Average Error: 59.6 → 27.2

Time: 19.0s

Precision: binary64

Cost: 1409

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

↓

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 9.05100518124077 \cdot 10^{+73}:\\ \;\;\;\;0.25 \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot h\right) \cdot M}{d} \cdot \frac{M}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)}{d \cdot d}\\ \end{array}\]

\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}\;D \cdot D \leq 9.05100518124077 \cdot 10^{+73}:\\
\;\;\;\;0.25 \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot h\right) \cdot M}{d} \cdot \frac{M}{d}\right)\\

\mathbf{else}:\\
\;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)}{d \cdot 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 (<= (* D D) 9.05100518124077e+73)
   (* 0.25 (* (/ (* (* (* D D) h) M) d) (/ M d)))
   (* 0.25 (/ (* M (* M (* D (* D h)))) (* 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 ((D * D) <= 9.05100518124077e+73) {
		tmp = 0.25 * (((((D * D) * h) * M) / d) * (M / d));
	} else {
		tmp = 0.25 * ((M * (M * (D * (D * h)))) / (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	62.9
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 2
Error	63.3
Cost	34624

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

Alternative 3
Error	60.4
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 4
Error	56.8
Cost	26304

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

Alternative 5
Error	56.8
Cost	26304

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

Alternative 6
Error	41.2
Cost	23744

\[\sqrt[3]{\frac{\left(\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right) \cdot w}{\left(d \cdot d\right) \cdot c0} \cdot \left(0.25 \cdot \frac{c0}{w}\right)} \cdot \left(\sqrt[3]{\frac{\left(\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right) \cdot w}{\left(d \cdot d\right) \cdot c0} \cdot \left(0.25 \cdot \frac{c0}{w}\right)} \cdot \sqrt[3]{\frac{\left(\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right) \cdot w}{\left(d \cdot d\right) \cdot c0} \cdot \left(0.25 \cdot \frac{c0}{w}\right)}\right)\]

Alternative 7
Error	60.7
Cost	22528

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

Alternative 8
Error	48.5
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 9
Error	35.4
Cost	21952

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

Alternative 10
Error	35.4
Cost	21440

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

Alternative 11
Error	35.4
Cost	20928

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

Alternative 12
Error	38.6
Cost	20608

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

Alternative 13
Error	35.4
Cost	20416

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

Alternative 14
Error	35.6
Cost	20096

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

Alternative 15
Error	52.6
Cost	20032

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

Alternative 16
Error	61.9
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 17
Error	52.1
Cost	14528

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

Alternative 18
Error	43.5
Cost	14464

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

Alternative 19
Error	46.5
Cost	14400

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

Alternative 20
Error	42.7
Cost	14400

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

Alternative 21
Error	48.9
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 22
Error	47.3
Cost	13888

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

Alternative 23
Error	42.5
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 24
Error	38.5
Cost	13760

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

Alternative 25
Error	60.8
Cost	9664

Alternative 26
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 27
Error	62.5
Cost	7936

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

Alternative 28
Error	51.4
Cost	7040

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

Alternative 29
Error	41.3
Cost	1728

\[\frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \left(\left(\left(\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right) \cdot w\right) \cdot \frac{1}{\left(d \cdot d\right) \cdot c0}\right)\right)\]

Alternative 30
Error	38.0
Cost	1600

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

Alternative 31
Error	41.2
Cost	1600

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

Alternative 32
Error	39.9
Cost	1600

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

Alternative 33
Error	39.3
Cost	1600

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

Alternative 34
Error	41.2
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 35
Error	39.0
Cost	1600

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

Alternative 36
Error	39.8
Cost	1600

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

Alternative 37
Error	42.7
Cost	1600

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

Alternative 38
Error	41.2
Cost	1472

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

Alternative 39
Error	38.0
Cost	1472

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

Alternative 40
Error	41.1
Cost	1472

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

Alternative 41
Error	59.5
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 42
Error	30.2
Cost	960

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

Alternative 43
Error	33.5
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 44
Error	32.5
Cost	960

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

Alternative 45
Error	35.1
Cost	960

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

Alternative 46
Error	32.4
Cost	960

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

Alternative 47
Error	33.1
Cost	960

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

Alternative 48
Error	32.5
Cost	960

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

Alternative 49
Error	32.0
Cost	960

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

Alternative 50
Error	28.2
Cost	960

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

Alternative 51
Error	29.0
Cost	960

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

Alternative 52
Error	27.6
Cost	960

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

Alternative 53
Error	35.4
Cost	960

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

Alternative 54
Error	61.9
Cost	64

\[1\]

Alternative 55
Error	31.4
Cost	64

\[0\]

Alternative 56
Error	61.9
Cost	64

\[-1\]

Derivation

Split input into 2 regimes
if (*.f64 D D) < 9.05100518124077008e73
1. Initial program 59.9
  \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
2. Taylor expanded around -inf 37.6
  \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{w \cdot \left({M}^{2} \cdot \left({D}^{2} \cdot h\right)\right)}{c0 \cdot {d}^{2}}\right)}\]
3. Simplified37.6
  \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{w \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)}{c0 \cdot \left(d \cdot d\right)}\right)}\]
4. Taylor expanded around 0 31.0
  \[\leadsto \color{blue}{0.25 \cdot \frac{{M}^{2} \cdot \left({D}^{2} \cdot h\right)}{{d}^{2}}}\]
5. Simplified31.0
  \[\leadsto \color{blue}{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{d \cdot d} \cdot 0.25}\]
6. Using strategy rm
7. Applied associate-*r*_binary64_101027.4
  \[\leadsto \frac{\color{blue}{\left(\left(h \cdot \left(D \cdot D\right)\right) \cdot M\right) \cdot M}}{d \cdot d} \cdot 0.25\]
8. Simplified27.4
  \[\leadsto \frac{\color{blue}{\left(M \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)} \cdot M}{d \cdot d} \cdot 0.25\]
9. Using strategy rm
10. Applied times-frac_binary64_107622.4
  \[\leadsto \color{blue}{\left(\frac{M \cdot \left(h \cdot \left(D \cdot D\right)\right)}{d} \cdot \frac{M}{d}\right)} \cdot 0.25\]
11. Simplified22.4
  \[\leadsto \color{blue}{0.25 \cdot \left(\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot M}{d} \cdot \frac{M}{d}\right)}\]
if 9.05100518124077008e73 < (*.f64 D D)
1. Initial program 58.6
  \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
2. Taylor expanded around -inf 52.0
  \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{w \cdot \left({M}^{2} \cdot \left({D}^{2} \cdot h\right)\right)}{c0 \cdot {d}^{2}}\right)}\]
3. Simplified52.0
  \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{w \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)}{c0 \cdot \left(d \cdot d\right)}\right)}\]
4. Taylor expanded around 0 48.4
  \[\leadsto \color{blue}{0.25 \cdot \frac{{M}^{2} \cdot \left({D}^{2} \cdot h\right)}{{d}^{2}}}\]
5. Simplified48.4
  \[\leadsto \color{blue}{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{d \cdot d} \cdot 0.25}\]
6. Using strategy rm
7. Applied associate-*r*_binary64_101047.6
  \[\leadsto \frac{\color{blue}{\left(\left(h \cdot \left(D \cdot D\right)\right) \cdot M\right) \cdot M}}{d \cdot d} \cdot 0.25\]
8. Simplified47.6
  \[\leadsto \frac{\color{blue}{\left(M \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)} \cdot M}{d \cdot d} \cdot 0.25\]
9. Using strategy rm
10. Applied associate-*r*_binary64_101041.5
  \[\leadsto \frac{\left(M \cdot \color{blue}{\left(\left(h \cdot D\right) \cdot D\right)}\right) \cdot M}{d \cdot d} \cdot 0.25\]
11. Simplified41.5
  \[\leadsto \color{blue}{0.25 \cdot \frac{M \cdot \left(M \cdot \left(D \cdot \left(h \cdot D\right)\right)\right)}{d \cdot d}}\]
Recombined 2 regimes into one program.
Final simplification27.2
\[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 9.05100518124077 \cdot 10^{+73}:\\ \;\;\;\;0.25 \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot h\right) \cdot M}{d} \cdot \frac{M}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)}{d \cdot d}\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(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 D D) < 9.05100518124077008e73`

`if 9.05100518124077008e73 < (*.f64 D D)`

Reproduce