Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 85.6%

Time: 13.9s

Alternatives: 11

Speedup: 1.7×

• 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: 81.0% 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: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\frac{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}{\frac{D}{\ell}}}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-206}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 M) (/ d M)) h) (/ D l)))))))
   (if (<= (/ h l) -1e-206)
     (* 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 / M) * (d / M)) / h) / (D / l))))));
	} else if ((h / l) <= -1e-206) {
		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 / M) * (d / M)) / h) / (D / l))))));
	} else if ((h / l) <= -1e-206) {
		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 / M) * (d / M)) / h) / (D / l))))))
	elif (h / l) <= -1e-206:
		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(0.25 * Float64(D / Float64(Float64(Float64(Float64(d / M) * Float64(d / M)) / h) / Float64(D / l)))))));
	elseif (Float64(h / l) <= -1e-206)
		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 / M) * (d / M)) / h) / (D / l))))));
	elseif ((h / l) <= -1e-206)
		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[(0.25 * N[(D / N[(N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-206], 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 39.1%

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

      1. *-commutative 39.1%

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

      2. times-frac 39.1%

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

   3. Simplified 39.1%

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

      1. *-commutative 39.1%

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

      2. frac-times 39.1%

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

      3. *-commutative 39.1%

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

      4. associate-*l/ 84.0%

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

      5. div-inv 84.0%

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

      6. associate-*l* 83.9%

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

      7. associate-/r* 83.9%

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

      8. metadata-eval 83.9%

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

   5. Applied egg-rr 83.9%

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

   6. Taylor expanded in h around 0 57.6%

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

      1. unpow2 57.6%

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

      2. associate-/l* 57.6%

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

      3. *-commutative 57.6%

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

      4. unpow2 57.6%

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

      5. associate-/l* 57.6%

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

      6. associate-/r/ 57.6%

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

      7. *-commutative 57.6%

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

      8. unpow2 57.6%

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

      9. associate-/l/ 62.2%

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

      10. associate-*l/ 66.9%

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

      11. *-commutative 66.9%

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

      12. associate-/l* 69.6%

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

   8. Simplified 70.7%

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

   if -inf.0 < (/.f64 h l) < -1.00000000000000003e-206

   1. Initial program 86.5%

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

      1. *-commutative 86.5%

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

      2. times-frac 87.4%

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

   3. Simplified 87.4%

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

   if -1.00000000000000003e-206 < (/.f64 h l)

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

      2. times-frac 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in M around 0 96.0%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\frac{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}{\frac{D}{\ell}}}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-206}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+160}:\\ \;\;\;\;w0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\frac{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}{\frac{D}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)))))
   (if (<= t_0 2e+160)
     (* w0 (sqrt t_0))
     (*
      w0
      (sqrt (- 1.0 (* 0.25 (/ D (/ (/ (* (/ d M) (/ d M)) h) (/ D l))))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (pow(((M * D) / (d * 2.0)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 2e+160) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (D / ((((d / M) * (d / M)) / h) / (D / l))))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((m * d) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l))
    if (t_0 <= 2d+160) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d / ((((d_1 / m) * (d_1 / m)) / h) / (d / l))))))
    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 t_0 = 1.0 - (Math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 2e+160) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D / ((((d / M) * (d / M)) / h) / (D / l))))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = 1.0 - (math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 2e+160:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D / ((((d / M) * (d / M)) / h) / (D / l))))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 2e+160)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D / Float64(Float64(Float64(Float64(d / M) * Float64(d / M)) / h) / Float64(D / l)))))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = 1.0 - ((((M * D) / (d * 2.0)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 2e+160)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (D / ((((d / M) * (d / M)) / h) / (D / l))))));
	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_] := Block[{t$95$0 = N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+160], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D / N[(N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 2.00000000000000001e160

   1. Initial program 99.4%

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

   if 2.00000000000000001e160 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 42.8%

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

      1. *-commutative 42.8%

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

      2. times-frac 44.1%

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

   3. Simplified 44.1%

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

      1. *-commutative 44.1%

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

      2. frac-times 42.8%

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

      3. *-commutative 42.8%

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

      4. associate-*l/ 70.0%

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

      5. div-inv 70.0%

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

      6. associate-*l* 72.5%

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

      7. associate-/r* 72.5%

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

      8. metadata-eval 72.5%

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

   5. Applied egg-rr 72.5%

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

   6. Taylor expanded in h around 0 52.7%

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

      1. unpow2 52.7%

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

      2. associate-/l* 54.0%

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

      3. *-commutative 54.0%

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

      4. unpow2 54.0%

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

      5. associate-/l* 52.7%

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

      6. associate-/r/ 56.6%

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

      7. *-commutative 56.6%

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

      8. unpow2 56.6%

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

      9. associate-/l/ 55.2%

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

      10. associate-*l/ 56.5%

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

      11. *-commutative 56.5%

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

      12. associate-/l* 61.3%

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

   8. Simplified 68.3%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{D}{\frac{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}{\frac{D}{\ell}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{+160}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\frac{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}{\frac{D}{\ell}}}}\\ \end{array} \]

Alternative 3: 86.5% accurate, 1.0× speedup

Math

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

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 (/ 0.5 d))) 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 * (0.5 / d))), 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 * (0.5d0 / d_1))) ** 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 * (0.5 / d))), 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 * (0.5 / d))), 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(M * Float64(D * Float64(0.5 / d))) ^ 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 * (0.5 / d))) ^ 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[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.4%

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

   1. *-commutative 82.4%

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

   2. times-frac 81.6%

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

3. Simplified 81.6%

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

   1. *-commutative 81.6%

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

   2. frac-times 82.4%

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

   3. *-commutative 82.4%

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

   4. associate-*l/ 90.5%

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

   5. div-inv 90.5%

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

   6. associate-*l* 91.7%

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

   7. associate-/r* 91.7%

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

   8. metadata-eval 91.7%

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

5. Applied egg-rr 91.7%

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

6. Final simplification 91.7%

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

Alternative 4: 81.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{M} \cdot \frac{d}{M}\\ \mathbf{if}\;D \leq -2.15 \cdot 10^{-203}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\frac{\frac{t_0}{h}}{\frac{D}{\ell}}}}\\ \mathbf{elif}\;D \leq 8 \cdot 10^{-116}:\\ \;\;\;\;w0\\ \mathbf{elif}\;D \leq 2.9 \cdot 10^{+136}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \frac{h}{t_0}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D}{\ell \cdot \frac{\frac{{\left(\frac{d}{M}\right)}^{2}}{h}}{D}} \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (/ d M) (/ d M))))
   (if (<= D -2.15e-203)
     (* w0 (sqrt (- 1.0 (* 0.25 (/ D (/ (/ t_0 h) (/ D l)))))))
     (if (<= D 8e-116)
       w0
       (if (<= D 2.9e+136)
         (* w0 (sqrt (- 1.0 (/ (* 0.25 (* (* D D) (/ h t_0))) l))))
         (*
          w0
          (+ 1.0 (* (/ D (* l (/ (/ (pow (/ d M) 2.0) h) D))) -0.125))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (d / M) * (d / M);
	double tmp;
	if (D <= -2.15e-203) {
		tmp = w0 * sqrt((1.0 - (0.25 * (D / ((t_0 / h) / (D / l))))));
	} else if (D <= 8e-116) {
		tmp = w0;
	} else if (D <= 2.9e+136) {
		tmp = w0 * sqrt((1.0 - ((0.25 * ((D * D) * (h / t_0))) / l)));
	} else {
		tmp = w0 * (1.0 + ((D / (l * ((pow((d / M), 2.0) / h) / D))) * -0.125));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (d_1 / m) * (d_1 / m)
    if (d <= (-2.15d-203)) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d / ((t_0 / h) / (d / l))))))
    else if (d <= 8d-116) then
        tmp = w0
    else if (d <= 2.9d+136) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * ((d * d) * (h / t_0))) / l)))
    else
        tmp = w0 * (1.0d0 + ((d / (l * ((((d_1 / m) ** 2.0d0) / h) / d))) * (-0.125d0)))
    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 t_0 = (d / M) * (d / M);
	double tmp;
	if (D <= -2.15e-203) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D / ((t_0 / h) / (D / l))))));
	} else if (D <= 8e-116) {
		tmp = w0;
	} else if (D <= 2.9e+136) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * ((D * D) * (h / t_0))) / l)));
	} else {
		tmp = w0 * (1.0 + ((D / (l * ((Math.pow((d / M), 2.0) / h) / D))) * -0.125));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = (d / M) * (d / M)
	tmp = 0
	if D <= -2.15e-203:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D / ((t_0 / h) / (D / l))))))
	elif D <= 8e-116:
		tmp = w0
	elif D <= 2.9e+136:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * ((D * D) * (h / t_0))) / l)))
	else:
		tmp = w0 * (1.0 + ((D / (l * ((math.pow((d / M), 2.0) / h) / D))) * -0.125))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(d / M) * Float64(d / M))
	tmp = 0.0
	if (D <= -2.15e-203)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D / Float64(Float64(t_0 / h) / Float64(D / l)))))));
	elseif (D <= 8e-116)
		tmp = w0;
	elseif (D <= 2.9e+136)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(D * D) * Float64(h / t_0))) / l))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D / Float64(l * Float64(Float64((Float64(d / M) ^ 2.0) / h) / D))) * -0.125)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (d / M) * (d / M);
	tmp = 0.0;
	if (D <= -2.15e-203)
		tmp = w0 * sqrt((1.0 - (0.25 * (D / ((t_0 / h) / (D / l))))));
	elseif (D <= 8e-116)
		tmp = w0;
	elseif (D <= 2.9e+136)
		tmp = w0 * sqrt((1.0 - ((0.25 * ((D * D) * (h / t_0))) / l)));
	else
		tmp = w0 * (1.0 + ((D / (l * ((((d / M) ^ 2.0) / h) / D))) * -0.125));
	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_] := Block[{t$95$0 = N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, -2.15e-203], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D / N[(N[(t$95$0 / h), $MachinePrecision] / N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 8e-116], w0, If[LessEqual[D, 2.9e+136], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(N[(D * D), $MachinePrecision] * N[(h / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(D / N[(l * N[(N[(N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision] / h), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if D < -2.15000000000000014e-203

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

      2. times-frac 80.4%

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

   3. Simplified 80.4%

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

      1. *-commutative 80.4%

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

      2. frac-times 78.4%

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

      3. *-commutative 78.4%

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

      4. associate-*l/ 89.5%

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

      5. div-inv 89.4%

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

      6. associate-*l* 91.2%

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

      7. associate-/r* 91.2%

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

      8. metadata-eval 91.2%

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

   5. Applied egg-rr 91.2%

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

   6. Taylor expanded in h around 0 65.7%

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

      1. unpow2 65.7%

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

      2. associate-/l* 67.9%

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

      3. *-commutative 67.9%

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

      4. unpow2 67.9%

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

      5. associate-/l* 67.8%

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

      6. associate-/r/ 69.9%

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

      7. *-commutative 69.9%

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

      8. unpow2 69.9%

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

      9. associate-/l/ 66.8%

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

      10. associate-*l/ 69.0%

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

      11. *-commutative 69.0%

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

      12. associate-/l* 75.8%

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

   8. Simplified 84.3%

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

   if -2.15000000000000014e-203 < D < 8e-116

   1. Initial program 89.5%

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

      1. *-commutative 89.5%

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

      2. times-frac 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in M around 0 91.8%

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

   if 8e-116 < D < 2.89999999999999974e136

   1. Initial program 79.5%

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

      1. *-commutative 79.5%

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

      2. times-frac 81.6%

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

   3. Simplified 81.6%

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

      1. *-commutative 81.6%

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

      2. frac-times 79.5%

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

      3. *-commutative 79.5%

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

      4. associate-*l/ 89.6%

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

      5. div-inv 89.6%

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

      6. associate-*l* 91.7%

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

      7. associate-/r* 91.7%

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

      8. metadata-eval 91.7%

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

   5. Applied egg-rr 91.7%

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

   6. Taylor expanded in h around 0 76.2%

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

      1. unpow2 76.2%

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

      2. associate-/l* 78.1%

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

      3. unpow2 78.1%

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

      4. associate-/l* 76.2%

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

      5. *-commutative 76.2%

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

      6. associate-*l/ 78.0%

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

      7. associate-*l/ 78.0%

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

      8. unpow2 78.0%

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

      9. *-commutative 78.0%

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

      10. unpow2 78.0%

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

      11. associate-/r/ 76.7%

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

      12. unpow2 76.7%

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

      13. times-frac 89.3%

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

   8. Simplified 89.3%

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

   if 2.89999999999999974e136 < D

   1. Initial program 82.3%

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

      1. *-commutative 82.3%

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

      2. times-frac 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in M around 0 25.9%

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

      1. *-commutative 25.9%

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

      2. *-commutative 25.9%

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

      3. times-frac 25.9%

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

      4. *-commutative 25.9%

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

      5. unpow2 25.9%

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

      6. unpow2 25.9%

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

      7. *-commutative 25.9%

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

      8. unpow2 25.9%

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

   6. Simplified 25.9%

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

   7. Taylor expanded in D around 0 25.9%

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

      1. *-commutative 25.9%

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

      2. unpow2 25.9%

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

      3. times-frac 25.9%

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

      4. unpow2 25.9%

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

      5. associate-/l* 25.9%

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

      6. associate-/r/ 25.9%

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

      7. associate-/l* 25.9%

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

      8. *-commutative 25.9%

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

      9. associate-/r/ 25.9%

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

      10. *-commutative 25.9%

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

      11. unpow2 25.9%

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

      12. associate-/l/ 19.5%

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

      13. associate-*r/ 38.2%

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

      14. associate-/l* 56.9%

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

   9. Simplified 63.4%

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

       1. associate-/r/ 79.4%

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

       2. pow2 79.4%

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

   11. Applied egg-rr 79.4%

       \[\leadsto w0 \cdot \left(1 + \frac{D}{\color{blue}{\frac{\frac{{\left(\frac{d}{M}\right)}^{2}}{h}}{D} \cdot \ell}} \cdot -0.125\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq -2.15 \cdot 10^{-203}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\frac{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}{\frac{D}{\ell}}}}\\ \mathbf{elif}\;D \leq 8 \cdot 10^{-116}:\\ \;\;\;\;w0\\ \mathbf{elif}\;D \leq 2.9 \cdot 10^{+136}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D}{\ell \cdot \frac{\frac{{\left(\frac{d}{M}\right)}^{2}}{h}}{D}} \cdot -0.125\right)\\ \end{array} \]

Alternative 5: 79.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq -1.1 \cdot 10^{-202}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\frac{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}{\frac{D}{\ell}}}}\\ \mathbf{elif}\;D \leq 1.8 \cdot 10^{-115}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D}{\ell \cdot \frac{\frac{{\left(\frac{d}{M}\right)}^{2}}{h}}{D}} \cdot -0.125\right)\\ \end{array} \end{array} \]

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 (<= D -1.1e-202)
   (* w0 (sqrt (- 1.0 (* 0.25 (/ D (/ (/ (* (/ d M) (/ d M)) h) (/ D l)))))))
   (if (<= D 1.8e-115)
     w0
     (* w0 (+ 1.0 (* (/ D (* l (/ (/ (pow (/ d M) 2.0) h) D))) -0.125))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= -1.1e-202) {
		tmp = w0 * sqrt((1.0 - (0.25 * (D / ((((d / M) * (d / M)) / h) / (D / l))))));
	} else if (D <= 1.8e-115) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((D / (l * ((pow((d / M), 2.0) / h) / D))) * -0.125));
	}
	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 <= (-1.1d-202)) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d / ((((d_1 / m) * (d_1 / m)) / h) / (d / l))))))
    else if (d <= 1.8d-115) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((d / (l * ((((d_1 / m) ** 2.0d0) / h) / d))) * (-0.125d0)))
    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 <= -1.1e-202) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D / ((((d / M) * (d / M)) / h) / (D / l))))));
	} else if (D <= 1.8e-115) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((D / (l * ((Math.pow((d / M), 2.0) / h) / D))) * -0.125));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= -1.1e-202:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D / ((((d / M) * (d / M)) / h) / (D / l))))))
	elif D <= 1.8e-115:
		tmp = w0
	else:
		tmp = w0 * (1.0 + ((D / (l * ((math.pow((d / M), 2.0) / h) / D))) * -0.125))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= -1.1e-202)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D / Float64(Float64(Float64(Float64(d / M) * Float64(d / M)) / h) / Float64(D / l)))))));
	elseif (D <= 1.8e-115)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D / Float64(l * Float64(Float64((Float64(d / M) ^ 2.0) / h) / D))) * -0.125)));
	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 <= -1.1e-202)
		tmp = w0 * sqrt((1.0 - (0.25 * (D / ((((d / M) * (d / M)) / h) / (D / l))))));
	elseif (D <= 1.8e-115)
		tmp = w0;
	else
		tmp = w0 * (1.0 + ((D / (l * ((((d / M) ^ 2.0) / h) / D))) * -0.125));
	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[D, -1.1e-202], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D / N[(N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 1.8e-115], w0, N[(w0 * N[(1.0 + N[(N[(D / N[(l * N[(N[(N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision] / h), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if D < -1.10000000000000004e-202

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

      2. times-frac 80.4%

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

   3. Simplified 80.4%

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

      1. *-commutative 80.4%

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

      2. frac-times 78.4%

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

      3. *-commutative 78.4%

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

      4. associate-*l/ 89.5%

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

      5. div-inv 89.4%

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

      6. associate-*l* 91.2%

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

      7. associate-/r* 91.2%

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

      8. metadata-eval 91.2%

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

   5. Applied egg-rr 91.2%

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

   6. Taylor expanded in h around 0 65.7%

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

      1. unpow2 65.7%

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

      2. associate-/l* 67.9%

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

      3. *-commutative 67.9%

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

      4. unpow2 67.9%

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

      5. associate-/l* 67.8%

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

      6. associate-/r/ 69.9%

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

      7. *-commutative 69.9%

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

      8. unpow2 69.9%

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

      9. associate-/l/ 66.8%

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

      10. associate-*l/ 69.0%

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

      11. *-commutative 69.0%

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

      12. associate-/l* 75.8%

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

   8. Simplified 84.3%

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

   if -1.10000000000000004e-202 < D < 1.80000000000000005e-115

   1. Initial program 89.5%

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

      1. *-commutative 89.5%

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

      2. times-frac 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in M around 0 91.8%

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

   if 1.80000000000000005e-115 < D

   1. Initial program 80.5%

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

      1. *-commutative 80.5%

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

      2. times-frac 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in M around 0 51.5%

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

      1. *-commutative 51.5%

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

      2. *-commutative 51.5%

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

      3. times-frac 53.8%

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

      4. *-commutative 53.8%

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

      5. unpow2 53.8%

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

      6. unpow2 53.8%

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

      7. *-commutative 53.8%

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

      8. unpow2 53.8%

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

   6. Simplified 53.8%

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

   7. Taylor expanded in D around 0 51.5%

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

      1. *-commutative 51.5%

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

      2. unpow2 51.5%

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

      3. times-frac 53.8%

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

      4. unpow2 53.8%

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

      5. associate-/l* 50.2%

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

      6. associate-/r/ 52.6%

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

      7. associate-/l* 56.3%

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

      8. *-commutative 56.3%

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

      9. associate-/r/ 56.3%

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

      10. *-commutative 56.3%

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

      11. unpow2 56.3%

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

      12. associate-/l/ 46.7%

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

      13. associate-*r/ 53.8%

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

      14. associate-/l* 64.4%

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

   9. Simplified 69.6%

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

       1. associate-/r/ 79.1%

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

       2. pow2 79.1%

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

   11. Applied egg-rr 79.1%

       \[\leadsto w0 \cdot \left(1 + \frac{D}{\color{blue}{\frac{\frac{{\left(\frac{d}{M}\right)}^{2}}{h}}{D} \cdot \ell}} \cdot -0.125\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq -1.1 \cdot 10^{-202}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\frac{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}{\frac{D}{\ell}}}}\\ \mathbf{elif}\;D \leq 1.8 \cdot 10^{-115}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D}{\ell \cdot \frac{\frac{{\left(\frac{d}{M}\right)}^{2}}{h}}{D}} \cdot -0.125\right)\\ \end{array} \]

Alternative 6: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;w0 \leq -6.6 \cdot 10^{+56}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D}{\frac{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}{\frac{D}{\ell}}} \cdot -0.125\right)\\ \mathbf{elif}\;w0 \leq 1.05 \cdot 10^{-57}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\ell} \cdot \frac{D}{\frac{{\left(\frac{d}{M}\right)}^{2}}{h}}\right)\right)\\ \end{array} \end{array} \]

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 (<= w0 -6.6e+56)
   (* w0 (+ 1.0 (* (/ D (/ (/ (* (/ d M) (/ d M)) h) (/ D l))) -0.125)))
   (if (<= w0 1.05e-57)
     w0
     (* w0 (+ 1.0 (* -0.125 (* (/ D l) (/ D (/ (pow (/ d M) 2.0) h)))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (w0 <= -6.6e+56) {
		tmp = w0 * (1.0 + ((D / ((((d / M) * (d / M)) / h) / (D / l))) * -0.125));
	} else if (w0 <= 1.05e-57) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D / l) * (D / (pow((d / M), 2.0) / h)))));
	}
	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 (w0 <= (-6.6d+56)) then
        tmp = w0 * (1.0d0 + ((d / ((((d_1 / m) * (d_1 / m)) / h) / (d / l))) * (-0.125d0)))
    else if (w0 <= 1.05d-57) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d / l) * (d / (((d_1 / m) ** 2.0d0) / h)))))
    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 (w0 <= -6.6e+56) {
		tmp = w0 * (1.0 + ((D / ((((d / M) * (d / M)) / h) / (D / l))) * -0.125));
	} else if (w0 <= 1.05e-57) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D / l) * (D / (Math.pow((d / M), 2.0) / h)))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if w0 <= -6.6e+56:
		tmp = w0 * (1.0 + ((D / ((((d / M) * (d / M)) / h) / (D / l))) * -0.125))
	elif w0 <= 1.05e-57:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((D / l) * (D / (math.pow((d / M), 2.0) / h)))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (w0 <= -6.6e+56)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D / Float64(Float64(Float64(Float64(d / M) * Float64(d / M)) / h) / Float64(D / l))) * -0.125)));
	elseif (w0 <= 1.05e-57)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / l) * Float64(D / Float64((Float64(d / M) ^ 2.0) / h))))));
	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 (w0 <= -6.6e+56)
		tmp = w0 * (1.0 + ((D / ((((d / M) * (d / M)) / h) / (D / l))) * -0.125));
	elseif (w0 <= 1.05e-57)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((D / l) * (D / (((d / M) ^ 2.0) / h)))));
	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[w0, -6.6e+56], N[(w0 * N[(1.0 + N[(N[(D / N[(N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w0, 1.05e-57], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / l), $MachinePrecision] * N[(D / N[(N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if w0 < -6.60000000000000004e56

   1. Initial program 89.3%

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

      1. *-commutative 89.3%

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

      2. times-frac 87.5%

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

   3. Simplified 87.5%

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

   4. Taylor expanded in M around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. *-commutative 71.4%

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

      3. times-frac 71.4%

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

      4. *-commutative 71.4%

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

      5. unpow2 71.4%

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

      6. unpow2 71.4%

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

      7. *-commutative 71.4%

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

      8. unpow2 71.4%

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

   6. Simplified 71.4%

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

   7. Taylor expanded in D around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. unpow2 71.4%

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

      3. times-frac 71.4%

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

      4. unpow2 71.4%

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

      5. associate-/l* 66.1%

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

      6. associate-/r/ 67.9%

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

      7. associate-/l* 73.2%

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

      8. *-commutative 73.2%

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

      9. associate-/r/ 73.2%

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

      10. *-commutative 73.2%

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

      11. unpow2 73.2%

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

      12. associate-/l/ 67.9%

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

      13. associate-*r/ 69.6%

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

      14. associate-/l* 76.9%

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

   9. Simplified 80.7%

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

   if -6.60000000000000004e56 < w0 < 1.05e-57

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

      2. times-frac 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in M around 0 79.5%

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

   if 1.05e-57 < w0

   1. Initial program 82.1%

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

      1. *-commutative 82.1%

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

      2. times-frac 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in M around 0 56.5%

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

      1. *-commutative 56.5%

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

      2. *-commutative 56.5%

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

      3. times-frac 56.2%

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

      4. *-commutative 56.2%

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

      5. unpow2 56.2%

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

      6. unpow2 56.2%

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

      7. *-commutative 56.2%

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

      8. unpow2 56.2%

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

   6. Simplified 56.2%

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

   7. Taylor expanded in D around 0 56.5%

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

      1. *-commutative 56.5%

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

      2. unpow2 56.5%

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

      3. times-frac 56.2%

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

      4. unpow2 56.2%

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

      5. associate-/l* 52.6%

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

      6. associate-/r/ 54.0%

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

      7. associate-/l* 57.5%

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

      8. *-commutative 57.5%

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

      9. associate-/r/ 58.8%

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

      10. *-commutative 58.8%

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

      11. unpow2 58.8%

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

      12. associate-/l/ 54.8%

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

      13. associate-*r/ 61.2%

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

      14. associate-/l* 64.0%

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

   9. Simplified 78.4%

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

       1. associate-/r/ 78.4%

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

       2. pow2 78.4%

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

   11. Applied egg-rr 78.4%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\left(\frac{D}{\frac{{\left(\frac{d}{M}\right)}^{2}}{h}} \cdot \frac{D}{\ell}\right)} \cdot -0.125\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \leq -6.6 \cdot 10^{+56}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D}{\frac{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}{\frac{D}{\ell}}} \cdot -0.125\right)\\ \mathbf{elif}\;w0 \leq 1.05 \cdot 10^{-57}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\ell} \cdot \frac{D}{\frac{{\left(\frac{d}{M}\right)}^{2}}{h}}\right)\right)\\ \end{array} \]

Alternative 7: 77.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+138}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D}{\ell \cdot \frac{\frac{{\left(\frac{d}{M}\right)}^{2}}{h}}{D}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= l 1e+138)
   (* w0 (+ 1.0 (* (/ D (* l (/ (/ (pow (/ d M) 2.0) 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 (l <= 1e+138) {
		tmp = w0 * (1.0 + ((D / (l * ((pow((d / M), 2.0) / 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 (l <= 1d+138) then
        tmp = w0 * (1.0d0 + ((d / (l * ((((d_1 / m) ** 2.0d0) / h) / d))) * (-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 (l <= 1e+138) {
		tmp = w0 * (1.0 + ((D / (l * ((Math.pow((d / M), 2.0) / 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 l <= 1e+138:
		tmp = w0 * (1.0 + ((D / (l * ((math.pow((d / M), 2.0) / 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 (l <= 1e+138)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D / Float64(l * Float64(Float64((Float64(d / M) ^ 2.0) / 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 (l <= 1e+138)
		tmp = w0 * (1.0 + ((D / (l * ((((d / M) ^ 2.0) / 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[l, 1e+138], N[(w0 * N[(1.0 + N[(N[(D / N[(l * N[(N[(N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision] / h), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if l < 1e138

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. times-frac 82.2%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in M around 0 56.4%

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

      1. *-commutative 56.4%

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

      2. *-commutative 56.4%

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

      3. times-frac 58.6%

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

      4. *-commutative 58.6%

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

      5. unpow2 58.6%

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

      6. unpow2 58.6%

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

      7. *-commutative 58.6%

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

      8. unpow2 58.6%

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

   6. Simplified 58.6%

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

   7. Taylor expanded in D around 0 56.4%

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

      1. *-commutative 56.4%

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

      2. unpow2 56.4%

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

      3. times-frac 58.6%

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

      4. unpow2 58.6%

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

      5. associate-/l* 54.1%

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

      6. associate-/r/ 54.7%

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

      7. associate-/l* 59.1%

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

      8. *-commutative 59.1%

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

      9. associate-/r/ 60.0%

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

      10. *-commutative 60.0%

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

      11. unpow2 60.0%

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

      12. associate-/l/ 55.5%

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

      13. associate-*r/ 57.7%

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

      14. associate-/l* 64.5%

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

   9. Simplified 75.2%

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

       1. associate-/r/ 80.4%

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

       2. pow2 80.4%

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

   11. Applied egg-rr 80.4%

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

   if 1e138 < l

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

      2. times-frac 78.0%

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

   3. Simplified 78.0%

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

   4. Taylor expanded in M around 0 85.9%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+138}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D}{\ell \cdot \frac{\frac{{\left(\frac{d}{M}\right)}^{2}}{h}}{D}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 8: 74.4% accurate, 8.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;w0 \leq -6.2 \cdot 10^{+57} \lor \neg \left(w0 \leq 1.06 \cdot 10^{-57}\right):\\ \;\;\;\;w0 \cdot \left(1 + \frac{D}{\frac{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}{\frac{D}{\ell}}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (or (<= w0 -6.2e+57) (not (<= w0 1.06e-57)))
   (* w0 (+ 1.0 (* (/ D (/ (/ (* (/ d M) (/ d M)) h) (/ D l))) -0.125)))
   w0))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((w0 <= -6.2e+57) || !(w0 <= 1.06e-57)) {
		tmp = w0 * (1.0 + ((D / ((((d / M) * (d / M)) / h) / (D / l))) * -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 ((w0 <= (-6.2d+57)) .or. (.not. (w0 <= 1.06d-57))) then
        tmp = w0 * (1.0d0 + ((d / ((((d_1 / m) * (d_1 / m)) / h) / (d / l))) * (-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 ((w0 <= -6.2e+57) || !(w0 <= 1.06e-57)) {
		tmp = w0 * (1.0 + ((D / ((((d / M) * (d / M)) / h) / (D / l))) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (w0 <= -6.2e+57) or not (w0 <= 1.06e-57):
		tmp = w0 * (1.0 + ((D / ((((d / M) * (d / M)) / h) / (D / l))) * -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 ((w0 <= -6.2e+57) || !(w0 <= 1.06e-57))
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D / Float64(Float64(Float64(Float64(d / M) * Float64(d / M)) / h) / Float64(D / l))) * -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 ((w0 <= -6.2e+57) || ~((w0 <= 1.06e-57)))
		tmp = w0 * (1.0 + ((D / ((((d / M) * (d / M)) / h) / (D / l))) * -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[Or[LessEqual[w0, -6.2e+57], N[Not[LessEqual[w0, 1.06e-57]], $MachinePrecision]], N[(w0 * N[(1.0 + N[(N[(D / N[(N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if w0 < -6.20000000000000026e57 or 1.0600000000000001e-57 < w0

   1. Initial program 85.1%

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

      1. *-commutative 85.1%

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

      2. times-frac 83.6%

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

   3. Simplified 83.6%

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

   4. Taylor expanded in M around 0 62.8%

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

      1. *-commutative 62.8%

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

      2. *-commutative 62.8%

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

      3. times-frac 62.6%

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

      4. *-commutative 62.6%

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

      5. unpow2 62.6%

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

      6. unpow2 62.6%

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

      7. *-commutative 62.6%

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

      8. unpow2 62.6%

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

   6. Simplified 62.6%

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

   7. Taylor expanded in D around 0 62.8%

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

      1. *-commutative 62.8%

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

      2. unpow2 62.8%

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

      3. times-frac 62.6%

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

      4. unpow2 62.6%

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

      5. associate-/l* 58.3%

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

      6. associate-/r/ 59.8%

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

      7. associate-/l* 64.1%

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

      8. *-commutative 64.1%

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

      9. associate-/r/ 64.9%

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

      10. *-commutative 64.9%

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

      11. unpow2 64.9%

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

      12. associate-/l/ 60.3%

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

      13. associate-*r/ 64.8%

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

      14. associate-/l* 69.4%

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

   9. Simplified 79.4%

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

   if -6.20000000000000026e57 < w0 < 1.0600000000000001e-57

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

      2. times-frac 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in M around 0 79.5%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \leq -6.2 \cdot 10^{+57} \lor \neg \left(w0 \leq 1.06 \cdot 10^{-57}\right):\\ \;\;\;\;w0 \cdot \left(1 + \frac{D}{\frac{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}{\frac{D}{\ell}}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 9: 73.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{-67}:\\ \;\;\;\;w0\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-96}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= d -2.1e-67)
   w0
   (if (<= d 1.35e-96)
     (* w0 (+ 1.0 (* -0.125 (* (* (/ D d) (/ D d)) (* h (/ M (/ l M)))))))
     w0)))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= -2.1e-67) {
		tmp = w0;
	} else if (d <= 1.35e-96) {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * (h * (M / (l / 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 <= (-2.1d-67)) then
        tmp = w0
    else if (d_1 <= 1.35d-96) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / d_1) * (d / d_1)) * (h * (m / (l / 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 <= -2.1e-67) {
		tmp = w0;
	} else if (d <= 1.35e-96) {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * (h * (M / (l / M))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= -2.1e-67:
		tmp = w0
	elif d <= 1.35e-96:
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * (h * (M / (l / 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 <= -2.1e-67)
		tmp = w0;
	elseif (d <= 1.35e-96)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(h * Float64(M / Float64(l / 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 <= -2.1e-67)
		tmp = w0;
	elseif (d <= 1.35e-96)
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * (h * (M / (l / 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[LessEqual[d, -2.1e-67], w0, If[LessEqual[d, 1.35e-96], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if d < -2.1000000000000002e-67 or 1.35e-96 < d

   1. Initial program 83.8%

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

      1. *-commutative 83.8%

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

      2. times-frac 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in M around 0 85.0%

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

   if -2.1000000000000002e-67 < d < 1.35e-96

   1. Initial program 78.9%

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

      1. *-commutative 78.9%

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

      2. times-frac 76.2%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in M around 0 39.9%

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

      1. *-commutative 39.9%

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

      2. *-commutative 39.9%

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

      3. times-frac 46.7%

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

      4. *-commutative 46.7%

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

      5. unpow2 46.7%

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

      6. unpow2 46.7%

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

      7. *-commutative 46.7%

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

      8. unpow2 46.7%

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

   6. Simplified 46.7%

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

   7. Taylor expanded in M around 0 46.7%

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

      1. unpow2 46.7%

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

      2. associate-*l/ 46.7%

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

      3. *-commutative 46.7%

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

      4. associate-/l* 49.5%

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

   9. Simplified 49.5%

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

       1. times-frac 75.4%

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

   11. Applied egg-rr 75.4%

       \[\leadsto w0 \cdot \left(1 + \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right) \cdot -0.125\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{-67}:\\ \;\;\;\;w0\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-96}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 10: 68.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -2.55 \cdot 10^{+269}:\\ \;\;\;\;-0.125 \cdot \left(h \cdot \frac{w0}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{\frac{D}{\frac{\ell}{D}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= M -2.55e+269)
   (* -0.125 (* h (/ w0 (/ (* (/ d M) (/ d M)) (/ D (/ l D))))))
   w0))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -2.55e+269) {
		tmp = -0.125 * (h * (w0 / (((d / M) * (d / M)) / (D / (l / 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 (m <= (-2.55d+269)) then
        tmp = (-0.125d0) * (h * (w0 / (((d_1 / m) * (d_1 / m)) / (d / (l / d)))))
    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 (M <= -2.55e+269) {
		tmp = -0.125 * (h * (w0 / (((d / M) * (d / M)) / (D / (l / D)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= -2.55e+269:
		tmp = -0.125 * (h * (w0 / (((d / M) * (d / M)) / (D / (l / D)))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= -2.55e+269)
		tmp = Float64(-0.125 * Float64(h * Float64(w0 / Float64(Float64(Float64(d / M) * Float64(d / M)) / Float64(D / Float64(l / 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 (M <= -2.55e+269)
		tmp = -0.125 * (h * (w0 / (((d / M) * (d / M)) / (D / (l / 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[M, -2.55e+269], N[(-0.125 * N[(h * N[(w0 / N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if M < -2.55000000000000006e269

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. times-frac 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in M around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-commutative 100.0%

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

      6. fma-def 100.0%

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

      7. times-frac 100.0%

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

      8. unpow2 100.0%

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

      9. associate-/l* 100.0%

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

      10. associate-/l* 100.0%

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

      11. unpow2 100.0%

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

      12. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in D around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. unpow2 100.0%

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

      3. *-commutative 100.0%

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

      4. unpow2 100.0%

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

      5. associate-*r/ 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/r* 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in M around 0 100.0%

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

       1. times-frac 100.0%

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

       2. *-commutative 100.0%

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

       3. unpow2 100.0%

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

       4. associate-*r* 100.0%

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

       5. associate-/l* 100.0%

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

       6. unpow2 100.0%

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

       7. unpow2 100.0%

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

       8. times-frac 100.0%

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

       9. unpow2 100.0%

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

       10. associate-*l/ 100.0%

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

       11. *-commutative 100.0%

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

       12. *-commutative 100.0%

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

       13. associate-*l* 100.0%

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

       14. *-commutative 100.0%

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

       15. associate-*r/ 100.0%

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

       16. *-commutative 100.0%

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

       17. associate-/l* 100.0%

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

   12. Simplified 100.0%

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

       1. pow2 100.0%

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

   14. Applied egg-rr 100.0%

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

   if -2.55000000000000006e269 < M

   1. Initial program 82.3%

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

      1. *-commutative 82.3%

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

      2. times-frac 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in M around 0 75.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.55 \cdot 10^{+269}:\\ \;\;\;\;-0.125 \cdot \left(h \cdot \frac{w0}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{\frac{D}{\frac{\ell}{D}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 11: 67.9% 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 82.4%

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

   1. *-commutative 82.4%

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

   2. times-frac 81.6%

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

3. Simplified 81.6%

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

4. Taylor expanded in M around 0 75.2%

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

5. Final simplification 75.2%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023178 
(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))))))