Average Error: 59.6 → 27.7

Time: 24.6s

Precision: binary64

Cost: 21314

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

↓

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 5.8123201035385834 \cdot 10^{+51}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \frac{D \cdot \left(D \cdot h\right)}{d}}{d}\\ \mathbf{elif}\;M \cdot M \leq 1.1046437598229507 \cdot 10^{+252}:\\ \;\;\;\;0.25 \cdot \frac{\left(\left(M \cdot M\right) \cdot \frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{D \cdot D}{\sqrt[3]{d}}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right)}{d}\\ \end{array}\]

\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 5.8123201035385834 \cdot 10^{+51}:\\
\;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \frac{D \cdot \left(D \cdot h\right)}{d}}{d}\\

\mathbf{elif}\;M \cdot M \leq 1.1046437598229507 \cdot 10^{+252}:\\
\;\;\;\;0.25 \cdot \frac{\left(\left(M \cdot M\right) \cdot \frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{D \cdot D}{\sqrt[3]{d}}}{d}\\

\mathbf{else}:\\
\;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right)}{d}\\

\end{array}

(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) 5.8123201035385834e+51)
   (* 0.25 (/ (* (* M M) (/ (* D (* D h)) d)) d))
   (if (<= (* M M) 1.1046437598229507e+252)
     (*
      0.25
      (/ (* (* (* M M) (/ h (* (cbrt d) (cbrt d)))) (/ (* D D) (cbrt d))) d))
     (* 0.25 (/ (* M (* M (/ (* h (* D D)) d))) d)))))

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) <= 5.8123201035385834e+51) {
		tmp = 0.25 * (((M * M) * ((D * (D * h)) / d)) / d);
	} else if ((M * M) <= 1.1046437598229507e+252) {
		tmp = 0.25 * ((((M * M) * (h / (cbrt(d) * cbrt(d)))) * ((D * D) / cbrt(d))) / d);
	} else {
		tmp = 0.25 * ((M * (M * ((h * (D * D)) / d))) / d);
	}
	return tmp;
}

Error

The X axis uses an exponential scale — Bits error versus `c0`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	60.4
Cost	30656

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

Alternative 2
Error	62.8
Cost	28736

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

Alternative 3
Error	62.8
Cost	28736

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

Alternative 4
Error	57.0
Cost	26304

\[0.25 \cdot e^{\log h + 2 \cdot \left(\log \left(M \cdot D\right) - \log d\right)}\]

Alternative 5
Error	41.8
Cost	23104

\[\frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \left(\sqrt[3]{\frac{\left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right) \cdot w}{\left(d \cdot d\right) \cdot c0}} \cdot \left(\sqrt[3]{\frac{\left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right) \cdot w}{\left(d \cdot d\right) \cdot c0}} \cdot \sqrt[3]{\frac{\left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right) \cdot w}{\left(d \cdot d\right) \cdot c0}}\right)\right)\right)\]

Alternative 6
Error	62.6
Cost	22464

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

Alternative 7
Error	49.2
Cost	22080

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

Alternative 8
Error	36.0
Cost	21952

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

Alternative 9
Error	41.8
Cost	21568

\[\frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{\left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right) \cdot w}{\sqrt[3]{\left(d \cdot d\right) \cdot c0} \cdot \left(\sqrt[3]{\left(d \cdot d\right) \cdot c0} \cdot \sqrt[3]{\left(d \cdot d\right) \cdot c0}\right)}\right)\]

Alternative 10
Error	39.1
Cost	20608

\[\log \left({\left(e^{\frac{\left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right) \cdot w}{\left(d \cdot d\right) \cdot c0}}\right)}^{\left(0.25 \cdot \frac{c0}{w}\right)}\right)\]

Alternative 11
Error	32.4
Cost	20416

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

Alternative 12
Error	32.9
Cost	20416

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

Alternative 13
Error	33.1
Cost	20416

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

Alternative 14
Error	52.8
Cost	20032

\[0.25 \cdot \frac{e^{\log h + 2 \cdot \log \left(M \cdot D\right)}}{d \cdot d}\]

Alternative 15
Error	48.5
Cost	20032

\[0.25 \cdot \frac{e^{2 \cdot \log \left(M \cdot D\right) + \log \left(\frac{h}{d}\right)}}{d}\]

Alternative 16
Error	47.9
Cost	20032

\[0.25 \cdot e^{\log \left(h \cdot \left(D \cdot D\right)\right) + 2 \cdot \log \left(\frac{M}{d}\right)}\]

Alternative 17
Error	61.7
Cost	15616

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

Alternative 18
Error	37.4
Cost	14656

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

Alternative 19
Error	52.5
Cost	14528

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

Alternative 20
Error	43.2
Cost	14400

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

Alternative 21
Error	36.1
Cost	14272

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

Alternative 22
Error	47.3
Cost	13888

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

Alternative 23
Error	48.2
Cost	13888

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

Alternative 24
Error	48.2
Cost	13888

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

Alternative 25
Error	49.3
Cost	13888

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

Alternative 26
Error	34.9
Cost	13824

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

Alternative 27
Error	39.0
Cost	13760

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

Alternative 28
Error	40.5
Cost	13760

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

Alternative 29
Error	42.9
Cost	13760

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

Alternative 30
Error	60.3
Cost	9664

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

Alternative 31
Error	60.3
Cost	9664

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

Alternative 32
Error	59.6
Cost	9664

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

Alternative 33
Error	62.6
Cost	7936

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

Alternative 34
Error	52.0
Cost	7040

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

Alternative 35
Error	41.8
Cost	1600

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

Alternative 36
Error	41.2
Cost	1600

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

Alternative 37
Error	38.5
Cost	1600

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

Alternative 38
Error	39.7
Cost	1600

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

Alternative 39
Error	39.7
Cost	1600

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

Alternative 40
Error	43.6
Cost	1600

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

Alternative 41
Error	40.5
Cost	1600

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

Alternative 42
Error	43.3
Cost	1600

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

Alternative 43
Error	41.9
Cost	1600

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

Alternative 44
Error	38.5
Cost	1472

\[c0 \cdot \left(\frac{\left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right) \cdot w}{\left(d \cdot d\right) \cdot c0} \cdot \frac{0.25}{w}\right)\]

Alternative 45
Error	41.8
Cost	1472

\[\frac{\left(\left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right) \cdot w\right) \cdot \left(0.25 \cdot \frac{c0}{w}\right)}{\left(d \cdot d\right) \cdot c0}\]

Alternative 46
Error	59.6
Cost	1344

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

Alternative 47
Error	36.2
Cost	1088

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

Alternative 48
Error	32.9
Cost	1088

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

Alternative 49
Error	35.5
Cost	960

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

Alternative 50
Error	32.9
Cost	960

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

Alternative 51
Error	33.7
Cost	960

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

Alternative 52
Error	32.9
Cost	960

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

Alternative 53
Error	32.9
Cost	960

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

Alternative 54
Error	36.0
Cost	960

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

Alternative 55
Error	32.1
Cost	960

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

Alternative 56
Error	30.5
Cost	960

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

Alternative 57
Error	29.3
Cost	960

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

Alternative 58
Error	32.1
Cost	960

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

Alternative 59
Error	34.1
Cost	960

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

Alternative 60
Error	33.2
Cost	960

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

Alternative 61
Error	61.9
Cost	64

\[1\]

Alternative 62
Error	32.3
Cost	64

\[0\]

Alternative 63
Error	61.9
Cost	64

\[-1\]

Derivation

Split input into 3 regimes
if (*.f64 M M) < 5.81232010353858337e51
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 37.5
  \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{w \cdot \left({M}^{2} \cdot \left({D}^{2} \cdot h\right)\right)}{c0 \cdot {d}^{2}}\right)}\]
3. Simplified37.5
  \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{w \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)}{c0 \cdot \left(d \cdot d\right)}\right)}\]
4. Taylor expanded around 0 30.7
  \[\leadsto \color{blue}{0.25 \cdot \frac{{M}^{2} \cdot \left({D}^{2} \cdot h\right)}{{d}^{2}}}\]
5. Simplified30.7
  \[\leadsto \color{blue}{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{d \cdot d} \cdot 0.25}\]
6. Using strategy rm
7. Applied associate-/r*_binary64_70427.5
  \[\leadsto \color{blue}{\frac{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{d}}{d}} \cdot 0.25\]
8. Simplified27.1
  \[\leadsto \frac{\color{blue}{\left(M \cdot M\right) \cdot \frac{h \cdot \left(D \cdot D\right)}{d}}}{d} \cdot 0.25\]
9. Using strategy rm
10. Applied associate-*r*_binary64_70024.5
  \[\leadsto \frac{\left(M \cdot M\right) \cdot \frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{d}}{d} \cdot 0.25\]
11. Simplified24.5
  \[\leadsto \color{blue}{0.25 \cdot \frac{\left(M \cdot M\right) \cdot \frac{D \cdot \left(h \cdot D\right)}{d}}{d}}\]
if 5.81232010353858337e51 < (*.f64 M M) < 1.10464375982295069e252
1. Initial program 62.6
  \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
2. Taylor expanded around -inf 40.4
  \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{w \cdot \left({M}^{2} \cdot \left({D}^{2} \cdot h\right)\right)}{c0 \cdot {d}^{2}}\right)}\]
3. Simplified40.4
  \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{w \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)}{c0 \cdot \left(d \cdot d\right)}\right)}\]
4. Taylor expanded around 0 33.5
  \[\leadsto \color{blue}{0.25 \cdot \frac{{M}^{2} \cdot \left({D}^{2} \cdot h\right)}{{d}^{2}}}\]
5. Simplified33.5
  \[\leadsto \color{blue}{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{d \cdot d} \cdot 0.25}\]
6. Using strategy rm
7. Applied associate-/r*_binary64_70431.3
  \[\leadsto \color{blue}{\frac{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{d}}{d}} \cdot 0.25\]
8. Simplified29.9
  \[\leadsto \frac{\color{blue}{\left(M \cdot M\right) \cdot \frac{h \cdot \left(D \cdot D\right)}{d}}}{d} \cdot 0.25\]
9. Using strategy rm
10. Applied add-cube-cbrt_binary64_79530.1
  \[\leadsto \frac{\left(M \cdot M\right) \cdot \frac{h \cdot \left(D \cdot D\right)}{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}}{d} \cdot 0.25\]
11. Applied times-frac_binary64_76628.8
  \[\leadsto \frac{\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \frac{D \cdot D}{\sqrt[3]{d}}\right)}}{d} \cdot 0.25\]
12. Applied associate-*r*_binary64_70028.5
  \[\leadsto \frac{\color{blue}{\left(\left(M \cdot M\right) \cdot \frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{D \cdot D}{\sqrt[3]{d}}}}{d} \cdot 0.25\]
13. Simplified28.5
  \[\leadsto \color{blue}{0.25 \cdot \frac{\left(\left(M \cdot M\right) \cdot \frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{D \cdot D}{\sqrt[3]{d}}}{d}}\]
if 1.10464375982295069e252 < (*.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 59.4
  \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{w \cdot \left({M}^{2} \cdot \left({D}^{2} \cdot h\right)\right)}{c0 \cdot {d}^{2}}\right)}\]
3. Simplified59.4
  \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{w \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)}{c0 \cdot \left(d \cdot d\right)}\right)}\]
4. Taylor expanded around 0 58.3
  \[\leadsto \color{blue}{0.25 \cdot \frac{{M}^{2} \cdot \left({D}^{2} \cdot h\right)}{{d}^{2}}}\]
5. Simplified58.3
  \[\leadsto \color{blue}{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{d \cdot d} \cdot 0.25}\]
6. Using strategy rm
7. Applied associate-/r*_binary64_70457.8
  \[\leadsto \color{blue}{\frac{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{d}}{d}} \cdot 0.25\]
8. Simplified57.6
  \[\leadsto \frac{\color{blue}{\left(M \cdot M\right) \cdot \frac{h \cdot \left(D \cdot D\right)}{d}}}{d} \cdot 0.25\]
9. Using strategy rm
10. Applied associate-*l*_binary64_70139.5
  \[\leadsto \frac{\color{blue}{M \cdot \left(M \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right)}}{d} \cdot 0.25\]
11. Simplified39.5
  \[\leadsto \color{blue}{0.25 \cdot \frac{M \cdot \left(M \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right)}{d}}\]
Recombined 3 regimes into one program.
Final simplification27.7
\[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 5.8123201035385834 \cdot 10^{+51}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \frac{D \cdot \left(D \cdot h\right)}{d}}{d}\\ \mathbf{elif}\;M \cdot M \leq 1.1046437598229507 \cdot 10^{+252}:\\ \;\;\;\;0.25 \cdot \frac{\left(\left(M \cdot M\right) \cdot \frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{D \cdot D}{\sqrt[3]{d}}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right)}{d}\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(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))))))

Error

Try it out

Alternatives

Error

Derivation

`if (*.f64 M M) < 5.81232010353858337e51`

`if 5.81232010353858337e51 < (*.f64 M M) < 1.10464375982295069e252`

`if 1.10464375982295069e252 < (*.f64 M M)`

Reproduce