Henrywood and Agarwal, Equation (9a)

Average Error: 14.6 → 9.6

Time: 17.5s

Precision: binary64

Cost: 8392

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \left(M \cdot \frac{M \cdot \frac{h}{d}}{d}\right)\right) \cdot -0.25}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-255}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \frac{2}{M \cdot \frac{D}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

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
 (if (<= (/ h l) (- INFINITY))
   (* w0 (sqrt (+ 1.0 (* (* (/ D (/ l D)) (* M (/ (* M (/ h d)) d))) -0.25))))
   (if (<= (/ h l) -5e-255)
     (*
      w0
      (sqrt
       (- 1.0 (/ (* M (* 0.5 (/ D d))) (* (/ l h) (/ 2.0 (* M (/ D d))))))))
     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 tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 + (((D / (l / D)) * (M * ((M * (h / d)) / d))) * -0.25)));
	} else if ((h / l) <= -5e-255) {
		tmp = w0 * sqrt((1.0 - ((M * (0.5 * (D / d))) / ((l / h) * (2.0 / (M * (D / d)))))));
	} 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 tmp;
	if ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 + (((D / (l / D)) * (M * ((M * (h / d)) / d))) * -0.25)));
	} else if ((h / l) <= -5e-255) {
		tmp = w0 * Math.sqrt((1.0 - ((M * (0.5 * (D / d))) / ((l / h) * (2.0 / (M * (D / d)))))));
	} 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):
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 + (((D / (l / D)) * (M * ((M * (h / d)) / d))) * -0.25)))
	elif (h / l) <= -5e-255:
		tmp = w0 * math.sqrt((1.0 - ((M * (0.5 * (D / d))) / ((l / h) * (2.0 / (M * (D / d)))))))
	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)
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(D / Float64(l / D)) * Float64(M * Float64(Float64(M * Float64(h / d)) / d))) * -0.25))));
	elseif (Float64(h / l) <= -5e-255)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M * Float64(0.5 * Float64(D / d))) / Float64(Float64(l / h) * Float64(2.0 / Float64(M * Float64(D / d))))))));
	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)
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 + (((D / (l / D)) * (M * ((M * (h / d)) / d))) * -0.25)));
	elseif ((h / l) <= -5e-255)
		tmp = w0 * sqrt((1.0 - ((M * (0.5 * (D / d))) / ((l / h) * (2.0 / (M * (D / d)))))));
	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_] := If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(M * N[(N[(M * N[(h / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -5e-255], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(l / h), $MachinePrecision] * N[(2.0 / N[(M * N[(D / d), $MachinePrecision]), $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

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

   3. Taylor expanded in w0 around 0 41.7

      \[\leadsto \color{blue}{\sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}} \cdot w0} \]

   4. Simplified 40.4

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{\frac{d \cdot d}{h}}\right) \cdot -0.25}} \]
      Proof

      [Start] 41.7	\[ \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}} \cdot w0 \]
      *-commutative [=>] 41.7	\[ \color{blue}{w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}}} \]
      cancel-sign-sub-inv [=>] 41.7	\[ w0 \cdot \sqrt{\color{blue}{1 + \left(-0.25\right) \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}}} \]
      metadata-eval [=>] 41.7	\[ w0 \cdot \sqrt{1 + \color{blue}{-0.25} \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}} \]
      *-commutative [<=] 41.7	\[ w0 \cdot \sqrt{1 + -0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\color{blue}{\ell \cdot {d}^{2}}}} \]
      *-commutative [=>] 41.7	\[ w0 \cdot \sqrt{1 + \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}} \cdot -0.25}} \]
      times-frac [=>] 42.1	\[ w0 \cdot \sqrt{1 + \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)} \cdot -0.25} \]
      unpow2 [=>] 42.1	\[ w0 \cdot \sqrt{1 + \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right) \cdot -0.25} \]
      associate-/l* [=>] 41.8	\[ w0 \cdot \sqrt{1 + \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right) \cdot -0.25} \]
      *-commutative [<=] 41.8	\[ w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right) \cdot -0.25} \]
      associate-/l* [=>] 40.4	\[ w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\frac{{M}^{2}}{\frac{{d}^{2}}{h}}}\right) \cdot -0.25} \]
      unpow2 [=>] 40.4	\[ w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot M}}{\frac{{d}^{2}}{h}}\right) \cdot -0.25} \]
      unpow2 [=>] 40.4	\[ w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{\frac{\color{blue}{d \cdot d}}{h}}\right) \cdot -0.25} \]

   5. Taylor expanded in M around 0 41.8

      \[\leadsto w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2}}}\right) \cdot -0.25} \]

   6. Simplified 36.0

      \[\leadsto w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(M \cdot \frac{M \cdot \frac{h}{d}}{d}\right)}\right) \cdot -0.25} \]
      Proof

      [Start] 41.8	\[ w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot -0.25} \]
      *-commutative [=>] 41.8	\[ w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}}\right) \cdot -0.25} \]
      unpow2 [=>] 41.8	\[ w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{{d}^{2}}\right) \cdot -0.25} \]
      *-commutative [<=] 41.8	\[ w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot h}}{{d}^{2}}\right) \cdot -0.25} \]
      unpow2 [=>] 41.8	\[ w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right) \cdot -0.25} \]
      associate-*r/ [<=] 40.7	\[ w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right)}\right) \cdot -0.25} \]
      associate-*r* [<=] 38.8	\[ w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{h}{d \cdot d}\right)\right)}\right) \cdot -0.25} \]
      associate-/r* [=>] 38.0	\[ w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \left(M \cdot \left(M \cdot \color{blue}{\frac{\frac{h}{d}}{d}}\right)\right)\right) \cdot -0.25} \]
      associate-*r/ [=>] 36.0	\[ w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \left(M \cdot \color{blue}{\frac{M \cdot \frac{h}{d}}{d}}\right)\right) \cdot -0.25} \]

   if -inf.0 < (/.f64 h l) < -4.9999999999999996e-255

   1. Initial program 14.3

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

   2. Simplified 14.2

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

      [Start] 14.3	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-/l* [=>] 14.2	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Applied egg-rr 12.9

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

   if -4.9999999999999996e-255 < (/.f64 h l)

   1. Initial program 9.1

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

   2. Simplified 9.0

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

      [Start] 9.1	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      times-frac [=>] 9.0	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Taylor expanded in M around 0 3.8

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

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \left(M \cdot \frac{M \cdot \frac{h}{d}}{d}\right)\right) \cdot -0.25}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-255}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \frac{2}{M \cdot \frac{D}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternatives

Alternative 1
Error	9.7
Cost	8264

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \left(M \cdot \frac{M \cdot \frac{h}{d}}{d}\right)\right) \cdot -0.25}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-255}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \left(t_0 \cdot \frac{\frac{h}{\ell}}{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Error	8.3
Cost	8000

\[\begin{array}{l} t_0 := 2 \cdot \frac{\frac{d}{D}}{M}\\ w0 \cdot \sqrt{1 - \frac{\frac{1}{t_0}}{\frac{t_0 \cdot \ell}{h}}} \end{array} \]

Alternative 3
Error	8.4
Cost	7872

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

Alternative 4
Error	14.1
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023054 
(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))))))