 10^{+154} \lor \neg \left(M \leq 1.1307209009648695 \cdot 10^{+142}\right):\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{\frac{h \cdot \left(D \cdot D\right)}{d}} \cdot \frac{M}{\sqrt{d}}\right) \cdot \left(\sqrt{\frac{h \cdot \left(D \cdot D\right)}{d}} \cdot \frac{M}{\sqrt{d}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\sqrt[3]{\frac{h \cdot \left(D \cdot D\right)}{d} \cdot \frac{M \cdot M}{d}} \cdot \left(\sqrt[3]{\frac{h \cdot \left(D \cdot D\right)}{d} \cdot \frac{M \cdot M}{d}} \cdot \sqrt[3]{\frac{h \cdot \left(D \cdot D\right)}{d} \cdot \frac{M \cdot M}{d}}\right)\right)\\ \end{array}\]

Alternative 9
Error	41.3
Cost	27521

\[\begin{array}{l} \mathbf{if}\;d \leq -8.948325751090156 \cdot 10^{-85}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{M \cdot M}{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - \sqrt{\frac{c0}{h \cdot w} \cdot \frac{c0 \cdot {d}^{4}}{\left(h \cdot w\right) \cdot {D}^{4}} - M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{\frac{h \cdot \left(D \cdot D\right)}{d}} \cdot \frac{M}{\sqrt{d}}\right) \cdot \left(\sqrt{\frac{h \cdot \left(D \cdot D\right)}{d}} \cdot \frac{M}{\sqrt{d}}\right)\right)\\ \end{array}\]

Alternative 10
Error	47.1
Cost	30977

\[\begin{array}{l} \mathbf{if}\;d \leq -1.8297134400212148 \cdot 10^{-258}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \sqrt[3]{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}} \cdot \left(\sqrt[3]{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}} \cdot \sqrt[3]{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{\frac{h \cdot \left(D \cdot D\right)}{d}} \cdot \frac{M}{\sqrt{d}}\right) \cdot \left(\sqrt{\frac{h \cdot \left(D \cdot D\right)}{d}} \cdot \frac{M}{\sqrt{d}}\right)\right)\\ \end{array}\]

Alternative 11
Error	60.5
Cost	30656

\[\frac{c0}{2 \cdot w} \cdot \left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \sqrt[3]{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}} \cdot \left(\sqrt[3]{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}} \cdot \sqrt[3]{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}}\right)\right)\]

Alternative 12
Error	62.8
Cost	34688

\[\frac{c0}{2 \cdot w} \cdot \sqrt[3]{{\left(\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} + \sqrt{\frac{c0}{h \cdot w} \cdot \frac{c0 \cdot {d}^{4}}{\left(h \cdot w\right) \cdot {D}^{4}} - M \cdot M}\right)}^{3}}\]

Alternative 13
Error	62.8
Cost	84032

\[\frac{c0}{2 \cdot w} \cdot \left(\sqrt[3]{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} + \sqrt{\frac{c0}{h \cdot w} \cdot \frac{c0 \cdot {d}^{4}}{\left(h \cdot w\right) \cdot {D}^{4}} - M \cdot M}} \cdot \left(\sqrt[3]{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} + \sqrt{\frac{c0}{h \cdot w} \cdot \frac{c0 \cdot {d}^{4}}{\left(h \cdot w\right) \cdot {D}^{4}} - M \cdot M}} \cdot \sqrt[3]{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} + \sqrt{\frac{c0}{h \cdot w} \cdot \frac{c0 \cdot {d}^{4}}{\left(h \cdot w\right) \cdot {D}^{4}} - M \cdot M}}\right)\right)\]

Derivation

1. Split input into 2 regimes

2. if D < -2.92387074850490543e63 or 1.6115875034803501e146 < D

   1. Initial program 41.8

      \[0\]

   if -2.92387074850490543e63 < D < 1.6115875034803501e146

   1. Initial program 59.5

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

   2. Taylor expanded around -inf 38.1

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

   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 31.9

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

   5. Simplified 31.9

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

   7. Applied times-frac_binary64_1107 28.4

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

   9. Applied *-un-lft-identity_binary64_1101 28.4

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

   10. Applied times-frac_binary64_1107 24.8

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

   11. Applied associate-*r*_binary64_1041 23.0

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

   12. Simplified 23.0

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

   13. Simplified 23.0

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

4. Final simplification 25.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq -2.9238707485049054 \cdot 10^{+63}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 1.6115875034803501 \cdot 10^{+146}:\\ \;\;\;\;0.25 \cdot \left(\left(M \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right) \cdot \frac{M}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(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))))))