Henrywood and Agarwal, Equation (9a)

Average Error: 13.8 → 9.2

Time: 19.0s

Precision: binary64

Cost: 8264

\[ \begin{array}{c}[M, D] = \mathsf{sort}([M, D])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \frac{D \cdot \frac{M}{d \cdot d}}{\ell \cdot 4}\right) \cdot \left(D \cdot M\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq 2 \cdot 10^{-171}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \frac{t_0}{4 \cdot \frac{\ell}{h}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

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
 (let* ((t_0 (* D (/ M d))))
   (if (<= (/ h l) (- INFINITY))
     (* w0 (sqrt (- 1.0 (* (* h (/ (* D (/ M (* d d))) (* l 4.0))) (* D M)))))
     (if (<= (/ h l) 2e-171)
       (* w0 (sqrt (- 1.0 (* t_0 (/ t_0 (* 4.0 (/ l h)))))))
       w0))))

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) {
	double t_0 = D * (M / d);
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - ((h * ((D * (M / (d * d))) / (l * 4.0))) * (D * M))));
	} else if ((h / l) <= 2e-171) {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 / (4.0 * (l / h))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

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) {
	double t_0 = D * (M / d);
	double tmp;
	if ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - ((h * ((D * (M / (d * d))) / (l * 4.0))) * (D * M))));
	} else if ((h / l) <= 2e-171) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_0 / (4.0 * (l / h))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

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):
	t_0 = D * (M / d)
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 - ((h * ((D * (M / (d * d))) / (l * 4.0))) * (D * M))))
	elif (h / l) <= 2e-171:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_0 / (4.0 * (l / h))))))
	else:
		tmp = w0
	return tmp

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)
	t_0 = Float64(D * Float64(M / d))
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(Float64(D * Float64(M / Float64(d * d))) / Float64(l * 4.0))) * Float64(D * M)))));
	elseif (Float64(h / l) <= 2e-171)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 / Float64(4.0 * Float64(l / h)))))));
	else
		tmp = w0;
	end
	return tmp
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_2 = code(w0, M, D, h, l, d)
	t_0 = D * (M / d);
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 - ((h * ((D * (M / (d * d))) / (l * 4.0))) * (D * M))));
	elseif ((h / l) <= 2e-171)
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 / (4.0 * (l / h))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
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_] := Block[{t$95$0 = N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(N[(D * N[(M / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], 2e-171], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 / N[(4.0 * N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*l/ [<=] 64.0	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2 \cdot d} \cdot D\right)}}^{2} \cdot \frac{h}{\ell}} \]
      *-commutative [=>] 64.0	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Applied egg-rr 61.7

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\ell}{h} \cdot \left(\left(d \cdot d\right) \cdot 4\right)}}} \]

   4. Simplified 61.7

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{D \cdot M}{\left(\frac{\ell}{h} \cdot 4\right) \cdot \left(d \cdot d\right)} \cdot \left(D \cdot M\right)}} \]
      Proof

      [Start] 61.7	\[ w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\ell}{h} \cdot \left(\left(d \cdot d\right) \cdot 4\right)}} \]
      associate-/l* [=>] 61.7	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{D \cdot M}{\frac{\frac{\ell}{h} \cdot \left(\left(d \cdot d\right) \cdot 4\right)}{D \cdot M}}}} \]
      associate-/r/ [=>] 61.7	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{D \cdot M}{\frac{\ell}{h} \cdot \left(\left(d \cdot d\right) \cdot 4\right)} \cdot \left(D \cdot M\right)}} \]
      *-commutative [=>] 61.7	\[ w0 \cdot \sqrt{1 - \frac{D \cdot M}{\frac{\ell}{h} \cdot \color{blue}{\left(4 \cdot \left(d \cdot d\right)\right)}} \cdot \left(D \cdot M\right)} \]
      unpow2 [<=] 61.7	\[ w0 \cdot \sqrt{1 - \frac{D \cdot M}{\frac{\ell}{h} \cdot \left(4 \cdot \color{blue}{{d}^{2}}\right)} \cdot \left(D \cdot M\right)} \]
      associate-*r* [=>] 61.7	\[ w0 \cdot \sqrt{1 - \frac{D \cdot M}{\color{blue}{\left(\frac{\ell}{h} \cdot 4\right) \cdot {d}^{2}}} \cdot \left(D \cdot M\right)} \]
      unpow2 [=>] 61.7	\[ w0 \cdot \sqrt{1 - \frac{D \cdot M}{\left(\frac{\ell}{h} \cdot 4\right) \cdot \color{blue}{\left(d \cdot d\right)}} \cdot \left(D \cdot M\right)} \]

   5. Applied egg-rr 31.8

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{D \cdot \frac{M}{d \cdot d}}{\ell \cdot 4} \cdot h\right)} \cdot \left(D \cdot M\right)} \]

   if -inf.0 < (/.f64 h l) < 2e-171

   1. Initial program 12.3

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

   2. Simplified 12.5

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

      [Start] 12.3	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      times-frac [=>] 12.5	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Applied egg-rr 12.5

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\frac{\ell}{h} \cdot 4}}} \]

   4. Applied egg-rr 10.5

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M}{d} \cdot D}{\frac{\ell}{h} \cdot 4} \cdot \left(\frac{M}{d} \cdot D\right)}} \]

   if 2e-171 < (/.f64 h l)

   1. Initial program 6.6

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

   2. Simplified 6.8

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

      [Start] 6.6	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*l/ [<=] 6.8	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2 \cdot d} \cdot D\right)}}^{2} \cdot \frac{h}{\ell}} \]
      *-commutative [=>] 6.8	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Taylor expanded in D around 0 0.7

      \[\leadsto \color{blue}{w0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \frac{D \cdot \frac{M}{d \cdot d}}{\ell \cdot 4}\right) \cdot \left(D \cdot M\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq 2 \cdot 10^{-171}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot \frac{M}{d}\right) \cdot \frac{D \cdot \frac{M}{d}}{4 \cdot \frac{\ell}{h}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternatives

Alternative 1
Error	12.5
Cost	8264

\[\begin{array}{l} \mathbf{if}\;D \cdot M \leq -1 \cdot 10^{+281}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(M \cdot \left(\frac{h}{\ell} \cdot M\right)\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right)}\\ \mathbf{elif}\;D \cdot M \leq -500:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot M\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{h}{d \cdot \left(\ell \cdot d\right)}}{4}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Error	9.9
Cost	8264

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right) \cdot -0.25}\\ \mathbf{elif}\;\frac{h}{\ell} \leq 2 \cdot 10^{-171}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \left(t_0 \cdot \frac{\frac{h}{\ell}}{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Error	9.2
Cost	8264

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \frac{D \cdot \frac{M}{d \cdot d}}{\ell \cdot 4}\right) \cdot \left(D \cdot M\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq 2 \cdot 10^{-171}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \left(t_0 \cdot \frac{\frac{h}{\ell}}{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4
Error	12.8
Cost	8004

\[\begin{array}{l} \mathbf{if}\;D \cdot M \leq -500:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot M\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{h}{d \cdot \left(\ell \cdot d\right)}}{4}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5
Error	13.5
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023039 
(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))))))