Henrywood and Agarwal, Equation (9a)

Average Accuracy: 81.0% → 89.1%

Time: 27.8s

Precision: binary64

Cost: 7872

• Unsound rule application detected in e-graph. Results may not be sound.

Math

\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

↓

\[w0 \cdot \sqrt{1 - \frac{\frac{h}{\frac{\frac{\frac{d}{M}}{0.5}}{D}}}{\ell \cdot \frac{\frac{2}{\frac{M}{d}}}{D}}} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

↓

(FPCore (w0 M D h l d)
 :precision binary64
 (*
  w0
  (sqrt (- 1.0 (/ (/ h (/ (/ (/ d M) 0.5) D)) (* l (/ (/ 2.0 (/ M d)) D)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

↓

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((h / (((d / M) / 0.5) / D)) / (l * ((2.0 / (M / d)) / D)))));
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

↓

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((h / (((d_1 / m) / 0.5d0) / d)) / (l * ((2.0d0 / (m / d_1)) / d)))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

↓

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - ((h / (((d / M) / 0.5) / D)) / (l * ((2.0 / (M / d)) / D)))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

↓

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - ((h / (((d / M) / 0.5) / D)) / (l * ((2.0 / (M / d)) / D)))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

↓

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / Float64(Float64(Float64(d / M) / 0.5) / D)) / Float64(l * Float64(Float64(2.0 / Float64(M / d)) / D))))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

↓

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((h / (((d / M) / 0.5) / D)) / (l * ((2.0 / (M / d)) / D)))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / N[(N[(N[(d / M), $MachinePrecision] / 0.5), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision] / N[(l * N[(N[(2.0 / N[(M / d), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

2. Simplified 81.0%

   \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}}} \]
   Proof

   [Start] 81.0	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   associate-/l* [=>] 81.0	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}}^{2} \cdot \frac{h}{\ell}} \]

3. Applied egg-rr 80.6%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + 1\right) - 1\right)}} \]
   Proof

   [Start] 81.0	\[ w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}} \]
   expm1-log1p-u [=>] 61.3	\[ w0 \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]
   expm1-udef [=>] 61.3	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)}} \]
   log1p-udef [=>] 61.3	\[ w0 \cdot \sqrt{1 - \left(e^{\color{blue}{\log \left(1 + {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}\right)}} - 1\right)} \]
   add-exp-log [<=] 81.0	\[ w0 \cdot \sqrt{1 - \left(\color{blue}{\left(1 + {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)} \]
   +-commutative [=>] 81.0	\[ w0 \cdot \sqrt{1 - \left(\color{blue}{\left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell} + 1\right)} - 1\right)} \]
   div-inv [=>] 80.6	\[ w0 \cdot \sqrt{1 - \left(\left({\color{blue}{\left(M \cdot \frac{1}{\frac{2 \cdot d}{D}}\right)}}^{2} \cdot \frac{h}{\ell} + 1\right) - 1\right)} \]
   associate-/l* [=>] 80.6	\[ w0 \cdot \sqrt{1 - \left(\left({\left(M \cdot \frac{1}{\color{blue}{\frac{2}{\frac{D}{d}}}}\right)}^{2} \cdot \frac{h}{\ell} + 1\right) - 1\right)} \]
   associate-/r/ [=>] 80.6	\[ w0 \cdot \sqrt{1 - \left(\left({\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2} \cdot \frac{h}{\ell} + 1\right) - 1\right)} \]
   metadata-eval [=>] 80.6	\[ w0 \cdot \sqrt{1 - \left(\left({\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + 1\right) - 1\right)} \]

4. Simplified 85.5%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{h \cdot \frac{{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}}} \]
   Proof

   [Start] 80.6	\[ w0 \cdot \sqrt{1 - \left(\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + 1\right) - 1\right)} \]
   associate--l+ [=>] 80.6	\[ w0 \cdot \sqrt{1 - \color{blue}{\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \left(1 - 1\right)\right)}} \]
   metadata-eval [=>] 80.6	\[ w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]
   +-rgt-identity [=>] 80.6	\[ w0 \cdot \sqrt{1 - \color{blue}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}}} \]
   associate-*r/ [=>] 84.7	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}} \]
   associate-*l/ [<=] 85.1	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot h}} \]
   *-commutative [=>] 85.1	\[ w0 \cdot \sqrt{1 - \color{blue}{h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}} \]
   *-commutative [=>] 85.1	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\color{blue}{\left(\left(0.5 \cdot \frac{D}{d}\right) \cdot M\right)}}^{2}}{\ell}} \]
   associate-*l* [=>] 85.1	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\color{blue}{\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}}^{2}}{\ell}} \]
   associate-*l/ [=>] 85.1	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(0.5 \cdot \color{blue}{\frac{D \cdot M}{d}}\right)}^{2}}{\ell}} \]
   *-commutative [<=] 85.1	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(0.5 \cdot \frac{\color{blue}{M \cdot D}}{d}\right)}^{2}}{\ell}} \]
   associate-*l/ [<=] 85.5	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(0.5 \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)}^{2}}{\ell}} \]
   *-commutative [=>] 85.5	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(0.5 \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right)}^{2}}{\ell}} \]

5. Applied egg-rr 88.8%

   \[\leadsto w0 \cdot \sqrt{1 - h \cdot \color{blue}{\left(\frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{1} \cdot \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\ell}\right)}} \]
   Proof

   [Start] 85.5	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}} \]
   unpow2 [=>] 85.5	\[ w0 \cdot \sqrt{1 - h \cdot \frac{\color{blue}{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}}{\ell}} \]
   *-un-lft-identity [=>] 85.5	\[ w0 \cdot \sqrt{1 - h \cdot \frac{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}{\color{blue}{1 \cdot \ell}}} \]
   times-frac [=>] 88.8	\[ w0 \cdot \sqrt{1 - h \cdot \color{blue}{\left(\frac{0.5 \cdot \left(D \cdot \frac{M}{d}\right)}{1} \cdot \frac{0.5 \cdot \left(D \cdot \frac{M}{d}\right)}{\ell}\right)}} \]
   *-commutative [=>] 88.8	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot 0.5}}{1} \cdot \frac{0.5 \cdot \left(D \cdot \frac{M}{d}\right)}{\ell}\right)} \]
   associate-*l* [=>] 88.8	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\color{blue}{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}}{1} \cdot \frac{0.5 \cdot \left(D \cdot \frac{M}{d}\right)}{\ell}\right)} \]
   *-commutative [=>] 88.8	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{1} \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot 0.5}}{\ell}\right)} \]
   associate-*l* [=>] 88.8	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{1} \cdot \frac{\color{blue}{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}}{\ell}\right)} \]

6. Applied egg-rr 89.3%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}{\ell \cdot \frac{\frac{2}{\frac{M}{d}}}{D}}}} \]
   Proof

   [Start] 88.8	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{1} \cdot \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\ell}\right)} \]
   /-rgt-identity [=>] 88.8	\[ w0 \cdot \sqrt{1 - h \cdot \left(\color{blue}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)} \cdot \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\ell}\right)} \]
   associate-*r* [=>] 89.2	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)\right) \cdot \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\ell}}} \]
   clear-num [=>] 89.2	\[ w0 \cdot \sqrt{1 - \left(h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}}}} \]
   un-div-inv [=>] 89.3	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}{\frac{\ell}{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}}}} \]
   div-inv [=>] 89.3	\[ w0 \cdot \sqrt{1 - \frac{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}{\color{blue}{\ell \cdot \frac{1}{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}}}} \]
   *-commutative [=>] 89.3	\[ w0 \cdot \sqrt{1 - \frac{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}{\ell \cdot \frac{1}{\color{blue}{\left(\frac{M}{d} \cdot 0.5\right) \cdot D}}}} \]
   associate-/r* [=>] 89.3	\[ w0 \cdot \sqrt{1 - \frac{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}{\ell \cdot \color{blue}{\frac{\frac{1}{\frac{M}{d} \cdot 0.5}}{D}}}} \]
   *-commutative [=>] 89.3	\[ w0 \cdot \sqrt{1 - \frac{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}{\ell \cdot \frac{\frac{1}{\color{blue}{0.5 \cdot \frac{M}{d}}}}{D}}} \]
   associate-/r* [=>] 89.3	\[ w0 \cdot \sqrt{1 - \frac{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}{\ell \cdot \frac{\color{blue}{\frac{\frac{1}{0.5}}{\frac{M}{d}}}}{D}}} \]
   metadata-eval [=>] 89.3	\[ w0 \cdot \sqrt{1 - \frac{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}{\ell \cdot \frac{\frac{\color{blue}{2}}{\frac{M}{d}}}{D}}} \]

7. Taylor expanded in h around 0 82.9%

   \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{0.5 \cdot \frac{D \cdot \left(h \cdot M\right)}{d}}}{\ell \cdot \frac{\frac{2}{\frac{M}{d}}}{D}}} \]

8. Simplified 89.4%

   \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{h}{\frac{\frac{\frac{d}{M}}{0.5}}{D}}}}{\ell \cdot \frac{\frac{2}{\frac{M}{d}}}{D}}} \]
   Proof

   [Start] 82.9	\[ w0 \cdot \sqrt{1 - \frac{0.5 \cdot \frac{D \cdot \left(h \cdot M\right)}{d}}{\ell \cdot \frac{\frac{2}{\frac{M}{d}}}{D}}} \]
   *-commutative [=>] 82.9	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{D \cdot \left(h \cdot M\right)}{d} \cdot 0.5}}{\ell \cdot \frac{\frac{2}{\frac{M}{d}}}{D}}} \]
   associate-*l/ [=>] 82.9	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{\left(D \cdot \left(h \cdot M\right)\right) \cdot 0.5}{d}}}{\ell \cdot \frac{\frac{2}{\frac{M}{d}}}{D}}} \]
   associate-*r* [=>] 82.4	\[ w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(\left(D \cdot h\right) \cdot M\right)} \cdot 0.5}{d}}{\ell \cdot \frac{\frac{2}{\frac{M}{d}}}{D}}} \]
   *-commutative [<=] 82.4	\[ w0 \cdot \sqrt{1 - \frac{\frac{\left(\color{blue}{\left(h \cdot D\right)} \cdot M\right) \cdot 0.5}{d}}{\ell \cdot \frac{\frac{2}{\frac{M}{d}}}{D}}} \]
   associate-*r* [<=] 82.4	\[ w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(h \cdot D\right) \cdot \left(M \cdot 0.5\right)}}{d}}{\ell \cdot \frac{\frac{2}{\frac{M}{d}}}{D}}} \]
   associate-/l* [=>] 85.4	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{h \cdot D}{\frac{d}{M \cdot 0.5}}}}{\ell \cdot \frac{\frac{2}{\frac{M}{d}}}{D}}} \]
   associate-/l* [=>] 89.4	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{h}{\frac{\frac{d}{M \cdot 0.5}}{D}}}}{\ell \cdot \frac{\frac{2}{\frac{M}{d}}}{D}}} \]
   associate-/r* [=>] 89.4	\[ w0 \cdot \sqrt{1 - \frac{\frac{h}{\frac{\color{blue}{\frac{\frac{d}{M}}{0.5}}}{D}}}{\ell \cdot \frac{\frac{2}{\frac{M}{d}}}{D}}} \]

9. Final simplification 89.4%

   \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{h}{\frac{\frac{\frac{d}{M}}{0.5}}{D}}}{\ell \cdot \frac{\frac{2}{\frac{M}{d}}}{D}}} \]

Alternatives

Alternative 1
Accuracy	80.2%
Cost	8524

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\frac{\frac{h}{\frac{\frac{\ell}{D}}{w0 \cdot D}}}{\frac{d}{M}}}{\frac{d}{M}}, w0\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-88}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(-0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq 2 \cdot 10^{-144}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\frac{h}{d} \cdot \left(M \cdot D\right)}{d \cdot \frac{\frac{\ell}{M}}{D}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Accuracy	74.0%
Cost	8276

\[\begin{array}{l} \mathbf{if}\;M \leq -5 \cdot 10^{+167}:\\ \;\;\;\;w0 \cdot \left(1 + {\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot -0.125}{d \cdot \left(d \cdot \ell\right)}\right)\\ \mathbf{elif}\;M \leq -4.7 \cdot 10^{+55}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\left(M \cdot D\right) \cdot \frac{M \cdot D}{\ell}\right) \cdot \frac{h}{d \cdot d}\right)}\\ \mathbf{elif}\;M \leq -4.2:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(0.25 \cdot \frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{M \cdot M}}\right)}\\ \mathbf{elif}\;M \leq 5 \cdot 10^{-215}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 5 \cdot 10^{-39}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{\frac{h}{d} \cdot \left(M \cdot D\right)}{d \cdot \frac{\frac{\ell}{M}}{D}}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{\ell}{h \cdot M}}\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	87.0%
Cost	8137

\[\begin{array}{l} \mathbf{if}\;D \leq -7.8 \cdot 10^{-143} \lor \neg \left(D \leq 9.2 \cdot 10^{-228}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D \cdot \left(h \cdot \frac{M}{d \cdot 2}\right)}{\ell \cdot \frac{\frac{2}{\frac{M}{d}}}{D}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4
Accuracy	75.1%
Cost	8012

\[\begin{array}{l} \mathbf{if}\;M \leq -4.7 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\frac{\frac{h}{\frac{\frac{\ell}{D}}{w0 \cdot D}}}{\frac{d}{M}}}{\frac{d}{M}}, w0\right)\\ \mathbf{elif}\;M \leq 3 \cdot 10^{-214}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 3.8 \cdot 10^{-47}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{\frac{h}{d} \cdot \left(M \cdot D\right)}{d \cdot \frac{\frac{\ell}{M}}{D}}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{\ell}{h \cdot M}}\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	81.5%
Cost	8009

\[\begin{array}{l} \mathbf{if}\;D \leq -2.85 \cdot 10^{-92} \lor \neg \left(D \leq 2.85 \cdot 10^{-64}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\frac{h}{d} \cdot \left(M \cdot D\right)}{d \cdot \frac{\frac{\ell}{M}}{D}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 6
Accuracy	77.6%
Cost	7881

\[\begin{array}{l} \mathbf{if}\;D \leq -2.85 \cdot 10^{-92} \lor \neg \left(D \leq 5.6 \cdot 10^{-66}\right):\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{\frac{h}{d} \cdot \left(M \cdot D\right)}{d \cdot \frac{\frac{\ell}{M}}{D}}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 7
Accuracy	76.0%
Cost	7880

\[\begin{array}{l} \mathbf{if}\;D \leq -1.45 \cdot 10^{-91}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{D \cdot \left(h \cdot \frac{M}{d}\right)}{\frac{d}{M} \cdot \frac{\frac{\ell}{D}}{w0}}, w0\right)\\ \mathbf{elif}\;D \leq 3.6 \cdot 10^{-65}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{\frac{h}{d} \cdot \left(M \cdot D\right)}{d \cdot \frac{\frac{\ell}{M}}{D}}, 1\right)\\ \end{array} \]

Alternative 8
Accuracy	74.6%
Cost	7880

\[\begin{array}{l} \mathbf{if}\;D \leq -6 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\frac{D}{\frac{\frac{\ell}{D}}{w0}} \cdot \left(h \cdot \frac{M}{d}\right)}{\frac{d}{M}}, w0\right)\\ \mathbf{elif}\;D \leq 1.36 \cdot 10^{-64}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{\frac{h}{d} \cdot \left(M \cdot D\right)}{d \cdot \frac{\frac{\ell}{M}}{D}}, 1\right)\\ \end{array} \]

Alternative 9
Accuracy	89.1%
Cost	7872

\[w0 \cdot \sqrt{1 - \frac{h \cdot \left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}{\ell \cdot \frac{\frac{2}{\frac{M}{d}}}{D}}} \]

Alternative 10
Accuracy	73.3%
Cost	7816

\[\begin{array}{l} t_0 := {\left(M \cdot D\right)}^{2}\\ \mathbf{if}\;D \leq -2.9 \cdot 10^{-238}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{h \cdot t_0}{d \cdot \ell} \cdot \frac{-0.125}{d}\right)\\ \mathbf{elif}\;D \leq 10^{-53}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + t_0 \cdot \frac{h \cdot -0.125}{d \cdot \left(d \cdot \ell\right)}\right)\\ \end{array} \]

Alternative 11
Accuracy	73.4%
Cost	7684

\[\begin{array}{l} \mathbf{if}\;D \leq -1.8 \cdot 10^{-238}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{h \cdot {\left(M \cdot D\right)}^{2}}{d \cdot \ell} \cdot \frac{-0.125}{d}\right)\\ \mathbf{elif}\;D \leq 1.95 \cdot 10^{-21}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{\ell}{h \cdot M}}\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	73.5%
Cost	1609

\[\begin{array}{l} \mathbf{if}\;D \leq -4.6 \cdot 10^{-89} \lor \neg \left(D \leq 2.15 \cdot 10^{-20}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{\ell}{h \cdot M}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 13
Accuracy	64.8%
Cost	1348

\[\begin{array}{l} \mathbf{if}\;D \leq -1.9 \cdot 10^{+44}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{w0}{d} \cdot \frac{h}{\frac{d}{M \cdot M}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 14
Accuracy	64.9%
Cost	1348

\[\begin{array}{l} \mathbf{if}\;D \leq -1.75 \cdot 10^{+46}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D}{\ell} \cdot \frac{\left(D \cdot \frac{h}{d}\right) \cdot \left(w0 \cdot \left(M \cdot M\right)\right)}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 15
Accuracy	65.0%
Cost	1348

\[\begin{array}{l} \mathbf{if}\;D \leq -1.75 \cdot 10^{+46}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{h}{d} \cdot \frac{M \cdot \left(w0 \cdot M\right)}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 16
Accuracy	60.8%
Cost	1348

\[\begin{array}{l} \mathbf{if}\;D \leq -8.6 \cdot 10^{-47}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \frac{\frac{D}{\ell}}{\frac{\frac{d}{M} \cdot \frac{d}{h}}{w0 \cdot M}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 17
Accuracy	68.0%
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023160 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))