Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.5% → 86.7%

Time: 15.7s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

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))));
}

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

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))));
}

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))))

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

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

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]

Initial Program: 81.5% accurate, 1.0× speedup

Math

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

FPCore

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

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))));
}

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

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))));
}

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))))

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

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

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]

Alternative 1: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 + -0.125 \cdot \frac{1}{\frac{\frac{\ell}{h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}{w0}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -4 \cdot 10^{-258}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) (- INFINITY))
   (+ w0 (* -0.125 (/ 1.0 (/ (/ l (* h (pow (* D (/ M d)) 2.0))) w0))))
   (if (<= (/ h l) -4e-258)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
     w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 + (-0.125 * (1.0 / ((l / (h * pow((D * (M / d)), 2.0))) / w0)));
	} else if ((h / l) <= -4e-258) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

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 + (-0.125 * (1.0 / ((l / (h * Math.pow((D * (M / d)), 2.0))) / w0)));
	} else if ((h / l) <= -4e-258) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0 + (-0.125 * (1.0 / ((l / (h * math.pow((D * (M / d)), 2.0))) / w0)))
	elif (h / l) <= -4e-258:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 + Float64(-0.125 * Float64(1.0 / Float64(Float64(l / Float64(h * (Float64(D * Float64(M / d)) ^ 2.0))) / w0))));
	elseif (Float64(h / l) <= -4e-258)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0 + (-0.125 * (1.0 / ((l / (h * ((D * (M / d)) ^ 2.0))) / w0)));
	elseif ((h / l) <= -4e-258)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 + N[(-0.125 * N[(1.0 / N[(N[(l / N[(h * N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -4e-258], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.4%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in D around 0 44.9%

      \[\leadsto \color{blue}{w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 28.3%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)\right)} \]

      2. expm1-udef 28.3%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)} - 1\right)} \]

      3. associate-*r* 33.8%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell}\right)} - 1\right) \]

      4. pow-prod-down 44.9%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} - 1\right) \]

      5. *-commutative 44.9%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\color{blue}{\ell \cdot {d}^{2}}}\right)} - 1\right) \]

   6. Applied egg-rr 44.9%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 44.9%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}\right)\right)} \]

      2. expm1-log1p 67.1%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}} \]

      3. associate-*r* 72.7%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot w0}}{\ell \cdot {d}^{2}} \]

      4. *-commutative 72.7%

         \[\leadsto w0 + -0.125 \cdot \frac{\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot w0}{\color{blue}{{d}^{2} \cdot \ell}} \]

      5. unpow2 72.7%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      6. swap-sqr 56.0%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)} \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      7. unpow2 56.0%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\left(\color{blue}{{D}^{2}} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      8. unpow2 56.0%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\left({D}^{2} \cdot \color{blue}{{M}^{2}}\right) \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      9. associate-*r* 56.0%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)} \cdot w0}{{d}^{2} \cdot \ell} \]

      10. associate-*r* 50.5%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{{D}^{2} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

      11. times-frac 50.5%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w0}{\ell}\right)} \]

      12. associate-*r/ 50.5%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{\frac{{D}^{2}}{{d}^{2}} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)}{\ell}} \]

   8. Simplified 73.2%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell}} \]
   9. Step-by-step derivation

      1. clear-num 73.2%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{1}{\frac{\ell}{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}}} \]

      2. inv-pow 73.2%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{{\left(\frac{\ell}{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}\right)}^{-1}} \]

      3. *-commutative 73.2%

         \[\leadsto w0 + -0.125 \cdot {\left(\frac{\ell}{{\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2} \cdot \left(h \cdot w0\right)}\right)}^{-1} \]

   10. Applied egg-rr 73.2%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{{\left(\frac{\ell}{{\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(h \cdot w0\right)}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 73.2%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{1}{\frac{\ell}{{\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(h \cdot w0\right)}}} \]

       2. associate-*r* 84.3%

          \[\leadsto w0 + -0.125 \cdot \frac{1}{\frac{\ell}{\color{blue}{\left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot h\right) \cdot w0}}} \]

       3. *-commutative 84.3%

          \[\leadsto w0 + -0.125 \cdot \frac{1}{\frac{\ell}{\color{blue}{\left(h \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)} \cdot w0}} \]

       4. associate-/r* 84.3%

          \[\leadsto w0 + -0.125 \cdot \frac{1}{\color{blue}{\frac{\frac{\ell}{h \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}}{w0}}} \]

       5. *-commutative 84.3%

          \[\leadsto w0 + -0.125 \cdot \frac{1}{\frac{\frac{\ell}{h \cdot {\color{blue}{\left(\frac{D}{d} \cdot M\right)}}^{2}}}{w0}} \]

       6. associate-*l/ 84.3%

          \[\leadsto w0 + -0.125 \cdot \frac{1}{\frac{\frac{\ell}{h \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}}}{w0}} \]

       7. associate-*r/ 84.3%

          \[\leadsto w0 + -0.125 \cdot \frac{1}{\frac{\frac{\ell}{h \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}}}{w0}} \]

   12. Simplified 84.3%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{1}{\frac{\frac{\ell}{h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}{w0}}} \]

   if -inf.0 < (/.f64 h l) < -3.99999999999999982e-258

   1. Initial program 82.1%

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

   3. Simplified 83.0%

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

   if -3.99999999999999982e-258 < (/.f64 h l)

   1. Initial program 85.2%

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

   3. Simplified 84.4%

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

   4. Taylor expanded in D around 0 93.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 + -0.125 \cdot \frac{1}{\frac{\frac{\ell}{h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}{w0}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -4 \cdot 10^{-258}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2: 86.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D M) (* d 2.0)) 2.0) (/ h l)) -10.0)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ D (* (/ d M) 2.0)) 2.0)))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((D * M) / (d * 2.0)), 2.0) * (h / l)) <= -10.0) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D / ((d / M) * 2.0)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((((d * m) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= (-10.0d0)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d / ((d_1 / m) * 2.0d0)) ** 2.0d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((D * M) / (d * 2.0)), 2.0) * (h / l)) <= -10.0) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D / ((d / M) * 2.0)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((D * M) / (d * 2.0)), 2.0) * (h / l)) <= -10.0:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((D / ((d / M) * 2.0)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -10.0)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D / Float64(Float64(d / M) * 2.0)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((D * M) / (d * 2.0)) ^ 2.0) * (h / l)) <= -10.0)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D / ((d / M) * 2.0)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -10.0], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D / N[(N[(d / M), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -10

   1. Initial program 69.7%

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

   3. Simplified 71.0%

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

      1. *-commutative 71.0%

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

      2. clear-num 70.9%

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

      3. frac-times 71.0%

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

      4. *-un-lft-identity 71.0%

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

   5. Applied egg-rr 71.0%

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

   if -10 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 87.1%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in D around 0 96.3%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

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

Alternative 3: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ w0 \cdot \sqrt{1 - {\left(\frac{\ell}{\left(0.25 \cdot {\left(\frac{D}{\frac{d}{M}}\right)}^{2}\right) \cdot h}\right)}^{-1}} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (pow (/ l (* (* 0.25 (pow (/ D (/ d M)) 2.0)) h)) -1.0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - pow((l / ((0.25 * pow((D / (d / M)), 2.0)) * h)), -1.0)));
}

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 - ((l / ((0.25d0 * ((d / (d_1 / m)) ** 2.0d0)) * h)) ** (-1.0d0))))
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((l / ((0.25 * Math.pow((D / (d / M)), 2.0)) * h)), -1.0)));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - math.pow((l / ((0.25 * math.pow((D / (d / M)), 2.0)) * h)), -1.0)))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - (Float64(l / Float64(Float64(0.25 * (Float64(D / Float64(d / M)) ^ 2.0)) * h)) ^ -1.0))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((l / ((0.25 * ((D / (d / M)) ^ 2.0)) * h)) ^ -1.0)));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[Power[N[(l / N[(N[(0.25 * N[Power[N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.2%

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

3. Simplified 81.3%

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

   1. associate-*r/ 87.0%

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

   2. frac-times 86.9%

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

   3. *-commutative 86.9%

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

   4. *-un-lft-identity 86.9%

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

   5. times-frac 86.9%

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

   6. metadata-eval 86.9%

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

   7. *-commutative 86.9%

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

5. Applied egg-rr 86.9%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2} \cdot h}{\ell}}} \]
6. Step-by-step derivation

   1. clear-num 86.9%

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

   2. inv-pow 86.9%

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

   3. unpow-prod-down 86.9%

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

   4. metadata-eval 86.9%

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

   5. associate-/l* 87.0%

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

7. Applied egg-rr 87.0%

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

8. Final simplification 87.0%

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

Alternative 4: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{h \cdot \frac{0.25}{{\left(\frac{d}{D \cdot M}\right)}^{2}}}}} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (+ 1.0 (/ -1.0 (/ l (* h (/ 0.25 (pow (/ d (* D M)) 2.0)))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 + (-1.0 / (l / (h * (0.25 / pow((d / (D * M)), 2.0)))))));
}

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 + ((-1.0d0) / (l / (h * (0.25d0 / ((d_1 / (d * m)) ** 2.0d0)))))))
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 + (-1.0 / (l / (h * (0.25 / Math.pow((d / (D * M)), 2.0)))))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 + (-1.0 / (l / (h * (0.25 / math.pow((d / (D * M)), 2.0)))))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 + Float64(-1.0 / Float64(l / Float64(h * Float64(0.25 / (Float64(d / Float64(D * M)) ^ 2.0))))))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 + (-1.0 / (l / (h * (0.25 / ((d / (D * M)) ^ 2.0)))))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[(h * N[(0.25 / N[Power[N[(d / N[(D * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.2%

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

3. Simplified 81.3%

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

   1. associate-*r/ 87.0%

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

   2. frac-times 86.9%

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

   3. *-commutative 86.9%

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

   4. *-un-lft-identity 86.9%

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

   5. times-frac 86.9%

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

   6. metadata-eval 86.9%

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

   7. *-commutative 86.9%

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

5. Applied egg-rr 86.9%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2} \cdot h}{\ell}}} \]
6. Step-by-step derivation

   1. clear-num 86.9%

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

   2. inv-pow 86.9%

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

   3. unpow-prod-down 86.9%

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

   4. metadata-eval 86.9%

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

   5. associate-/l* 87.0%

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

7. Applied egg-rr 87.0%

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

   1. unpow2 87.0%

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

   2. clear-num 87.0%

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

   3. frac-times 86.2%

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

   4. *-un-lft-identity 86.2%

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

9. Applied egg-rr 86.2%

   \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{\ell}{\left(0.25 \cdot \color{blue}{\frac{D}{\frac{\frac{d}{M}}{D} \cdot \frac{d}{M}}}\right) \cdot h}\right)}^{-1}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 85.7%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\frac{\ell}{\left(0.25 \cdot \frac{D}{\frac{\frac{d}{M}}{D} \cdot \frac{d}{M}}\right) \cdot h}\right)}^{-1}}\right)\right)} \]

    2. expm1-udef 85.8%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\frac{\ell}{\left(0.25 \cdot \frac{D}{\frac{\frac{d}{M}}{D} \cdot \frac{d}{M}}\right) \cdot h}\right)}^{-1}}\right)} - 1\right)} \]

11. Applied egg-rr 81.2%

    \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{1}{\frac{\frac{\ell}{\frac{0.25 \cdot D}{\frac{{\left(\frac{d}{M}\right)}^{2}}{D}}}}{h}}}\right)} - 1\right)} \]
12. Step-by-step derivation

    1. expm1-def 81.2%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{1}{\frac{\frac{\ell}{\frac{0.25 \cdot D}{\frac{{\left(\frac{d}{M}\right)}^{2}}{D}}}}{h}}}\right)\right)} \]

    2. expm1-log1p 81.4%

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

    3. sub-neg 81.4%

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

    4. distribute-neg-frac 81.4%

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

    5. metadata-eval 81.4%

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

    6. associate-/l/ 82.2%

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

    7. associate-/l* 82.2%

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

    8. associate-/l/ 72.6%

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

    9. unpow2 72.6%

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

    10. times-frac 87.0%

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

    11. associate-/r* 86.2%

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

    12. *-commutative 86.2%

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

    13. associate-/r* 87.0%

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

    14. *-commutative 87.0%

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

    15. unpow1 87.0%

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

    16. pow-plus 87.0%

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

    17. metadata-eval 87.0%

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

13. Simplified 87.0%

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

14. Final simplification 87.0%

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

Alternative 5: 86.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - ((h * ((0.5d0 * ((d * m) / d_1)) ** 2.0d0)) / l)))
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 - ((h * Math.pow((0.5 * ((D * M) / d)), 2.0)) / l)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Simplified 81.3%

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

   1. associate-*r/ 87.0%

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

   2. frac-times 86.9%

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

   3. *-commutative 86.9%

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

   4. *-un-lft-identity 86.9%

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

   5. times-frac 86.9%

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

   6. metadata-eval 86.9%

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

   7. *-commutative 86.9%

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

5. Applied egg-rr 86.9%

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

6. Final simplification 86.9%

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

Alternative 6: 79.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 + -0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(w0 \cdot h\right)}{\ell}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-142}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(w0 \cdot {\left(\frac{d}{D \cdot M}\right)}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) (- INFINITY))
   (+ w0 (* -0.125 (/ (* (pow (* D (/ M d)) 2.0) (* w0 h)) l)))
   (if (<= (/ h l) -1e-142)
     (+ w0 (* -0.125 (* (/ h l) (* w0 (pow (/ d (* D M)) -2.0)))))
     w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 + (-0.125 * ((pow((D * (M / d)), 2.0) * (w0 * h)) / l));
	} else if ((h / l) <= -1e-142) {
		tmp = w0 + (-0.125 * ((h / l) * (w0 * pow((d / (D * M)), -2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

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 + (-0.125 * ((Math.pow((D * (M / d)), 2.0) * (w0 * h)) / l));
	} else if ((h / l) <= -1e-142) {
		tmp = w0 + (-0.125 * ((h / l) * (w0 * Math.pow((d / (D * M)), -2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0 + (-0.125 * ((math.pow((D * (M / d)), 2.0) * (w0 * h)) / l))
	elif (h / l) <= -1e-142:
		tmp = w0 + (-0.125 * ((h / l) * (w0 * math.pow((d / (D * M)), -2.0))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64((Float64(D * Float64(M / d)) ^ 2.0) * Float64(w0 * h)) / l)));
	elseif (Float64(h / l) <= -1e-142)
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(h / l) * Float64(w0 * (Float64(d / Float64(D * M)) ^ -2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0 + (-0.125 * ((((D * (M / d)) ^ 2.0) * (w0 * h)) / l));
	elseif ((h / l) <= -1e-142)
		tmp = w0 + (-0.125 * ((h / l) * (w0 * ((d / (D * M)) ^ -2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 + N[(-0.125 * N[(N[(N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(w0 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-142], N[(w0 + N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[(w0 * N[Power[N[(d / N[(D * M), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.4%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in D around 0 44.9%

      \[\leadsto \color{blue}{w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 28.3%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)\right)} \]

      2. expm1-udef 28.3%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)} - 1\right)} \]

      3. associate-*r* 33.8%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell}\right)} - 1\right) \]

      4. pow-prod-down 44.9%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} - 1\right) \]

      5. *-commutative 44.9%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\color{blue}{\ell \cdot {d}^{2}}}\right)} - 1\right) \]

   6. Applied egg-rr 44.9%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 44.9%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}\right)\right)} \]

      2. expm1-log1p 67.1%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}} \]

      3. associate-*r* 72.7%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot w0}}{\ell \cdot {d}^{2}} \]

      4. *-commutative 72.7%

         \[\leadsto w0 + -0.125 \cdot \frac{\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot w0}{\color{blue}{{d}^{2} \cdot \ell}} \]

      5. unpow2 72.7%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      6. swap-sqr 56.0%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)} \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      7. unpow2 56.0%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\left(\color{blue}{{D}^{2}} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      8. unpow2 56.0%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\left({D}^{2} \cdot \color{blue}{{M}^{2}}\right) \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      9. associate-*r* 56.0%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)} \cdot w0}{{d}^{2} \cdot \ell} \]

      10. associate-*r* 50.5%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{{D}^{2} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

      11. times-frac 50.5%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w0}{\ell}\right)} \]

      12. associate-*r/ 50.5%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{\frac{{D}^{2}}{{d}^{2}} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)}{\ell}} \]

   8. Simplified 73.2%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell}} \]

   9. Taylor expanded in D around 0 44.9%

      \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2}}}}{\ell} \]
   10. Step-by-step derivation

       1. *-commutative 44.9%

          \[\leadsto w0 + -0.125 \cdot \frac{\frac{{D}^{2} \cdot \color{blue}{\left(\left(h \cdot w0\right) \cdot {M}^{2}\right)}}{{d}^{2}}}{\ell} \]

       2. associate-*r/ 44.9%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{{D}^{2} \cdot \frac{\left(h \cdot w0\right) \cdot {M}^{2}}{{d}^{2}}}}{\ell} \]

       3. associate-/l* 50.5%

          \[\leadsto w0 + -0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\frac{h \cdot w0}{\frac{{d}^{2}}{{M}^{2}}}}}{\ell} \]

       4. associate-*r/ 50.2%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\frac{{D}^{2} \cdot \left(h \cdot w0\right)}{\frac{{d}^{2}}{{M}^{2}}}}}{\ell} \]

       5. associate-*l/ 50.5%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2}}} \cdot \left(h \cdot w0\right)}}{\ell} \]

       6. unpow2 50.5%

          \[\leadsto w0 + -0.125 \cdot \frac{\frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{{M}^{2}}} \cdot \left(h \cdot w0\right)}{\ell} \]

       7. unpow2 50.5%

          \[\leadsto w0 + -0.125 \cdot \frac{\frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{{M}^{2}}} \cdot \left(h \cdot w0\right)}{\ell} \]

       8. unpow2 50.5%

          \[\leadsto w0 + -0.125 \cdot \frac{\frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{M \cdot M}}} \cdot \left(h \cdot w0\right)}{\ell} \]

       9. times-frac 67.1%

          \[\leadsto w0 + -0.125 \cdot \frac{\frac{D \cdot D}{\color{blue}{\frac{d}{M} \cdot \frac{d}{M}}} \cdot \left(h \cdot w0\right)}{\ell} \]

       10. times-frac 73.2%

           \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left(\frac{D}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right)} \cdot \left(h \cdot w0\right)}{\ell} \]

       11. associate-/l* 73.2%

           \[\leadsto w0 + -0.125 \cdot \frac{\left(\color{blue}{\frac{D \cdot M}{d}} \cdot \frac{D}{\frac{d}{M}}\right) \cdot \left(h \cdot w0\right)}{\ell} \]

       12. times-frac 67.7%

           \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\frac{\left(D \cdot M\right) \cdot D}{d \cdot \frac{d}{M}}} \cdot \left(h \cdot w0\right)}{\ell} \]

   11. Simplified 73.2%

       \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left(h \cdot w0\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}{\ell} \]

   if -inf.0 < (/.f64 h l) < -1e-142

   1. Initial program 82.3%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in D around 0 41.3%

      \[\leadsto \color{blue}{w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 29.3%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)\right)} \]

      2. expm1-udef 29.3%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)} - 1\right)} \]

      3. associate-*r* 29.3%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell}\right)} - 1\right) \]

      4. pow-prod-down 32.4%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} - 1\right) \]

      5. *-commutative 32.4%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\color{blue}{\ell \cdot {d}^{2}}}\right)} - 1\right) \]

   6. Applied egg-rr 32.4%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 32.4%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}\right)\right)} \]

      2. expm1-log1p 47.4%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}} \]

      3. associate-*r* 51.4%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot w0}}{\ell \cdot {d}^{2}} \]

      4. *-commutative 51.4%

         \[\leadsto w0 + -0.125 \cdot \frac{\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot w0}{\color{blue}{{d}^{2} \cdot \ell}} \]

      5. unpow2 51.4%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      6. swap-sqr 47.2%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)} \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      7. unpow2 47.2%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\left(\color{blue}{{D}^{2}} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      8. unpow2 47.2%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\left({D}^{2} \cdot \color{blue}{{M}^{2}}\right) \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      9. associate-*r* 45.3%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)} \cdot w0}{{d}^{2} \cdot \ell} \]

      10. associate-*r* 45.1%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{{D}^{2} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

      11. times-frac 45.1%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w0}{\ell}\right)} \]

      12. associate-*r/ 46.1%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{\frac{{D}^{2}}{{d}^{2}} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)}{\ell}} \]

   8. Simplified 56.7%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell}} \]

   9. Taylor expanded in D around 0 41.3%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   10. Step-by-step derivation

       1. *-commutative 41.3%

          \[\leadsto w0 + -0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(\left(h \cdot w0\right) \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell} \]

       2. *-commutative 41.3%

          \[\leadsto w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left(\left(h \cdot w0\right) \cdot {M}^{2}\right)}{\color{blue}{\ell \cdot {d}^{2}}} \]

       3. times-frac 40.2%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{\left(h \cdot w0\right) \cdot {M}^{2}}{{d}^{2}}\right)} \]

       4. associate-*l/ 42.3%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \frac{\left(h \cdot w0\right) \cdot {M}^{2}}{{d}^{2}}}{\ell}} \]

   11. Simplified 60.7%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)} \]

   12. Taylor expanded in D around 0 41.3%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   13. Step-by-step derivation

       1. associate-*r* 41.3%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

       2. times-frac 40.3%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h \cdot w0}{\ell}\right)} \]

   14. Simplified 65.7%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left(w0 \cdot {\left(\frac{d}{D \cdot M}\right)}^{-2}\right)\right)} \]

   if -1e-142 < (/.f64 h l)

   1. Initial program 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in D around 0 91.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 + -0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(w0 \cdot h\right)}{\ell}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-142}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(w0 \cdot {\left(\frac{d}{D \cdot M}\right)}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 7: 78.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-142}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(w0 \cdot {\left(\frac{d}{D \cdot M}\right)}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -1e-142)
   (+ w0 (* -0.125 (* (/ h l) (* w0 (pow (/ d (* D M)) -2.0)))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e-142) {
		tmp = w0 + (-0.125 * ((h / l) * (w0 * pow((d / (D * M)), -2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((h / l) <= (-1d-142)) then
        tmp = w0 + ((-0.125d0) * ((h / l) * (w0 * ((d_1 / (d * m)) ** (-2.0d0)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e-142) {
		tmp = w0 + (-0.125 * ((h / l) * (w0 * Math.pow((d / (D * M)), -2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -1e-142:
		tmp = w0 + (-0.125 * ((h / l) * (w0 * math.pow((d / (D * M)), -2.0))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -1e-142)
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(h / l) * Float64(w0 * (Float64(d / Float64(D * M)) ^ -2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -1e-142)
		tmp = w0 + (-0.125 * ((h / l) * (w0 * ((d / (D * M)) ^ -2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e-142], N[(w0 + N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[(w0 * N[Power[N[(d / N[(D * M), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1e-142

   1. Initial program 77.6%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in D around 0 41.9%

      \[\leadsto \color{blue}{w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 29.2%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)\right)} \]

      2. expm1-udef 29.2%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)} - 1\right)} \]

      3. associate-*r* 30.0%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell}\right)} - 1\right) \]

      4. pow-prod-down 34.2%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} - 1\right) \]

      5. *-commutative 34.2%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\color{blue}{\ell \cdot {d}^{2}}}\right)} - 1\right) \]

   6. Applied egg-rr 34.2%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 34.2%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}\right)\right)} \]

      2. expm1-log1p 50.3%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}} \]

      3. associate-*r* 54.6%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot w0}}{\ell \cdot {d}^{2}} \]

      4. *-commutative 54.6%

         \[\leadsto w0 + -0.125 \cdot \frac{\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot w0}{\color{blue}{{d}^{2} \cdot \ell}} \]

      5. unpow2 54.6%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      6. swap-sqr 48.5%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)} \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      7. unpow2 48.5%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\left(\color{blue}{{D}^{2}} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      8. unpow2 48.5%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\left({D}^{2} \cdot \color{blue}{{M}^{2}}\right) \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      9. associate-*r* 46.9%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)} \cdot w0}{{d}^{2} \cdot \ell} \]

      10. associate-*r* 45.9%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{{D}^{2} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

      11. times-frac 45.9%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w0}{\ell}\right)} \]

      12. associate-*r/ 46.8%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{\frac{{D}^{2}}{{d}^{2}} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)}{\ell}} \]

   8. Simplified 59.1%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell}} \]

   9. Taylor expanded in D around 0 41.9%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   10. Step-by-step derivation

       1. *-commutative 41.9%

          \[\leadsto w0 + -0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(\left(h \cdot w0\right) \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell} \]

       2. *-commutative 41.9%

          \[\leadsto w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left(\left(h \cdot w0\right) \cdot {M}^{2}\right)}{\color{blue}{\ell \cdot {d}^{2}}} \]

       3. times-frac 40.0%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{\left(h \cdot w0\right) \cdot {M}^{2}}{{d}^{2}}\right)} \]

       4. associate-*l/ 42.7%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \frac{\left(h \cdot w0\right) \cdot {M}^{2}}{{d}^{2}}}{\ell}} \]

   11. Simplified 61.7%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)} \]

   12. Taylor expanded in D around 0 41.9%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   13. Step-by-step derivation

       1. associate-*r* 42.6%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

       2. times-frac 40.9%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h \cdot w0}{\ell}\right)} \]

   14. Simplified 63.5%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left(w0 \cdot {\left(\frac{d}{D \cdot M}\right)}^{-2}\right)\right)} \]

   if -1e-142 < (/.f64 h l)

   1. Initial program 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in D around 0 91.1%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-142}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(w0 \cdot {\left(\frac{d}{D \cdot M}\right)}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 8: 77.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-120}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{h \cdot {\left(\frac{d}{D \cdot M}\right)}^{-2}}{\frac{\ell}{w0}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -2e-120)
   (+ w0 (* -0.125 (/ (* h (pow (/ d (* D M)) -2.0)) (/ l w0))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -2e-120) {
		tmp = w0 + (-0.125 * ((h * pow((d / (D * M)), -2.0)) / (l / w0)));
	} else {
		tmp = w0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((h / l) <= (-2d-120)) then
        tmp = w0 + ((-0.125d0) * ((h * ((d_1 / (d * m)) ** (-2.0d0))) / (l / w0)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -2e-120) {
		tmp = w0 + (-0.125 * ((h * Math.pow((d / (D * M)), -2.0)) / (l / w0)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -2e-120:
		tmp = w0 + (-0.125 * ((h * math.pow((d / (D * M)), -2.0)) / (l / w0)))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -2e-120)
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(h * (Float64(d / Float64(D * M)) ^ -2.0)) / Float64(l / w0))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -2e-120)
		tmp = w0 + (-0.125 * ((h * ((d / (D * M)) ^ -2.0)) / (l / w0)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -2e-120], N[(w0 + N[(-0.125 * N[(N[(h * N[Power[N[(d / N[(D * M), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(l / w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.99999999999999996e-120

   1. Initial program 79.5%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in D around 0 42.9%

      \[\leadsto \color{blue}{w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 29.9%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)\right)} \]

      2. expm1-udef 29.9%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)} - 1\right)} \]

      3. associate-*r* 30.7%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell}\right)} - 1\right) \]

      4. pow-prod-down 35.1%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} - 1\right) \]

      5. *-commutative 35.1%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\color{blue}{\ell \cdot {d}^{2}}}\right)} - 1\right) \]

   6. Applied egg-rr 35.1%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 35.1%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}\right)\right)} \]

      2. expm1-log1p 51.5%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}} \]

      3. associate-*r* 55.9%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot w0}}{\ell \cdot {d}^{2}} \]

      4. *-commutative 55.9%

         \[\leadsto w0 + -0.125 \cdot \frac{\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot w0}{\color{blue}{{d}^{2} \cdot \ell}} \]

      5. unpow2 55.9%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      6. swap-sqr 49.7%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)} \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      7. unpow2 49.7%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\left(\color{blue}{{D}^{2}} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      8. unpow2 49.7%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\left({D}^{2} \cdot \color{blue}{{M}^{2}}\right) \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      9. associate-*r* 48.0%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)} \cdot w0}{{d}^{2} \cdot \ell} \]

      10. associate-*r* 47.0%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{{D}^{2} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

      11. times-frac 47.0%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w0}{\ell}\right)} \]

      12. associate-*r/ 47.9%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{\frac{{D}^{2}}{{d}^{2}} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)}{\ell}} \]

   8. Simplified 60.6%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell}} \]

   9. Taylor expanded in D around 0 42.9%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   10. Step-by-step derivation

       1. *-commutative 42.9%

          \[\leadsto w0 + -0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(\left(h \cdot w0\right) \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell} \]

       2. *-commutative 42.9%

          \[\leadsto w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left(\left(h \cdot w0\right) \cdot {M}^{2}\right)}{\color{blue}{\ell \cdot {d}^{2}}} \]

       3. times-frac 41.0%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{\left(h \cdot w0\right) \cdot {M}^{2}}{{d}^{2}}\right)} \]

       4. associate-*l/ 43.7%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \frac{\left(h \cdot w0\right) \cdot {M}^{2}}{{d}^{2}}}{\ell}} \]

   11. Simplified 63.2%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ 66.7%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot h}{\frac{\ell}{w0}}} \]

       2. *-commutative 66.7%

          \[\leadsto w0 + -0.125 \cdot \frac{{\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2} \cdot h}{\frac{\ell}{w0}} \]

       3. clear-num 66.7%

          \[\leadsto w0 + -0.125 \cdot \frac{{\left(\color{blue}{\frac{1}{\frac{d}{M}}} \cdot D\right)}^{2} \cdot h}{\frac{\ell}{w0}} \]

       4. associate-/r/ 66.7%

          \[\leadsto w0 + -0.125 \cdot \frac{{\color{blue}{\left(\frac{1}{\frac{\frac{d}{M}}{D}}\right)}}^{2} \cdot h}{\frac{\ell}{w0}} \]

       5. inv-pow 66.7%

          \[\leadsto w0 + -0.125 \cdot \frac{{\color{blue}{\left({\left(\frac{\frac{d}{M}}{D}\right)}^{-1}\right)}}^{2} \cdot h}{\frac{\ell}{w0}} \]

       6. pow-pow 66.7%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{{\left(\frac{\frac{d}{M}}{D}\right)}^{\left(-1 \cdot 2\right)}} \cdot h}{\frac{\ell}{w0}} \]

       7. associate-/l/ 66.7%

          \[\leadsto w0 + -0.125 \cdot \frac{{\color{blue}{\left(\frac{d}{D \cdot M}\right)}}^{\left(-1 \cdot 2\right)} \cdot h}{\frac{\ell}{w0}} \]

       8. metadata-eval 66.7%

          \[\leadsto w0 + -0.125 \cdot \frac{{\left(\frac{d}{D \cdot M}\right)}^{\color{blue}{-2}} \cdot h}{\frac{\ell}{w0}} \]

   13. Applied egg-rr 66.7%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{\left(\frac{d}{D \cdot M}\right)}^{-2} \cdot h}{\frac{\ell}{w0}}} \]

   if -1.99999999999999996e-120 < (/.f64 h l)

   1. Initial program 82.8%

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

   3. Simplified 82.8%

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

   4. Taylor expanded in D around 0 89.2%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-120}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{h \cdot {\left(\frac{d}{D \cdot M}\right)}^{-2}}{\frac{\ell}{w0}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 9: 81.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ w0 + -0.125 \cdot \frac{1}{\frac{\frac{\ell}{h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}{w0}} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (+ w0 (* -0.125 (/ 1.0 (/ (/ l (* h (pow (* D (/ M d)) 2.0))) w0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 + (-0.125 * (1.0 / ((l / (h * pow((D * (M / d)), 2.0))) / w0)));
}

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 + ((-0.125d0) * (1.0d0 / ((l / (h * ((d * (m / d_1)) ** 2.0d0))) / w0)))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 + (-0.125 * (1.0 / ((l / (h * Math.pow((D * (M / d)), 2.0))) / w0)));
}

Python

def code(w0, M, D, h, l, d):
	return w0 + (-0.125 * (1.0 / ((l / (h * math.pow((D * (M / d)), 2.0))) / w0)))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 + Float64(-0.125 * Float64(1.0 / Float64(Float64(l / Float64(h * (Float64(D * Float64(M / d)) ^ 2.0))) / w0))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 + (-0.125 * (1.0 / ((l / (h * ((D * (M / d)) ^ 2.0))) / w0)));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 + N[(-0.125 * N[(1.0 / N[(N[(l / N[(h * N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.2%

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

3. Simplified 81.3%

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

4. Taylor expanded in D around 0 46.7%

   \[\leadsto \color{blue}{w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
5. Step-by-step derivation

   1. expm1-log1p-u 40.6%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)\right)} \]

   2. expm1-udef 40.6%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)} - 1\right)} \]

   3. associate-*r* 41.3%

      \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell}\right)} - 1\right) \]

   4. pow-prod-down 52.4%

      \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} - 1\right) \]

   5. *-commutative 52.4%

      \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\color{blue}{\ell \cdot {d}^{2}}}\right)} - 1\right) \]

6. Applied egg-rr 52.4%

   \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 52.4%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}\right)\right)} \]

   2. expm1-log1p 60.5%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}} \]

   3. associate-*r* 66.9%

      \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot w0}}{\ell \cdot {d}^{2}} \]

   4. *-commutative 66.9%

      \[\leadsto w0 + -0.125 \cdot \frac{\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot w0}{\color{blue}{{d}^{2} \cdot \ell}} \]

   5. unpow2 66.9%

      \[\leadsto w0 + -0.125 \cdot \frac{\left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

   6. swap-sqr 53.9%

      \[\leadsto w0 + -0.125 \cdot \frac{\left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)} \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

   7. unpow2 53.9%

      \[\leadsto w0 + -0.125 \cdot \frac{\left(\left(\color{blue}{{D}^{2}} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

   8. unpow2 53.9%

      \[\leadsto w0 + -0.125 \cdot \frac{\left(\left({D}^{2} \cdot \color{blue}{{M}^{2}}\right) \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

   9. associate-*r* 53.0%

      \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)} \cdot w0}{{d}^{2} \cdot \ell} \]

   10. associate-*r* 51.4%

       \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{{D}^{2} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

   11. times-frac 51.0%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w0}{\ell}\right)} \]

   12. associate-*r/ 52.6%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{\frac{{D}^{2}}{{d}^{2}} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)}{\ell}} \]

8. Simplified 69.5%

   \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell}} \]
9. Step-by-step derivation

   1. clear-num 69.5%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{1}{\frac{\ell}{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}}} \]

   2. inv-pow 69.5%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{{\left(\frac{\ell}{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}\right)}^{-1}} \]

   3. *-commutative 69.5%

      \[\leadsto w0 + -0.125 \cdot {\left(\frac{\ell}{{\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2} \cdot \left(h \cdot w0\right)}\right)}^{-1} \]

10. Applied egg-rr 69.5%

    \[\leadsto w0 + -0.125 \cdot \color{blue}{{\left(\frac{\ell}{{\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(h \cdot w0\right)}\right)}^{-1}} \]
11. Step-by-step derivation

    1. unpow-1 69.5%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{1}{\frac{\ell}{{\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(h \cdot w0\right)}}} \]

    2. associate-*r* 79.1%

       \[\leadsto w0 + -0.125 \cdot \frac{1}{\frac{\ell}{\color{blue}{\left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot h\right) \cdot w0}}} \]

    3. *-commutative 79.1%

       \[\leadsto w0 + -0.125 \cdot \frac{1}{\frac{\ell}{\color{blue}{\left(h \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)} \cdot w0}} \]

    4. associate-/r* 79.9%

       \[\leadsto w0 + -0.125 \cdot \frac{1}{\color{blue}{\frac{\frac{\ell}{h \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}}{w0}}} \]

    5. *-commutative 79.9%

       \[\leadsto w0 + -0.125 \cdot \frac{1}{\frac{\frac{\ell}{h \cdot {\color{blue}{\left(\frac{D}{d} \cdot M\right)}}^{2}}}{w0}} \]

    6. associate-*l/ 79.7%

       \[\leadsto w0 + -0.125 \cdot \frac{1}{\frac{\frac{\ell}{h \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}}}{w0}} \]

    7. associate-*r/ 79.7%

       \[\leadsto w0 + -0.125 \cdot \frac{1}{\frac{\frac{\ell}{h \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}}}{w0}} \]

12. Simplified 79.7%

    \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{1}{\frac{\frac{\ell}{h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}{w0}}} \]

13. Final simplification 79.7%

    \[\leadsto w0 + -0.125 \cdot \frac{1}{\frac{\frac{\ell}{h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}{w0}} \]

Alternative 10: 72.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;M \leq 2.06 \cdot 10^{-7}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(t_0 \cdot t_0\right) \cdot \frac{h}{\frac{\ell}{w0}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* D (/ M d))))
   (if (<= M 2.06e-7) w0 (+ w0 (* -0.125 (* (* t_0 t_0) (/ h (/ l w0))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = D * (M / d);
	double tmp;
	if (M <= 2.06e-7) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((t_0 * t_0) * (h / (l / w0))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * (m / d_1)
    if (m <= 2.06d-7) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((t_0 * t_0) * (h / (l / w0))))
    end if
    code = tmp
end function

Java

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 (M <= 2.06e-7) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((t_0 * t_0) * (h / (l / w0))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = D * (M / d)
	tmp = 0
	if M <= 2.06e-7:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((t_0 * t_0) * (h / (l / w0))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(D * Float64(M / d))
	tmp = 0.0
	if (M <= 2.06e-7)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(t_0 * t_0) * Float64(h / Float64(l / w0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = D * (M / d);
	tmp = 0.0;
	if (M <= 2.06e-7)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((t_0 * t_0) * (h / (l / w0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 2.06e-7], w0, N[(w0 + N[(-0.125 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(h / N[(l / w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 2.05999999999999992e-7

   1. Initial program 80.9%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in D around 0 72.1%

      \[\leadsto \color{blue}{w0} \]

   if 2.05999999999999992e-7 < M

   1. Initial program 82.4%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in D around 0 43.3%

      \[\leadsto \color{blue}{w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 25.1%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)\right)} \]

      2. expm1-udef 25.1%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)} - 1\right)} \]

      3. associate-*r* 25.1%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell}\right)} - 1\right) \]

      4. pow-prod-down 38.2%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} - 1\right) \]

      5. *-commutative 38.2%

         \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\color{blue}{\ell \cdot {d}^{2}}}\right)} - 1\right) \]

   6. Applied egg-rr 38.2%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 38.2%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}\right)\right)} \]

      2. expm1-log1p 58.1%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}} \]

      3. associate-*r* 63.1%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot w0}}{\ell \cdot {d}^{2}} \]

      4. *-commutative 63.1%

         \[\leadsto w0 + -0.125 \cdot \frac{\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot w0}{\color{blue}{{d}^{2} \cdot \ell}} \]

      5. unpow2 63.1%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      6. swap-sqr 46.7%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)} \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      7. unpow2 46.7%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\left(\color{blue}{{D}^{2}} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      8. unpow2 46.7%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(\left({D}^{2} \cdot \color{blue}{{M}^{2}}\right) \cdot h\right) \cdot w0}{{d}^{2} \cdot \ell} \]

      9. associate-*r* 45.1%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)} \cdot w0}{{d}^{2} \cdot \ell} \]

      10. associate-*r* 43.4%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{{D}^{2} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

      11. times-frac 41.7%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w0}{\ell}\right)} \]

      12. associate-*r/ 43.3%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{\frac{{D}^{2}}{{d}^{2}} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)}{\ell}} \]

   8. Simplified 68.7%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell}} \]

   9. Taylor expanded in D around 0 43.3%

      \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   10. Step-by-step derivation

       1. *-commutative 43.3%

          \[\leadsto w0 + -0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(\left(h \cdot w0\right) \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell} \]

       2. *-commutative 43.3%

          \[\leadsto w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left(\left(h \cdot w0\right) \cdot {M}^{2}\right)}{\color{blue}{\ell \cdot {d}^{2}}} \]

       3. times-frac 39.9%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{\left(h \cdot w0\right) \cdot {M}^{2}}{{d}^{2}}\right)} \]

       4. associate-*l/ 41.6%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \frac{\left(h \cdot w0\right) \cdot {M}^{2}}{{d}^{2}}}{\ell}} \]

   11. Simplified 68.7%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)} \]
   12. Step-by-step derivation

       1. unpow2 68.7%

          \[\leadsto w0 + -0.125 \cdot \left(\color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \frac{h}{\frac{\ell}{w0}}\right) \]

   13. Applied egg-rr 68.7%

       \[\leadsto w0 + -0.125 \cdot \left(\color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \frac{h}{\frac{\ell}{w0}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.06 \cdot 10^{-7}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{h}{\frac{\ell}{w0}}\right)\\ \end{array} \]

Alternative 11: 69.2% accurate, 216.0× speedup

Math

\[\begin{array}{l} \\ w0 \end{array} \]

FPCore

(FPCore (w0 M D h l d) :precision binary64 w0)

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

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
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Python

def code(w0, M, D, h, l, d):
	return w0

Julia

function code(w0, M, D, h, l, d)
	return w0
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 81.2%

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

3. Simplified 81.3%

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

4. Taylor expanded in D around 0 66.1%

   \[\leadsto \color{blue}{w0} \]

5. Final simplification 66.1%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023298 
(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))))))