Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.3% → 88.4%

Time: 10.4s

Alternatives: 13

Speedup: 2.1×

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: 80.3% 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: 88.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{2} \cdot \frac{M\_m}{d}\\ \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq \infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D\_m}{\left(\frac{\ell}{\left(0.25 \cdot \left(M\_m \cdot M\_m\right)\right) \cdot D\_m} \cdot d\right) \cdot \frac{d}{h}}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* (/ D_m 2.0) (/ M_m d))))
   (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) INFINITY)
     (* w0 (sqrt (- 1.0 (* (* (/ h l) t_0) t_0))))
     (*
      w0
      (sqrt
       (-
        1.0
        (/ D_m (* (* (/ l (* (* 0.25 (* M_m M_m)) D_m)) d) (/ d h)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (D_m / 2.0) * (M_m / d);
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= ((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - (((h / l) * t_0) * t_0)));
	} else {
		tmp = w0 * sqrt((1.0 - (D_m / (((l / ((0.25 * (M_m * M_m)) * D_m)) * d) * (d / h)))));
	}
	return tmp;
}

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (D_m / 2.0) * (M_m / d);
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - (((h / l) * t_0) * t_0)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (D_m / (((l / ((0.25 * (M_m * M_m)) * D_m)) * d) * (d / h)))));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = (D_m / 2.0) * (M_m / d);
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= Inf)
		tmp = w0 * sqrt((1.0 - (((h / l) * t_0) * t_0)));
	else
		tmp = w0 * sqrt((1.0 - (D_m / (((l / ((0.25 * (M_m * M_m)) * D_m)) * d) * (d / h)))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], Infinity], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(D$95$m / N[(N[(N[(l / N[(N[(0.25 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < +inf.0

   1. Initial program 88.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*r* N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-1 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}\right) \cdot \frac{1}{\ell \cdot \color{blue}{\frac{1}{h}}}} \]

      8. div-inv N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}\right) \cdot \frac{1}{\color{blue}{\frac{\ell}{h}}}} \]

      9. clear-num N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}\right) \cdot \color{blue}{\frac{h}{\ell}}} \]

   6. Applied rewrites 89.9%

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

   if +inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*r* N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. frac-times N/A

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

      8. metadata-eval N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left|-1\right|} \cdot D}{\frac{\ell}{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)} \cdot \left(d \cdot {h}^{-1}\right)}} \]

      9. unpow1 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left|-1\right| \cdot \color{blue}{{D}^{1}}}{\frac{\ell}{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)} \cdot \left(d \cdot {h}^{-1}\right)}} \]

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left|-1\right| \cdot \sqrt{\color{blue}{D \cdot D}}}{\frac{\ell}{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)} \cdot \left(d \cdot {h}^{-1}\right)}} \]

      13. rem-sqrt-square-rev N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left|-1\right| \cdot \color{blue}{\left|D\right|}}{\frac{\ell}{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)} \cdot \left(d \cdot {h}^{-1}\right)}} \]

      14. fabs-mul N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left|-1 \cdot D\right|}}{\frac{\ell}{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)} \cdot \left(d \cdot {h}^{-1}\right)}} \]

      15. neg-mul-1 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left|\color{blue}{\mathsf{neg}\left(D\right)}\right|}{\frac{\ell}{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)} \cdot \left(d \cdot {h}^{-1}\right)}} \]

      16. neg-fabs N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left|D\right|}}{\frac{\ell}{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)} \cdot \left(d \cdot {h}^{-1}\right)}} \]

      17. rem-sqrt-square-rev N/A

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

      18. pow2 N/A

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

      19. sqrt-pow1 N/A

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

      20. metadata-eval N/A

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

      21. unpow1 N/A

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

      22. lower-/.f64 N/A

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

   6. Applied rewrites 65.3%

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

Alternative 2: 79.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot \frac{M\_m \cdot \left(\left(h \cdot M\_m\right) \cdot w0\right)}{\ell}, -0.5, w0\right)\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+56}:\\ \;\;\;\;w0 \cdot \sqrt{-0.25 \cdot \left(\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot \frac{D\_m}{\left(d \cdot d\right) \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 (- INFINITY))
     (fma (* (* D_m D_m) (/ (* M_m (* (* h M_m) w0)) l)) -0.5 w0)
     (if (<= t_0 -1e+56)
       (*
        w0
        (sqrt (* -0.25 (* (* (* (* M_m M_m) h) D_m) (/ D_m (* (* d d) l))))))
       (* w0 1.0)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((D_m * D_m) * ((M_m * ((h * M_m) * w0)) / l)), -0.5, w0);
	} else if (t_0 <= -1e+56) {
		tmp = w0 * sqrt((-0.25 * ((((M_m * M_m) * h) * D_m) * (D_m / ((d * d) * l)))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(Float64(D_m * D_m) * Float64(Float64(M_m * Float64(Float64(h * M_m) * w0)) / l)), -0.5, w0);
	elseif (t_0 <= -1e+56)
		tmp = Float64(w0 * sqrt(Float64(-0.25 * Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * Float64(D_m / Float64(Float64(d * d) * l))))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * N[(N[(h * M$95$m), $MachinePrecision] * w0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.5 + w0), $MachinePrecision], If[LessEqual[t$95$0, -1e+56], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -inf.0

   1. Initial program 58.1%

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

   4. Applied rewrites 47.6%

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

   5. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 53.2

          \[\leadsto \mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot w0}{\ell}, -0.5, w0\right) \]

   7. Applied rewrites 53.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot w0}{\ell}, -0.5, w0\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.3%

      \[\leadsto \mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{\ell}, -0.5, w0\right) \]

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.00000000000000009e56

   1. Initial program 99.3%

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

   3. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{-1}{4} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

      12. lower-*.f64 30.2

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

   5. Applied rewrites 30.2%

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

   7. Applied rewrites 30.6%

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

   if -1.00000000000000009e56 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.6%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 95.6%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -100:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\frac{D\_m}{d}}{\ell} \cdot \left(\left(h \cdot \left(\left(\frac{D\_m}{d} \cdot M\_m\right) \cdot M\_m\right)\right) \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -100.0)
   (* w0 (sqrt (* (/ (/ D_m d) l) (* (* h (* (* (/ D_m d) M_m) M_m)) -0.25))))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -100.0) {
		tmp = w0 * sqrt((((D_m / d) / l) * ((h * (((D_m / d) * M_m) * M_m)) * -0.25)));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-100.0d0)) then
        tmp = w0 * sqrt((((d_m / d) / l) * ((h * (((d_m / d) * m_m) * m_m)) * (-0.25d0))))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -100.0) {
		tmp = w0 * Math.sqrt((((D_m / d) / l) * ((h * (((D_m / d) * M_m) * M_m)) * -0.25)));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -100.0:
		tmp = w0 * math.sqrt((((D_m / d) / l) * ((h * (((D_m / d) * M_m) * M_m)) * -0.25)))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -100.0)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(D_m / d) / l) * Float64(Float64(h * Float64(Float64(Float64(D_m / d) * M_m) * M_m)) * -0.25))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -100.0)
		tmp = w0 * sqrt((((D_m / d) / l) * ((h * (((D_m / d) * M_m) * M_m)) * -0.25)));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -100.0], N[(w0 * N[Sqrt[N[(N[(N[(D$95$m / d), $MachinePrecision] / l), $MachinePrecision] * N[(N[(h * N[(N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -100

   1. Initial program 66.9%

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

   3. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{-1}{4} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

      12. lower-*.f64 43.9

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

   5. Applied rewrites 43.9%

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

   7. Applied rewrites 43.9%

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

   9. Applied rewrites 63.9%

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

   if -100 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.5%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.1%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(-0.25 \cdot \left(\left(M\_m \cdot \frac{D\_m}{d}\right) \cdot M\_m\right)\right) \cdot D\_m}{\ell \cdot d}, h, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -1e-12)
   (*
    w0
    (sqrt (fma (/ (* (* -0.25 (* (* M_m (/ D_m d)) M_m)) D_m) (* l d)) h 1.0)))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e-12) {
		tmp = w0 * sqrt(fma((((-0.25 * ((M_m * (D_m / d)) * M_m)) * D_m) / (l * d)), h, 1.0));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e-12)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(-0.25 * Float64(Float64(M_m * Float64(D_m / d)) * M_m)) * D_m) / Float64(l * d)), h, 1.0)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e-12], N[(w0 * N[Sqrt[N[(N[(N[(N[(-0.25 * N[(N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.9999999999999998e-13

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*r* N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 56.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell} \cdot \frac{\frac{D}{d}}{{h}^{-1}}\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell} \cdot \frac{\frac{D}{d}}{{h}^{-1}}\right)\right) + 1}} \]

      4. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\left(\mathsf{neg}\left(\color{blue}{\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell} \cdot \frac{\frac{D}{d}}{{h}^{-1}}}\right)\right) + 1} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell}\right)\right) \cdot \frac{\frac{D}{d}}{{h}^{-1}}} + 1} \]

      6. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell}\right)\right) \cdot \color{blue}{\frac{\frac{D}{d}}{{h}^{-1}}} + 1} \]

      7. div-inv N/A

         \[\leadsto w0 \cdot \sqrt{\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell}\right)\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{{h}^{-1}}\right)} + 1} \]

      8. lift-pow.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell}\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{1}{\color{blue}{{h}^{-1}}}\right) + 1} \]

      9. unpow-1 N/A

         \[\leadsto w0 \cdot \sqrt{\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell}\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{1}{\color{blue}{\frac{1}{h}}}\right) + 1} \]

      10. remove-double-div N/A

          \[\leadsto w0 \cdot \sqrt{\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell}\right)\right) \cdot \left(\frac{D}{d} \cdot \color{blue}{h}\right) + 1} \]

      11. associate-*r* N/A

          \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell}\right)\right) \cdot \frac{D}{d}\right) \cdot h} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell}\right)\right) \cdot \frac{D}{d}, h, 1\right)}} \]

   6. Applied rewrites 64.0%

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

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}}{\ell} \cdot \frac{D}{d}}, h, 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}}{\ell}} \cdot \frac{D}{d}, h, 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}}{\ell} \cdot \color{blue}{\frac{D}{d}}, h, 1\right)} \]

      4. frac-times N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right) \cdot D}{\ell \cdot d}}, h, 1\right)} \]

      5. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right) \cdot D}{\color{blue}{d \cdot \ell}}, h, 1\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right) \cdot D}{d \cdot \ell}}, h, 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right) \cdot D}}{d \cdot \ell}, h, 1\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right)} \cdot D}{d \cdot \ell}, h, 1\right)} \]

      9. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left(\frac{-1}{4} \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)\right)} \cdot D}{d \cdot \ell}, h, 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left(\frac{-1}{4} \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)\right)} \cdot D}{d \cdot \ell}, h, 1\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(\frac{-1}{4} \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot M\right)\right) \cdot D}{d \cdot \ell}, h, 1\right)} \]

      12. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(\frac{-1}{4} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot M\right)\right) \cdot D}{d \cdot \ell}, h, 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(\frac{-1}{4} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot M\right)\right) \cdot D}{d \cdot \ell}, h, 1\right)} \]

      14. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(\frac{-1}{4} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot M\right)\right) \cdot D}{\color{blue}{\ell \cdot d}}, h, 1\right)} \]

      15. lower-*.f64 63.8

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

   8. Applied rewrites 63.8%

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

   if -9.9999999999999998e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.5%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.2%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -100:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\left(\left(h \cdot \left(\left(\frac{D\_m}{d} \cdot M\_m\right) \cdot M\_m\right)\right) \cdot -0.25\right) \cdot D\_m}{\ell \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -100.0)
   (* w0 (sqrt (/ (* (* (* h (* (* (/ D_m d) M_m) M_m)) -0.25) D_m) (* l d))))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -100.0) {
		tmp = w0 * sqrt(((((h * (((D_m / d) * M_m) * M_m)) * -0.25) * D_m) / (l * d)));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-100.0d0)) then
        tmp = w0 * sqrt(((((h * (((d_m / d) * m_m) * m_m)) * (-0.25d0)) * d_m) / (l * d)))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -100.0) {
		tmp = w0 * Math.sqrt(((((h * (((D_m / d) * M_m) * M_m)) * -0.25) * D_m) / (l * d)));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -100.0:
		tmp = w0 * math.sqrt(((((h * (((D_m / d) * M_m) * M_m)) * -0.25) * D_m) / (l * d)))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -100.0)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(Float64(h * Float64(Float64(Float64(D_m / d) * M_m) * M_m)) * -0.25) * D_m) / Float64(l * d))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -100.0)
		tmp = w0 * sqrt(((((h * (((D_m / d) * M_m) * M_m)) * -0.25) * D_m) / (l * d)));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -100.0], N[(w0 * N[Sqrt[N[(N[(N[(N[(h * N[(N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -100

   1. Initial program 66.9%

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

   3. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{-1}{4} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

      12. lower-*.f64 43.9

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

   5. Applied rewrites 43.9%

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

   7. Applied rewrites 43.9%

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

   9. Applied rewrites 61.3%

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

   if -100 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.5%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.1%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 83.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -100:\\ \;\;\;\;w0 \cdot \sqrt{-0.25 \cdot \left(\frac{M\_m \cdot \left(M\_m \cdot \left(h \cdot D\_m\right)\right)}{d} \cdot \frac{D\_m}{\ell \cdot d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -100.0)
   (* w0 (sqrt (* -0.25 (* (/ (* M_m (* M_m (* h D_m))) d) (/ D_m (* l d))))))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -100.0) {
		tmp = w0 * sqrt((-0.25 * (((M_m * (M_m * (h * D_m))) / d) * (D_m / (l * d)))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-100.0d0)) then
        tmp = w0 * sqrt(((-0.25d0) * (((m_m * (m_m * (h * d_m))) / d) * (d_m / (l * d)))))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -100.0) {
		tmp = w0 * Math.sqrt((-0.25 * (((M_m * (M_m * (h * D_m))) / d) * (D_m / (l * d)))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -100.0:
		tmp = w0 * math.sqrt((-0.25 * (((M_m * (M_m * (h * D_m))) / d) * (D_m / (l * d)))))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -100.0)
		tmp = Float64(w0 * sqrt(Float64(-0.25 * Float64(Float64(Float64(M_m * Float64(M_m * Float64(h * D_m))) / d) * Float64(D_m / Float64(l * d))))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -100.0)
		tmp = w0 * sqrt((-0.25 * (((M_m * (M_m * (h * D_m))) / d) * (D_m / (l * d)))));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -100.0], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(N[(M$95$m * N[(M$95$m * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(D$95$m / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -100

   1. Initial program 66.9%

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

   3. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{-1}{4} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

      12. lower-*.f64 43.9

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

   5. Applied rewrites 43.9%

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

   7. Applied rewrites 50.4%

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

   9. Applied rewrites 57.8%

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

   if -100 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.5%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.1%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+72}:\\ \;\;\;\;w0 \cdot \sqrt{-0.25 \cdot \left(M\_m \cdot \left(\left(M\_m \cdot h\right) \cdot \left(\frac{D\_m}{d \cdot d} \cdot \frac{D\_m}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e+72)
   (* w0 (sqrt (* -0.25 (* M_m (* (* M_m h) (* (/ D_m (* d d)) (/ D_m l)))))))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+72) {
		tmp = w0 * sqrt((-0.25 * (M_m * ((M_m * h) * ((D_m / (d * d)) * (D_m / l))))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-2d+72)) then
        tmp = w0 * sqrt(((-0.25d0) * (m_m * ((m_m * h) * ((d_m / (d * d)) * (d_m / l))))))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+72) {
		tmp = w0 * Math.sqrt((-0.25 * (M_m * ((M_m * h) * ((D_m / (d * d)) * (D_m / l))))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+72:
		tmp = w0 * math.sqrt((-0.25 * (M_m * ((M_m * h) * ((D_m / (d * d)) * (D_m / l))))))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+72)
		tmp = Float64(w0 * sqrt(Float64(-0.25 * Float64(M_m * Float64(Float64(M_m * h) * Float64(Float64(D_m / Float64(d * d)) * Float64(D_m / l)))))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+72)
		tmp = w0 * sqrt((-0.25 * (M_m * ((M_m * h) * ((D_m / (d * d)) * (D_m / l))))));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+72], N[(w0 * N[Sqrt[N[(-0.25 * N[(M$95$m * N[(N[(M$95$m * h), $MachinePrecision] * N[(N[(D$95$m / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.99999999999999989e72

   1. Initial program 65.7%

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

   3. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{-1}{4} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

      12. lower-*.f64 45.5

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

   5. Applied rewrites 45.5%

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

   7. Applied rewrites 55.8%

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

   if -1.99999999999999989e72 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.7%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.6%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+56}:\\ \;\;\;\;w0 \cdot \sqrt{-0.25 \cdot \frac{M\_m \cdot \left(M\_m \cdot \left(\left(h \cdot D\_m\right) \cdot D\_m\right)\right)}{\left(d \cdot d\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -1e+56)
   (* w0 (sqrt (* -0.25 (/ (* M_m (* M_m (* (* h D_m) D_m))) (* (* d d) l)))))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+56) {
		tmp = w0 * sqrt((-0.25 * ((M_m * (M_m * ((h * D_m) * D_m))) / ((d * d) * l))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-1d+56)) then
        tmp = w0 * sqrt(((-0.25d0) * ((m_m * (m_m * ((h * d_m) * d_m))) / ((d * d) * l))))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+56) {
		tmp = w0 * Math.sqrt((-0.25 * ((M_m * (M_m * ((h * D_m) * D_m))) / ((d * d) * l))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+56:
		tmp = w0 * math.sqrt((-0.25 * ((M_m * (M_m * ((h * D_m) * D_m))) / ((d * d) * l))))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+56)
		tmp = Float64(w0 * sqrt(Float64(-0.25 * Float64(Float64(M_m * Float64(M_m * Float64(Float64(h * D_m) * D_m))) / Float64(Float64(d * d) * l)))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -1e+56)
		tmp = w0 * sqrt((-0.25 * ((M_m * (M_m * ((h * D_m) * D_m))) / ((d * d) * l))));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+56], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(M$95$m * N[(M$95$m * N[(N[(h * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.00000000000000009e56

   1. Initial program 66.5%

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

   3. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{-1}{4} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

      12. lower-*.f64 44.5

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

   5. Applied rewrites 44.5%

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

   7. Applied rewrites 50.6%

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

   9. Applied rewrites 51.9%

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

   if -1.00000000000000009e56 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.6%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 95.6%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+72}:\\ \;\;\;\;w0 \cdot \sqrt{-0.25 \cdot \frac{M\_m \cdot \left(M\_m \cdot \left(\left(D\_m \cdot D\_m\right) \cdot h\right)\right)}{\left(d \cdot d\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e+72)
   (* w0 (sqrt (* -0.25 (/ (* M_m (* M_m (* (* D_m D_m) h))) (* (* d d) l)))))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+72) {
		tmp = w0 * sqrt((-0.25 * ((M_m * (M_m * ((D_m * D_m) * h))) / ((d * d) * l))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-2d+72)) then
        tmp = w0 * sqrt(((-0.25d0) * ((m_m * (m_m * ((d_m * d_m) * h))) / ((d * d) * l))))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+72) {
		tmp = w0 * Math.sqrt((-0.25 * ((M_m * (M_m * ((D_m * D_m) * h))) / ((d * d) * l))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+72:
		tmp = w0 * math.sqrt((-0.25 * ((M_m * (M_m * ((D_m * D_m) * h))) / ((d * d) * l))))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+72)
		tmp = Float64(w0 * sqrt(Float64(-0.25 * Float64(Float64(M_m * Float64(M_m * Float64(Float64(D_m * D_m) * h))) / Float64(Float64(d * d) * l)))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+72)
		tmp = w0 * sqrt((-0.25 * ((M_m * (M_m * ((D_m * D_m) * h))) / ((d * d) * l))));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+72], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(M$95$m * N[(M$95$m * N[(N[(D$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.99999999999999989e72

   1. Initial program 65.7%

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

   3. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{-1}{4} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

      12. lower-*.f64 45.5

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

   5. Applied rewrites 45.5%

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

   7. Applied rewrites 50.7%

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

   if -1.99999999999999989e72 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.7%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.6%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot \frac{M\_m \cdot \left(\left(h \cdot M\_m\right) \cdot w0\right)}{\ell}, -0.5, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -5e+105)
   (fma (* (* D_m D_m) (/ (* M_m (* (* h M_m) w0)) l)) -0.5 w0)
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+105) {
		tmp = fma(((D_m * D_m) * ((M_m * ((h * M_m) * w0)) / l)), -0.5, w0);
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+105)
		tmp = fma(Float64(Float64(D_m * D_m) * Float64(Float64(M_m * Float64(Float64(h * M_m) * w0)) / l)), -0.5, w0);
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+105], N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * N[(N[(h * M$95$m), $MachinePrecision] * w0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.5 + w0), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.00000000000000046e105

   1. Initial program 64.0%

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 49.8

          \[\leadsto \mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot w0}{\ell}, -0.5, w0\right) \]

   7. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot w0}{\ell}, -0.5, w0\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 53.8%

      \[\leadsto \mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{\ell}, -0.5, w0\right) \]

   if -5.00000000000000046e105 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.9%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 92.7%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+158}:\\ \;\;\;\;\left(\left(D\_m \cdot D\_m\right) \cdot \frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot w0}{\ell}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -5e+158)
   (* (* (* D_m D_m) (/ (* (* (* M_m M_m) h) w0) l)) -0.5)
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+158) {
		tmp = ((D_m * D_m) * ((((M_m * M_m) * h) * w0) / l)) * -0.5;
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-5d+158)) then
        tmp = ((d_m * d_m) * ((((m_m * m_m) * h) * w0) / l)) * (-0.5d0)
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+158) {
		tmp = ((D_m * D_m) * ((((M_m * M_m) * h) * w0) / l)) * -0.5;
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+158:
		tmp = ((D_m * D_m) * ((((M_m * M_m) * h) * w0) / l)) * -0.5
	else:
		tmp = w0 * 1.0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+158)
		tmp = Float64(Float64(Float64(D_m * D_m) * Float64(Float64(Float64(Float64(M_m * M_m) * h) * w0) / l)) * -0.5);
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+158)
		tmp = ((D_m * D_m) * ((((M_m * M_m) * h) * w0) / l)) * -0.5;
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+158], N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * w0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.9999999999999996e158

   1. Initial program 63.0%

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

   4. Applied rewrites 45.1%

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

   5. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 51.1

          \[\leadsto \mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot w0}{\ell}, -0.5, w0\right) \]

   7. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot w0}{\ell}, -0.5, w0\right)} \]

   8. Taylor expanded in M around inf

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

   10. Applied rewrites 51.0%

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

   if -4.9999999999999996e158 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.0%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 91.7%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 85.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(\left(\frac{D\_m}{d} \cdot M\_m\right) \cdot M\_m\right) \cdot -0.25}{\ell} \cdot \frac{D\_m}{d}, h, 1\right)} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (*
  w0
  (sqrt (fma (* (/ (* (* (* (/ D_m d) M_m) M_m) -0.25) l) (/ D_m d)) h 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0 * sqrt(fma(((((((D_m / d) * M_m) * M_m) * -0.25) / l) * (D_m / d)), h, 1.0));
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	return Float64(w0 * sqrt(fma(Float64(Float64(Float64(Float64(Float64(Float64(D_m / d) * M_m) * M_m) * -0.25) / l) * Float64(D_m / d)), h, 1.0)))
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * -0.25), $MachinePrecision] / l), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. times-frac N/A

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

   11. associate-*r* N/A

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

   12. div-inv N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 75.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

      \[\leadsto w0 \cdot \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell} \cdot \frac{\frac{D}{d}}{{h}^{-1}}\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell} \cdot \frac{\frac{D}{d}}{{h}^{-1}}\right)\right) + 1}} \]

   4. lift-*.f64 N/A

      \[\leadsto w0 \cdot \sqrt{\left(\mathsf{neg}\left(\color{blue}{\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell} \cdot \frac{\frac{D}{d}}{{h}^{-1}}}\right)\right) + 1} \]

   5. distribute-lft-neg-in N/A

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell}\right)\right) \cdot \frac{\frac{D}{d}}{{h}^{-1}}} + 1} \]

   6. lift-/.f64 N/A

      \[\leadsto w0 \cdot \sqrt{\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell}\right)\right) \cdot \color{blue}{\frac{\frac{D}{d}}{{h}^{-1}}} + 1} \]

   7. div-inv N/A

      \[\leadsto w0 \cdot \sqrt{\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell}\right)\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{{h}^{-1}}\right)} + 1} \]

   8. lift-pow.f64 N/A

      \[\leadsto w0 \cdot \sqrt{\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell}\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{1}{\color{blue}{{h}^{-1}}}\right) + 1} \]

   9. unpow-1 N/A

      \[\leadsto w0 \cdot \sqrt{\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell}\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{1}{\color{blue}{\frac{1}{h}}}\right) + 1} \]

   10. remove-double-div N/A

       \[\leadsto w0 \cdot \sqrt{\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell}\right)\right) \cdot \left(\frac{D}{d} \cdot \color{blue}{h}\right) + 1} \]

   11. associate-*r* N/A

       \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell}\right)\right) \cdot \frac{D}{d}\right) \cdot h} + 1} \]

   12. lower-fma.f64 N/A

       \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\ell}\right)\right) \cdot \frac{D}{d}, h, 1\right)}} \]

6. Applied rewrites 83.5%

   \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot -0.25}{\ell} \cdot \frac{D}{d}, h, 1\right)}} \]
7. Add Preprocessing

Alternative 13: 67.8% accurate, 26.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ w0 \cdot 1 \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d) :precision binary64 (* w0 1.0))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0 * 1.0;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0 * 1.0d0
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0 * 1.0;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	return w0 * 1.0

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	return Float64(w0 * 1.0)
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp = code(w0, M_m, D_m, h, l, d)
	tmp = w0 * 1.0;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0 * 1.0), $MachinePrecision]

Derivation

1. Initial program 82.1%

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

3. Taylor expanded in M around 0

   \[\leadsto w0 \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 66.2%

   \[\leadsto w0 \cdot \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024305 
(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))))))