Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.2% → 87.0%

Time: 16.1s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.2% 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: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-309}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{h \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{{\left(\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot \sqrt{h}\right)}^{2}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 -2e-309)
   (* w0 (sqrt (+ 1.0 (/ -1.0 (/ l (* h (pow (* (* D M) (/ 0.5 d)) 2.0)))))))
   (*
    w0
    (sqrt
     (+ 1.0 (/ -1.0 (/ l (pow (* (* D (* M (/ 0.5 d))) (sqrt h)) 2.0))))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (h <= -2e-309) {
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (h * pow(((D * M) * (0.5 / d)), 2.0))))));
	} else {
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / pow(((D * (M * (0.5 / d))) * sqrt(h)), 2.0)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-2d-309)) then
        tmp = w0 * sqrt((1.0d0 + ((-1.0d0) / (l / (h * (((d * m) * (0.5d0 / d_1)) ** 2.0d0))))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-1.0d0) / (l / (((d * (m * (0.5d0 / d_1))) * sqrt(h)) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (h <= -2e-309) {
		tmp = w0 * Math.sqrt((1.0 + (-1.0 / (l / (h * Math.pow(((D * M) * (0.5 / d)), 2.0))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-1.0 / (l / Math.pow(((D * (M * (0.5 / d))) * Math.sqrt(h)), 2.0)))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if h <= -2e-309:
		tmp = w0 * math.sqrt((1.0 + (-1.0 / (l / (h * math.pow(((D * M) * (0.5 / d)), 2.0))))))
	else:
		tmp = w0 * math.sqrt((1.0 + (-1.0 / (l / math.pow(((D * (M * (0.5 / d))) * math.sqrt(h)), 2.0)))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (h <= -2e-309)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-1.0 / Float64(l / Float64(h * (Float64(Float64(D * M) * Float64(0.5 / d)) ^ 2.0)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-1.0 / Float64(l / (Float64(Float64(D * Float64(M * Float64(0.5 / d))) * sqrt(h)) ^ 2.0))))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (h <= -2e-309)
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (h * (((D * M) * (0.5 / d)) ^ 2.0))))));
	else
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (((D * (M * (0.5 / d))) * sqrt(h)) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[h, -2e-309], N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[(h * N[Power[N[(N[(D * M), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[Power[N[(N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -1.9999999999999988e-309

   1. Initial program 86.3%

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

   3. Simplified 87.1%

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

      1. associate-*r/ 93.3%

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

      2. frac-times 92.5%

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

      3. *-commutative 92.5%

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

      4. clear-num 92.5%

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

      5. *-commutative 92.5%

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

      6. *-commutative 92.5%

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

      7. frac-times 93.3%

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

      8. frac-times 92.5%

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

      9. *-commutative 92.5%

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

      10. div-inv 92.5%

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

      11. *-commutative 92.5%

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

      12. associate-/r* 92.5%

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

      13. metadata-eval 92.5%

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

   5. Applied egg-rr 92.5%

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

   if -1.9999999999999988e-309 < h

   1. Initial program 79.3%

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

   3. Simplified 78.5%

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

      1. associate-*r/ 81.4%

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

      2. frac-times 82.0%

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

      3. *-commutative 82.0%

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

      4. clear-num 82.0%

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

      5. *-commutative 82.0%

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

      6. *-commutative 82.0%

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

      7. frac-times 81.4%

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

      8. frac-times 82.0%

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

      9. *-commutative 82.0%

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

      10. div-inv 82.0%

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

      11. *-commutative 82.0%

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

      12. associate-/r* 82.0%

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

      13. metadata-eval 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. add-sqr-sqrt 82.0%

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

      2. pow2 82.0%

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

      3. *-commutative 82.0%

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

      4. sqrt-prod 82.0%

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

      5. unpow2 82.0%

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

      6. sqrt-prod 53.1%

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

      7. add-sqr-sqrt 86.5%

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

      8. associate-*l* 85.4%

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

   7. Applied egg-rr 85.4%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{\color{blue}{{\left(\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot \sqrt{h}\right)}^{2}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-309}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{h \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{{\left(\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot \sqrt{h}\right)}^{2}}}}\\ \end{array} \]

Alternative 2: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-308}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{h \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \left(\sqrt{h} \cdot \frac{D \cdot M}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 -1e-308)
   (* w0 (sqrt (+ 1.0 (/ -1.0 (/ l (* h (pow (* (* D M) (/ 0.5 d)) 2.0)))))))
   (* w0 (sqrt (- 1.0 (/ (pow (* 0.5 (* (sqrt h) (/ (* D M) d))) 2.0) l))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (h <= -1e-308) {
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (h * pow(((D * M) * (0.5 / d)), 2.0))))));
	} else {
		tmp = w0 * sqrt((1.0 - (pow((0.5 * (sqrt(h) * ((D * M) / d))), 2.0) / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-1d-308)) then
        tmp = w0 * sqrt((1.0d0 + ((-1.0d0) / (l / (h * (((d * m) * (0.5d0 / d_1)) ** 2.0d0))))))
    else
        tmp = w0 * sqrt((1.0d0 - (((0.5d0 * (sqrt(h) * ((d * m) / d_1))) ** 2.0d0) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (h <= -1e-308) {
		tmp = w0 * Math.sqrt((1.0 + (-1.0 / (l / (h * Math.pow(((D * M) * (0.5 / d)), 2.0))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow((0.5 * (Math.sqrt(h) * ((D * M) / d))), 2.0) / l)));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if h <= -1e-308:
		tmp = w0 * math.sqrt((1.0 + (-1.0 / (l / (h * math.pow(((D * M) * (0.5 / d)), 2.0))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (math.pow((0.5 * (math.sqrt(h) * ((D * M) / d))), 2.0) / l)))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (h <= -1e-308)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-1.0 / Float64(l / Float64(h * (Float64(Float64(D * M) * Float64(0.5 / d)) ^ 2.0)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(0.5 * Float64(sqrt(h) * Float64(Float64(D * M) / d))) ^ 2.0) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (h <= -1e-308)
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (h * (((D * M) * (0.5 / d)) ^ 2.0))))));
	else
		tmp = w0 * sqrt((1.0 - (((0.5 * (sqrt(h) * ((D * M) / d))) ^ 2.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[h, -1e-308], N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[(h * N[Power[N[(N[(D * M), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(0.5 * N[(N[Sqrt[h], $MachinePrecision] * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -9.9999999999999991e-309

   1. Initial program 86.3%

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

   3. Simplified 87.1%

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

      1. associate-*r/ 93.3%

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

      2. frac-times 92.5%

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

      3. *-commutative 92.5%

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

      4. clear-num 92.5%

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

      5. *-commutative 92.5%

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

      6. *-commutative 92.5%

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

      7. frac-times 93.3%

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

      8. frac-times 92.5%

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

      9. *-commutative 92.5%

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

      10. div-inv 92.5%

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

      11. *-commutative 92.5%

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

      12. associate-/r* 92.5%

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

      13. metadata-eval 92.5%

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

   5. Applied egg-rr 92.5%

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

   if -9.9999999999999991e-309 < h

   1. Initial program 79.3%

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

   3. Simplified 78.5%

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

      1. *-commutative 78.5%

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

      2. frac-times 79.3%

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

      3. *-commutative 79.3%

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

      4. associate-*l/ 82.0%

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

      5. *-commutative 82.0%

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

      6. frac-times 81.4%

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

      7. frac-times 82.0%

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

      8. *-commutative 82.0%

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

      9. div-inv 82.0%

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

      10. *-commutative 82.0%

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

      11. associate-/r* 82.0%

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

      12. metadata-eval 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. add-sqr-sqrt 82.0%

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

      2. pow2 82.0%

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

      3. *-commutative 82.0%

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

      4. sqrt-prod 82.0%

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

      5. unpow2 82.0%

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

      6. sqrt-prod 53.1%

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

      7. add-sqr-sqrt 86.5%

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

      8. associate-*l* 85.4%

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

   7. Applied egg-rr 85.4%

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

      1. associate-*l* 85.4%

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

   9. Simplified 85.4%

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

   10. Taylor expanded in D around 0 86.5%

       \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(0.5 \cdot \left(\frac{D \cdot M}{d} \cdot \sqrt{h}\right)\right)}}^{2}}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-308}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{h \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \left(\sqrt{h} \cdot \frac{D \cdot M}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 3: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-308}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{h \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(D \cdot \left(\sqrt{h} \cdot \frac{M}{\frac{d}{0.5}}\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 -1e-308)
   (* w0 (sqrt (+ 1.0 (/ -1.0 (/ l (* h (pow (* (* D M) (/ 0.5 d)) 2.0)))))))
   (* w0 (sqrt (- 1.0 (/ (pow (* D (* (sqrt h) (/ M (/ d 0.5)))) 2.0) l))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (h <= -1e-308) {
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (h * pow(((D * M) * (0.5 / d)), 2.0))))));
	} else {
		tmp = w0 * sqrt((1.0 - (pow((D * (sqrt(h) * (M / (d / 0.5)))), 2.0) / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-1d-308)) then
        tmp = w0 * sqrt((1.0d0 + ((-1.0d0) / (l / (h * (((d * m) * (0.5d0 / d_1)) ** 2.0d0))))))
    else
        tmp = w0 * sqrt((1.0d0 - (((d * (sqrt(h) * (m / (d_1 / 0.5d0)))) ** 2.0d0) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (h <= -1e-308) {
		tmp = w0 * Math.sqrt((1.0 + (-1.0 / (l / (h * Math.pow(((D * M) * (0.5 / d)), 2.0))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow((D * (Math.sqrt(h) * (M / (d / 0.5)))), 2.0) / l)));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if h <= -1e-308:
		tmp = w0 * math.sqrt((1.0 + (-1.0 / (l / (h * math.pow(((D * M) * (0.5 / d)), 2.0))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (math.pow((D * (math.sqrt(h) * (M / (d / 0.5)))), 2.0) / l)))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (h <= -1e-308)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-1.0 / Float64(l / Float64(h * (Float64(Float64(D * M) * Float64(0.5 / d)) ^ 2.0)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(D * Float64(sqrt(h) * Float64(M / Float64(d / 0.5)))) ^ 2.0) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (h <= -1e-308)
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (h * (((D * M) * (0.5 / d)) ^ 2.0))))));
	else
		tmp = w0 * sqrt((1.0 - (((D * (sqrt(h) * (M / (d / 0.5)))) ^ 2.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[h, -1e-308], N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[(h * N[Power[N[(N[(D * M), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(D * N[(N[Sqrt[h], $MachinePrecision] * N[(M / N[(d / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -9.9999999999999991e-309

   1. Initial program 86.3%

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

   3. Simplified 87.1%

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

      1. associate-*r/ 93.3%

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

      2. frac-times 92.5%

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

      3. *-commutative 92.5%

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

      4. clear-num 92.5%

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

      5. *-commutative 92.5%

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

      6. *-commutative 92.5%

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

      7. frac-times 93.3%

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

      8. frac-times 92.5%

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

      9. *-commutative 92.5%

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

      10. div-inv 92.5%

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

      11. *-commutative 92.5%

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

      12. associate-/r* 92.5%

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

      13. metadata-eval 92.5%

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

   5. Applied egg-rr 92.5%

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

   if -9.9999999999999991e-309 < h

   1. Initial program 79.3%

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

   3. Simplified 78.5%

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

      1. *-commutative 78.5%

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

      2. frac-times 79.3%

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

      3. *-commutative 79.3%

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

      4. associate-*l/ 82.0%

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

      5. *-commutative 82.0%

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

      6. frac-times 81.4%

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

      7. frac-times 82.0%

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

      8. *-commutative 82.0%

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

      9. div-inv 82.0%

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

      10. *-commutative 82.0%

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

      11. associate-/r* 82.0%

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

      12. metadata-eval 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. add-sqr-sqrt 82.0%

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

      2. pow2 82.0%

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

      3. *-commutative 82.0%

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

      4. sqrt-prod 82.0%

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

      5. unpow2 82.0%

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

      6. sqrt-prod 53.1%

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

      7. add-sqr-sqrt 86.5%

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

      8. associate-*l* 85.4%

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

   7. Applied egg-rr 85.4%

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

      1. associate-*l* 85.4%

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

   9. Simplified 85.4%

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

       1. expm1-log1p-u 67.6%

          \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{h}\right)\right)\right)\right)}}^{2}}{\ell}} \]

       2. expm1-udef 64.5%

          \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(e^{\mathsf{log1p}\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{h}\right)\right)} - 1\right)}}^{2}}{\ell}} \]

   11. Applied egg-rr 64.5%

       \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(e^{\mathsf{log1p}\left(D \cdot \left(\frac{M \cdot 0.5}{d} \cdot \sqrt{h}\right)\right)} - 1\right)}}^{2}}{\ell}} \]
   12. Step-by-step derivation

       1. expm1-def 67.6%

          \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(D \cdot \left(\frac{M \cdot 0.5}{d} \cdot \sqrt{h}\right)\right)\right)\right)}}^{2}}{\ell}} \]

       2. expm1-log1p 85.4%

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

       3. *-commutative 85.4%

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

       4. associate-/l* 85.4%

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

   13. Simplified 85.4%

       \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(D \cdot \left(\sqrt{h} \cdot \frac{M}{\frac{d}{0.5}}\right)\right)}}^{2}}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-308}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{h \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(D \cdot \left(\sqrt{h} \cdot \frac{M}{\frac{d}{0.5}}\right)\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 4: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-308}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{h \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{h}\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 -1e-308)
   (* w0 (sqrt (+ 1.0 (/ -1.0 (/ l (* h (pow (* (* D M) (/ 0.5 d)) 2.0)))))))
   (* w0 (sqrt (- 1.0 (/ (pow (* D (* (* M (/ 0.5 d)) (sqrt h))) 2.0) l))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (h <= -1e-308) {
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (h * pow(((D * M) * (0.5 / d)), 2.0))))));
	} else {
		tmp = w0 * sqrt((1.0 - (pow((D * ((M * (0.5 / d)) * sqrt(h))), 2.0) / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-1d-308)) then
        tmp = w0 * sqrt((1.0d0 + ((-1.0d0) / (l / (h * (((d * m) * (0.5d0 / d_1)) ** 2.0d0))))))
    else
        tmp = w0 * sqrt((1.0d0 - (((d * ((m * (0.5d0 / d_1)) * sqrt(h))) ** 2.0d0) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (h <= -1e-308) {
		tmp = w0 * Math.sqrt((1.0 + (-1.0 / (l / (h * Math.pow(((D * M) * (0.5 / d)), 2.0))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow((D * ((M * (0.5 / d)) * Math.sqrt(h))), 2.0) / l)));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if h <= -1e-308:
		tmp = w0 * math.sqrt((1.0 + (-1.0 / (l / (h * math.pow(((D * M) * (0.5 / d)), 2.0))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (math.pow((D * ((M * (0.5 / d)) * math.sqrt(h))), 2.0) / l)))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (h <= -1e-308)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-1.0 / Float64(l / Float64(h * (Float64(Float64(D * M) * Float64(0.5 / d)) ^ 2.0)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(D * Float64(Float64(M * Float64(0.5 / d)) * sqrt(h))) ^ 2.0) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (h <= -1e-308)
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (h * (((D * M) * (0.5 / d)) ^ 2.0))))));
	else
		tmp = w0 * sqrt((1.0 - (((D * ((M * (0.5 / d)) * sqrt(h))) ^ 2.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[h, -1e-308], N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[(h * N[Power[N[(N[(D * M), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(D * N[(N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -9.9999999999999991e-309

   1. Initial program 86.3%

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

   3. Simplified 87.1%

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

      1. associate-*r/ 93.3%

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

      2. frac-times 92.5%

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

      3. *-commutative 92.5%

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

      4. clear-num 92.5%

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

      5. *-commutative 92.5%

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

      6. *-commutative 92.5%

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

      7. frac-times 93.3%

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

      8. frac-times 92.5%

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

      9. *-commutative 92.5%

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

      10. div-inv 92.5%

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

      11. *-commutative 92.5%

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

      12. associate-/r* 92.5%

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

      13. metadata-eval 92.5%

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

   5. Applied egg-rr 92.5%

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

   if -9.9999999999999991e-309 < h

   1. Initial program 79.3%

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

   3. Simplified 78.5%

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

      1. *-commutative 78.5%

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

      2. frac-times 79.3%

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

      3. *-commutative 79.3%

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

      4. associate-*l/ 82.0%

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

      5. *-commutative 82.0%

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

      6. frac-times 81.4%

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

      7. frac-times 82.0%

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

      8. *-commutative 82.0%

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

      9. div-inv 82.0%

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

      10. *-commutative 82.0%

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

      11. associate-/r* 82.0%

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

      12. metadata-eval 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. add-sqr-sqrt 82.0%

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

      2. pow2 82.0%

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

      3. *-commutative 82.0%

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

      4. sqrt-prod 82.0%

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

      5. unpow2 82.0%

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

      6. sqrt-prod 53.1%

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

      7. add-sqr-sqrt 86.5%

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

      8. associate-*l* 85.4%

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

   7. Applied egg-rr 85.4%

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

      1. associate-*l* 85.4%

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

   9. Simplified 85.4%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{{\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{h}\right)\right)}^{2}}}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-308}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{h \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{h}\right)\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 5: 86.2% accurate, 1.0× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
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 (/ 2.0 D))) 2.0) l))))))

C

M = abs(M);
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 * (2.0 / D))), 2.0) / l))));
}

Fortran

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

Java

M = Math.abs(M);
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 * (2.0 / D))), 2.0) / l))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: M should be positive before calling this function
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[(h * N[(N[Power[N[(M / N[(d * N[(2.0 / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.8%

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

3. Simplified 82.9%

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

   1. frac-times 82.8%

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

   2. *-commutative 82.8%

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

   3. clear-num 82.8%

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

   4. un-div-inv 84.0%

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

   5. *-commutative 84.0%

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

   6. frac-times 84.0%

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

   7. frac-times 84.0%

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

   8. *-commutative 84.0%

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

   9. div-inv 84.0%

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

   10. *-commutative 84.0%

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

   11. associate-/r* 84.0%

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

   12. metadata-eval 84.0%

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

5. Applied egg-rr 84.0%

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

   1. associate-/r/ 87.6%

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

   2. *-commutative 87.6%

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

   3. *-commutative 87.6%

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

   4. associate-*r* 87.8%

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

   5. associate-*r/ 87.8%

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

   6. associate-*r/ 87.7%

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

   7. /-rgt-identity 87.7%

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

   8. associate-/l* 87.7%

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

   9. metadata-eval 87.7%

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

   10. associate-*r/ 87.7%

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

   11. associate-/l* 87.6%

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

   12. associate-/l/ 87.5%

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

7. Simplified 87.5%

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

8. Final simplification 87.5%

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

Alternative 6: 72.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 520000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\ell} \cdot \left(\frac{h}{d} \cdot \frac{w0}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 520000.0)
   w0
   (+ w0 (* -0.125 (* (/ (* (* D M) (* D M)) l) (* (/ h d) (/ w0 d)))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 520000.0) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((((D * M) * (D * M)) / l) * ((h / d) * (w0 / d))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 <= 520000.0d0) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((((d * m) * (d * m)) / l) * ((h / d_1) * (w0 / d_1))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 520000.0) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((((D * M) * (D * M)) / l) * ((h / d) * (w0 / d))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 520000.0:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((((D * M) * (D * M)) / l) * ((h / d) * (w0 / d))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 520000.0)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(Float64(D * M) * Float64(D * M)) / l) * Float64(Float64(h / d) * Float64(w0 / d)))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 520000.0)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((((D * M) * (D * M)) / l) * ((h / d) * (w0 / d))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 520000.0], w0, N[(w0 + N[(-0.125 * N[(N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(h / d), $MachinePrecision] * N[(w0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 5.2e5

   1. Initial program 86.5%

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

   3. Simplified 85.6%

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

   4. Taylor expanded in D around 0 75.8%

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

   if 5.2e5 < D

   1. Initial program 70.5%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in D around 0 42.5%

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

      1. expm1-log1p-u 25.3%

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

      2. expm1-udef 24.9%

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

      3. associate-*r* 21.6%

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

      4. pow-prod-down 28.8%

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

      5. *-commutative 28.8%

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

      6. *-commutative 28.8%

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

   6. Applied egg-rr 28.8%

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

      1. expm1-def 29.1%

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

      2. expm1-log1p 48.2%

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

      3. times-frac 49.5%

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

      4. *-commutative 49.5%

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

   8. Simplified 49.5%

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

      1. *-commutative 49.5%

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

      2. unpow2 49.5%

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

      3. times-frac 59.0%

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

   10. Applied egg-rr 59.0%

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

       1. unpow2 59.0%

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

   12. Applied egg-rr 59.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 520000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\ell} \cdot \left(\frac{h}{d} \cdot \frac{w0}{d}\right)\right)\\ \end{array} \]

Alternative 7: 76.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 4.2 \cdot 10^{-51}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(w0 \cdot \frac{h}{d}\right)}{d \cdot \frac{\ell}{D \cdot M}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 4.2e-51)
   w0
   (+ w0 (* -0.125 (/ (* (* D M) (* w0 (/ h d))) (* d (/ l (* D M))))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 4.2e-51) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((D * M) * (w0 * (h / d))) / (d * (l / (D * M)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 4.2d-51) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((d * m) * (w0 * (h / d_1))) / (d_1 * (l / (d * m)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 4.2e-51) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((D * M) * (w0 * (h / d))) / (d * (l / (D * M)))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 4.2e-51:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (((D * M) * (w0 * (h / d))) / (d * (l / (D * M)))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 4.2e-51)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(D * M) * Float64(w0 * Float64(h / d))) / Float64(d * Float64(l / Float64(D * M))))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 4.2e-51)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((D * M) * (w0 * (h / d))) / (d * (l / (D * M)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 4.2e-51], w0, N[(w0 + N[(-0.125 * N[(N[(N[(D * M), $MachinePrecision] * N[(w0 * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l / N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 4.20000000000000003e-51

   1. Initial program 87.0%

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

   3. Simplified 86.0%

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

   4. Taylor expanded in D around 0 75.3%

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

   if 4.20000000000000003e-51 < D

   1. Initial program 71.1%

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

   3. Simplified 74.1%

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

   4. Taylor expanded in D around 0 45.0%

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

      1. expm1-log1p-u 29.9%

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

      2. expm1-udef 29.5%

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

      3. associate-*r* 26.6%

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

      4. pow-prod-down 32.9%

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

      5. *-commutative 32.9%

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

      6. *-commutative 32.9%

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

   6. Applied egg-rr 32.9%

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

      1. expm1-def 33.2%

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

      2. expm1-log1p 50.1%

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

      3. times-frac 51.2%

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

      4. *-commutative 51.2%

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

   8. Simplified 51.2%

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

      1. *-commutative 51.2%

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

      2. unpow2 51.2%

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

      3. times-frac 61.0%

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

   10. Applied egg-rr 61.0%

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

       1. *-commutative 61.0%

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

       2. associate-*l/ 61.0%

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

       3. pow2 61.0%

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

       4. associate-/l* 62.5%

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

       5. frac-times 65.7%

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

   12. Applied egg-rr 65.7%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 4.2 \cdot 10^{-51}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(w0 \cdot \frac{h}{d}\right)}{d \cdot \frac{\ell}{D \cdot M}}\\ \end{array} \]

Alternative 8: 67.9% accurate, 216.0× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
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

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

Fortran

NOTE: M should be positive before calling this function
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

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: M should be positive before calling this function
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 82.8%

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

3. Simplified 82.9%

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

4. Taylor expanded in D around 0 65.5%

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

5. Final simplification 65.5%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023306 
(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))))))