Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 86.4%

Time: 14.5s

Alternatives: 10

Speedup: 1.0×

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

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: 81.1% 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: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D \cdot 0.5}{\frac{d}{M}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) 2e-12)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ (* D 0.5) (/ d M)) 2.0)))))
   w0))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= 2e-12) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D * 0.5) / (d / M)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= 2d-12) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d * 0.5d0) / (d_1 / m)) ** 2.0d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= 2e-12) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D * 0.5) / (d / M)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= 2e-12:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D * 0.5) / (d / M)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= 2e-12)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D * 0.5) / Float64(d / M)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (d * 2.0)) ^ 2.0) * (h / l)) <= 2e-12)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D * 0.5) / (d / M)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 2e-12], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D * 0.5), $MachinePrecision] / N[(d / M), $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)) < 1.99999999999999996e-12

   1. Initial program 89.8%

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

      1. *-commutative 89.8%

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

      2. times-frac 89.3%

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

   3. Simplified 89.3%

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

      1. *-commutative 89.3%

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

      2. clear-num 89.4%

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

      3. un-div-inv 89.7%

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

      4. div-inv 89.7%

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

      5. metadata-eval 89.7%

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

   5. Applied egg-rr 89.7%

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

   if 1.99999999999999996e-12 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in M around 0 65.3%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D \cdot 0.5}{\frac{d}{M}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-0.25 \cdot \left(D \cdot D\right)}{\frac{\frac{d \cdot \left(\ell \cdot d\right)}{h}}{M \cdot M}}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) (- INFINITY))
   (* w0 (sqrt (+ 1.0 (/ (* -0.25 (* D D)) (/ (/ (* d (* l d)) h) (* M M))))))
   (if (<= (/ h l) -1e-321)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
     w0)))

C

assert(M < D);
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 + ((-0.25 * (D * D)) / (((d * (l * d)) / h) / (M * M)))));
	} else if ((h / l) <= -1e-321) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

assert M < D;
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 + ((-0.25 * (D * D)) / (((d * (l * d)) / h) / (M * M)))));
	} else if ((h / l) <= -1e-321) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 + ((-0.25 * (D * D)) / (((d * (l * d)) / h) / (M * M)))))
	elif (h / l) <= -1e-321:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
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(-0.25 * Float64(D * D)) / Float64(Float64(Float64(d * Float64(l * d)) / h) / Float64(M * M))))));
	elseif (Float64(h / l) <= -1e-321)
		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

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 + ((-0.25 * (D * D)) / (((d * (l * d)) / h) / (M * M)))));
	elseif ((h / l) <= -1e-321)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(-0.25 * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-321], 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 39.3%

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

      1. *-commutative 39.3%

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

      2. times-frac 39.3%

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

   3. Simplified 39.3%

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

   4. Taylor expanded in w0 around 0 52.6%

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

      1. *-commutative 52.6%

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

      2. cancel-sign-sub-inv 52.6%

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

      3. metadata-eval 52.6%

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

      4. *-commutative 52.6%

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

      5. associate-/l* 52.6%

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

      6. associate-*r/ 52.6%

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

      7. unpow2 52.6%

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

      8. associate-/r* 53.2%

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

      9. *-commutative 53.2%

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

      10. unpow2 53.2%

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

      11. associate-*l* 65.2%

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

      12. unpow2 65.2%

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

   6. Simplified 65.2%

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

   if -inf.0 < (/.f64 h l) < -9.98013e-322

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. times-frac 84.7%

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

   3. Simplified 84.7%

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

   if -9.98013e-322 < (/.f64 h l)

   1. Initial program 87.9%

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

      1. *-commutative 87.9%

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

      2. times-frac 87.9%

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

   3. Simplified 87.9%

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

   4. Taylor expanded in M around 0 95.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-0.25 \cdot \left(D \cdot D\right)}{\frac{\frac{d \cdot \left(\ell \cdot d\right)}{h}}{M \cdot M}}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3: 86.3% accurate, 1.0× speedup

Math

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

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (*
  w0
  (sqrt (+ 1.0 (/ -1.0 (* l (/ 1.0 (* h (pow (* (/ M d) (* D 0.5)) 2.0)))))))))

C

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

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 + ((-1.0d0) / (l * (1.0d0 / (h * (((m / d_1) * (d * 0.5d0)) ** 2.0d0)))))))
end function

Java

assert M < D;
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 * (1.0 / (h * Math.pow(((M / d) * (D * 0.5)), 2.0)))))));
}

Python

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

Julia

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

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 + (-1.0 / (l * (1.0 / (h * (((M / d) * (D * 0.5)) ^ 2.0)))))));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l * N[(1.0 / N[(h * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. *-commutative 81.7%

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

   2. times-frac 81.3%

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

3. Simplified 81.3%

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

   1. associate-*r/ 86.8%

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

   2. frac-times 87.2%

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

   3. *-commutative 87.2%

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

   4. clear-num 87.2%

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

   5. *-commutative 87.2%

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

   6. *-commutative 87.2%

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

   7. frac-times 86.8%

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

   8. div-inv 86.8%

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

   9. metadata-eval 86.8%

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

5. Applied egg-rr 86.8%

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

   1. div-inv 86.8%

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

7. Applied egg-rr 86.8%

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

8. Final simplification 86.8%

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

Alternative 4: 86.3% accurate, 1.0× speedup

Math

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

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (+ 1.0 (/ -1.0 (/ l (* h (pow (* (/ M d) (* D 0.5)) 2.0))))))))

C

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

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 + ((-1.0d0) / (l / (h * (((m / d_1) * (d * 0.5d0)) ** 2.0d0))))))
end function

Java

assert M < D;
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 * Math.pow(((M / d) * (D * 0.5)), 2.0))))));
}

Python

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

Julia

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

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 + (-1.0 / (l / (h * (((M / d) * (D * 0.5)) ^ 2.0))))));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[(h * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. *-commutative 81.7%

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

   2. times-frac 81.3%

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

3. Simplified 81.3%

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

   1. associate-*r/ 86.8%

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

   2. frac-times 87.2%

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

   3. *-commutative 87.2%

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

   4. clear-num 87.2%

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

   5. *-commutative 87.2%

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

   6. *-commutative 87.2%

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

   7. frac-times 86.8%

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

   8. div-inv 86.8%

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

   9. metadata-eval 86.8%

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

5. Applied egg-rr 86.8%

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

6. Final simplification 86.8%

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

Alternative 5: 86.3% accurate, 1.0× speedup

Math

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

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (/ (* h (pow (* (/ M d) (* D 0.5)) 2.0)) l)))))

C

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

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((h * (((m / d_1) * (d * 0.5d0)) ** 2.0d0)) / l)))
end function

Java

assert M < D;
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(((M / d) * (D * 0.5)), 2.0)) / l)));
}

Python

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

Julia

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

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((h * (((M / d) * (D * 0.5)) ^ 2.0)) / l)));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. *-commutative 81.7%

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

   2. times-frac 81.3%

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

3. Simplified 81.3%

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

   1. *-commutative 81.3%

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

   2. frac-times 81.7%

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

   3. *-commutative 81.7%

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

   4. associate-*l/ 87.2%

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

   5. *-commutative 87.2%

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

   6. frac-times 86.8%

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

   7. div-inv 86.8%

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

   8. metadata-eval 86.8%

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

5. Applied egg-rr 86.8%

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

6. Final simplification 86.8%

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

Alternative 6: 77.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-223}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\left(M \cdot M\right) \cdot \left(\frac{h \cdot D}{d} \cdot \frac{D}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -1e-223)
   (* w0 (sqrt (- 1.0 (* (/ 0.25 l) (* (* M M) (* (/ (* h D) d) (/ D d)))))))
   w0))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e-223) {
		tmp = w0 * sqrt((1.0 - ((0.25 / l) * ((M * M) * (((h * D) / d) * (D / d))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((h / l) <= (-1d-223)) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 / l) * ((m * m) * (((h * d) / d_1) * (d / d_1))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e-223) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 / l) * ((M * M) * (((h * D) / d) * (D / d))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -1e-223:
		tmp = w0 * math.sqrt((1.0 - ((0.25 / l) * ((M * M) * (((h * D) / d) * (D / d))))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -1e-223)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 / l) * Float64(Float64(M * M) * Float64(Float64(Float64(h * D) / d) * Float64(D / d)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -1e-223)
		tmp = w0 * sqrt((1.0 - ((0.25 / l) * ((M * M) * (((h * D) / d) * (D / d))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e-223], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 / l), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(N[(N[(h * D), $MachinePrecision] / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -9.9999999999999997e-224

   1. Initial program 77.4%

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

      1. *-commutative 77.4%

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

      2. times-frac 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in M around 0 49.2%

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

      1. associate-*r/ 49.2%

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

      2. *-commutative 49.2%

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

      3. times-frac 52.1%

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

      4. associate-*r* 53.8%

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

      5. unpow2 53.8%

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

      6. associate-*l* 55.2%

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

      7. unpow2 55.2%

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

      8. unpow2 55.2%

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

   6. Simplified 55.2%

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

   7. Taylor expanded in D around 0 52.1%

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

      1. unpow2 52.1%

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

      2. *-commutative 52.1%

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

      3. unpow2 52.1%

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

      4. associate-*l* 53.8%

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

      5. unpow2 53.8%

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

      6. associate-*l* 55.2%

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

      7. *-commutative 55.2%

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

      8. *-rgt-identity 55.2%

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

      9. associate-*r/ 55.2%

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

      10. associate-*l* 56.4%

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

      11. *-commutative 56.4%

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

      12. associate-*l* 60.2%

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

      13. *-commutative 60.2%

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

      14. associate-*r/ 60.2%

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

      15. *-commutative 60.2%

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

      16. *-lft-identity 60.2%

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

   9. Simplified 60.2%

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

       1. associate-*r/ 56.4%

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

       2. *-commutative 56.4%

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

   11. Applied egg-rr 56.4%

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

       1. times-frac 66.8%

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

       2. *-commutative 66.8%

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

   13. Applied egg-rr 66.8%

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

   if -9.9999999999999997e-224 < (/.f64 h l)

   1. Initial program 87.1%

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

      1. *-commutative 87.1%

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

      2. times-frac 87.0%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in M around 0 90.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-223}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\left(M \cdot M\right) \cdot \left(\frac{h \cdot D}{d} \cdot \frac{D}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 7: 72.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq -4.3 \cdot 10^{+52} \lor \neg \left(D \leq 1.1 \cdot 10^{+21}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{d \cdot d}\right) \cdot \frac{h}{\frac{\ell}{M \cdot M}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (or (<= D -4.3e+52) (not (<= D 1.1e+21)))
   (* w0 (+ 1.0 (* -0.125 (* (* D (/ D (* d d))) (/ h (/ l (* M M)))))))
   w0))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D <= -4.3e+52) || !(D <= 1.1e+21)) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / (d * d))) * (h / (l / (M * M))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((d <= (-4.3d+52)) .or. (.not. (d <= 1.1d+21))) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d / (d_1 * d_1))) * (h / (l / (m * m))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D <= -4.3e+52) || !(D <= 1.1e+21)) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / (d * d))) * (h / (l / (M * M))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (D <= -4.3e+52) or not (D <= 1.1e+21):
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / (d * d))) * (h / (l / (M * M))))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if ((D <= -4.3e+52) || !(D <= 1.1e+21))
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D / Float64(d * d))) * Float64(h / Float64(l / Float64(M * M)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((D <= -4.3e+52) || ~((D <= 1.1e+21)))
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / (d * d))) * (h / (l / (M * M))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[Or[LessEqual[D, -4.3e+52], N[Not[LessEqual[D, 1.1e+21]], $MachinePrecision]], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / N[(l / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if D < -4.3e52 or 1.1e21 < D

   1. Initial program 76.8%

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

      1. *-commutative 76.8%

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

      2. times-frac 77.6%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in M around 0 42.2%

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

      1. associate-*r/ 42.2%

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

      2. *-commutative 42.2%

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

      3. associate-*r/ 42.2%

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

      4. *-commutative 42.2%

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

      5. times-frac 39.5%

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

      6. unpow2 39.5%

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

      7. *-commutative 39.5%

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

      8. unpow2 39.5%

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

      9. unpow2 39.5%

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

   6. Simplified 39.5%

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

   7. Taylor expanded in D around 0 42.2%

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

      1. unpow2 42.2%

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

      2. times-frac 42.1%

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

      3. unpow2 42.1%

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

      4. *-commutative 42.1%

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

      5. unpow2 42.1%

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

      6. associate-*l/ 58.8%

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

      7. *-commutative 58.8%

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

      8. associate-/l* 58.8%

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

   9. Simplified 58.8%

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

   if -4.3e52 < D < 1.1e21

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. times-frac 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in M around 0 82.2%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq -4.3 \cdot 10^{+52} \lor \neg \left(D \leq 1.1 \cdot 10^{+21}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{d \cdot d}\right) \cdot \frac{h}{\frac{\ell}{M \cdot M}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 8: 72.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq -1.58 \cdot 10^{-12} \lor \neg \left(D \leq 5.1 \cdot 10^{+20}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{d \cdot d}\right) \cdot \frac{h}{\frac{\frac{\ell}{M}}{M}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (or (<= D -1.58e-12) (not (<= D 5.1e+20)))
   (* w0 (+ 1.0 (* -0.125 (* (* D (/ D (* d d))) (/ h (/ (/ l M) M))))))
   w0))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D <= -1.58e-12) || !(D <= 5.1e+20)) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / (d * d))) * (h / ((l / M) / M)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((d <= (-1.58d-12)) .or. (.not. (d <= 5.1d+20))) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d / (d_1 * d_1))) * (h / ((l / m) / m)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D <= -1.58e-12) || !(D <= 5.1e+20)) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / (d * d))) * (h / ((l / M) / M)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (D <= -1.58e-12) or not (D <= 5.1e+20):
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / (d * d))) * (h / ((l / M) / M)))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if ((D <= -1.58e-12) || !(D <= 5.1e+20))
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D / Float64(d * d))) * Float64(h / Float64(Float64(l / M) / M))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((D <= -1.58e-12) || ~((D <= 5.1e+20)))
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / (d * d))) * (h / ((l / M) / M)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[Or[LessEqual[D, -1.58e-12], N[Not[LessEqual[D, 5.1e+20]], $MachinePrecision]], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / N[(N[(l / M), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if D < -1.57999999999999993e-12 or 5.1e20 < D

   1. Initial program 77.7%

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

      1. *-commutative 77.7%

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

      2. times-frac 78.4%

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

   3. Simplified 78.4%

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

   4. Taylor expanded in M around 0 43.6%

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

      1. associate-*r/ 43.6%

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

      2. *-commutative 43.6%

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

      3. associate-*r/ 43.6%

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

      4. *-commutative 43.6%

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

      5. times-frac 41.2%

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

      6. unpow2 41.2%

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

      7. *-commutative 41.2%

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

      8. unpow2 41.2%

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

      9. unpow2 41.2%

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

   6. Simplified 41.2%

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

   7. Taylor expanded in D around 0 43.6%

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

      1. unpow2 43.6%

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

      2. times-frac 43.4%

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

      3. unpow2 43.4%

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

      4. *-commutative 43.4%

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

      5. unpow2 43.4%

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

      6. associate-*l/ 58.4%

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

      7. *-commutative 58.4%

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

      8. associate-/l* 58.4%

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

   9. Simplified 58.4%

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

   10. Taylor expanded in l around 0 58.4%

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

       1. unpow2 58.4%

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

       2. associate-/r* 61.8%

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

   12. Simplified 61.8%

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

   if -1.57999999999999993e-12 < D < 5.1e20

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. times-frac 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in M around 0 83.9%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq -1.58 \cdot 10^{-12} \lor \neg \left(D \leq 5.1 \cdot 10^{+20}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{d \cdot d}\right) \cdot \frac{h}{\frac{\frac{\ell}{M}}{M}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 9: 72.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+76}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\frac{D \cdot D}{\ell} \cdot \left(\left(M \cdot \frac{M}{d}\right) \cdot \frac{h}{d}\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -1e+76)
   (* w0 (+ 1.0 (* (* (/ (* D D) l) (* (* M (/ M d)) (/ h d))) -0.125)))
   w0))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e+76) {
		tmp = w0 * (1.0 + ((((D * D) / l) * ((M * (M / d)) * (h / d))) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((h / l) <= (-1d+76)) then
        tmp = w0 * (1.0d0 + ((((d * d) / l) * ((m * (m / d_1)) * (h / d_1))) * (-0.125d0)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e+76) {
		tmp = w0 * (1.0 + ((((D * D) / l) * ((M * (M / d)) * (h / d))) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -1e+76:
		tmp = w0 * (1.0 + ((((D * D) / l) * ((M * (M / d)) * (h / d))) * -0.125))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -1e+76)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(Float64(D * D) / l) * Float64(Float64(M * Float64(M / d)) * Float64(h / d))) * -0.125)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -1e+76)
		tmp = w0 * (1.0 + ((((D * D) / l) * ((M * (M / d)) * (h / d))) * -0.125));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e+76], N[(w0 * N[(1.0 + N[(N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1e76

   1. Initial program 70.0%

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

      1. *-commutative 70.0%

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

      2. times-frac 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in M around 0 50.7%

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

      1. associate-*r/ 50.7%

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

      2. *-commutative 50.7%

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

      3. associate-*r/ 50.7%

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

      4. *-commutative 50.7%

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

      5. times-frac 49.8%

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

      6. unpow2 49.8%

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

      7. *-commutative 49.8%

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

      8. unpow2 49.8%

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

      9. unpow2 49.8%

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

   6. Simplified 49.8%

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

   7. Taylor expanded in M around 0 49.8%

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

      1. unpow2 49.8%

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

      2. unpow2 49.8%

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

      3. times-frac 56.9%

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

      4. associate-/l* 62.3%

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

      5. associate-/r/ 62.3%

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

   9. Simplified 62.3%

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

   if -1e76 < (/.f64 h l)

   1. Initial program 86.6%

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

      1. *-commutative 86.6%

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

      2. times-frac 86.0%

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

   3. Simplified 86.0%

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

   4. Taylor expanded in M around 0 79.6%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+76}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\frac{D \cdot D}{\ell} \cdot \left(\left(M \cdot \frac{M}{d}\right) \cdot \frac{h}{d}\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 10: 68.6% accurate, 216.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d) :precision binary64 w0)

C

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

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

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

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return w0
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 81.7%

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

   1. *-commutative 81.7%

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

   2. times-frac 81.3%

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

3. Simplified 81.3%

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

4. Taylor expanded in M around 0 66.4%

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

5. Final simplification 66.4%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023176 
(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))))))