Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 88.3%

Time: 15.8s

Alternatives: 15

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.4% 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: 88.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2 \cdot 10^{-90}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M\_m \cdot D\_m}{2 \cdot d}, \frac{\frac{\left(M\_m \cdot D\_m\right) \cdot h}{2 \cdot d}}{-\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{D\_m}{d} \cdot \left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right)}{\ell} \cdot \frac{\frac{D\_m}{d}}{\frac{-1}{h}}}\\ \end{array} \end{array} \]

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
 (if (<= M_m 2e-90)
   (*
    w0
    (sqrt
     (fma
      (/ (* M_m D_m) (* 2.0 d))
      (/ (/ (* (* M_m D_m) h) (* 2.0 d)) (- l))
      1.0)))
   (*
    w0
    (sqrt
     (+
      1.0
      (*
       (/ (* (/ D_m d) (* (* M_m M_m) 0.25)) l)
       (/ (/ D_m d) (/ -1.0 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 tmp;
	if (M_m <= 2e-90) {
		tmp = w0 * sqrt(fma(((M_m * D_m) / (2.0 * d)), ((((M_m * D_m) * h) / (2.0 * d)) / -l), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 + ((((D_m / d) * ((M_m * M_m) * 0.25)) / l) * ((D_m / d) / (-1.0 / 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)
	tmp = 0.0
	if (M_m <= 2e-90)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M_m * D_m) / Float64(2.0 * d)), Float64(Float64(Float64(Float64(M_m * D_m) * h) / Float64(2.0 * d)) / Float64(-l)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(D_m / d) * Float64(Float64(M_m * M_m) * 0.25)) / l) * Float64(Float64(D_m / d) / Float64(-1.0 / h))))));
	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[M$95$m, 2e-90], N[(w0 * N[Sqrt[N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.99999999999999999e-90

   1. Initial program 82.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied egg-rr 90.3%

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

   if 1.99999999999999999e-90 < M

   1. Initial program 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. div-inv N/A

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

      15. times-frac N/A

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

      16. 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 egg-rr 68.4%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-90}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{\frac{\left(M \cdot D\right) \cdot h}{2 \cdot d}}{-\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\ell} \cdot \frac{\frac{D}{d}}{\frac{-1}{h}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+163}:\\ \;\;\;\;D\_m \cdot \left(w0 \cdot \sqrt{\frac{M\_m \cdot \left(-0.25 \cdot \left(M\_m \cdot h\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right)\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-5}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25 \cdot \frac{h}{d \cdot \ell}, M\_m \cdot \frac{D\_m \cdot \left(M\_m \cdot D\_m\right)}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 -2e+163)
     (* D_m (* w0 (sqrt (/ (* M_m (* -0.25 (* M_m h))) (* d (* d l))))))
     (if (<= t_0 -4e-5)
       (*
        w0
        (sqrt
         (fma (* -0.25 (/ h (* d l))) (* M_m (/ (* D_m (* M_m D_m)) d)) 1.0)))
       w0))))

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 <= -2e+163) {
		tmp = D_m * (w0 * sqrt(((M_m * (-0.25 * (M_m * h))) / (d * (d * l)))));
	} else if (t_0 <= -4e-5) {
		tmp = w0 * sqrt(fma((-0.25 * (h / (d * l))), (M_m * ((D_m * (M_m * D_m)) / d)), 1.0));
	} else {
		tmp = w0;
	}
	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 <= -2e+163)
		tmp = Float64(D_m * Float64(w0 * sqrt(Float64(Float64(M_m * Float64(-0.25 * Float64(M_m * h))) / Float64(d * Float64(d * l))))));
	elseif (t_0 <= -4e-5)
		tmp = Float64(w0 * sqrt(fma(Float64(-0.25 * Float64(h / Float64(d * l))), Float64(M_m * Float64(Float64(D_m * Float64(M_m * D_m)) / d)), 1.0)));
	else
		tmp = w0;
	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, -2e+163], N[(D$95$m * N[(w0 * N[Sqrt[N[(N[(M$95$m * N[(-0.25 * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -4e-5], N[(w0 * N[Sqrt[N[(N[(-0.25 * N[(h / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * N[(N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

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)) < -1.9999999999999999e163

   1. Initial program 60.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied egg-rr 69.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   6. Applied egg-rr 65.7%

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

   7. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 48.7

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

   9. Simplified 48.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-sqrt.f64 N/A

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

       10. *-commutative N/A

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

   11. Applied egg-rr 28.6%

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

   if -1.9999999999999999e163 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.00000000000000033e-5

   1. Initial program 99.5%

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

   3. Taylor expanded in w0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

   5. Simplified 37.8%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 49.4

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

   7. Applied egg-rr 49.4%

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

   9. Applied egg-rr 88.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. *-commutative N/A

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

       9. lift-/.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-/l* N/A

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

       12. associate-*r* N/A

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

       13. lower-fma.f64 N/A

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

   11. Applied egg-rr 82.5%

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

   if -4.00000000000000033e-5 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 97.0%

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

      1. *-rgt-identity 97.0

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

   7. Applied egg-rr 97.0%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+163}:\\ \;\;\;\;D \cdot \left(w0 \cdot \sqrt{\frac{M \cdot \left(-0.25 \cdot \left(M \cdot h\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right)\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{-5}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25 \cdot \frac{h}{d \cdot \ell}, M \cdot \frac{D \cdot \left(M \cdot D\right)}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+163}:\\ \;\;\;\;D\_m \cdot \left(w0 \cdot \sqrt{\frac{M\_m \cdot \left(-0.25 \cdot \left(M\_m \cdot h\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right)\\ \mathbf{elif}\;t\_0 \leq -40:\\ \;\;\;\;w0 \cdot \sqrt{\left(-0.25 \cdot \frac{h}{d \cdot \ell}\right) \cdot \left(M\_m \cdot \frac{D\_m \cdot \left(M\_m \cdot D\_m\right)}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 -2e+163)
     (* D_m (* w0 (sqrt (/ (* M_m (* -0.25 (* M_m h))) (* d (* d l))))))
     (if (<= t_0 -40.0)
       (*
        w0
        (sqrt (* (* -0.25 (/ h (* d l))) (* M_m (/ (* D_m (* M_m D_m)) d)))))
       w0))))

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 <= -2e+163) {
		tmp = D_m * (w0 * sqrt(((M_m * (-0.25 * (M_m * h))) / (d * (d * l)))));
	} else if (t_0 <= -40.0) {
		tmp = w0 * sqrt(((-0.25 * (h / (d * l))) * (M_m * ((D_m * (M_m * D_m)) / d))));
	} else {
		tmp = w0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)
    if (t_0 <= (-2d+163)) then
        tmp = d_m * (w0 * sqrt(((m_m * ((-0.25d0) * (m_m * h))) / (d * (d * l)))))
    else if (t_0 <= (-40.0d0)) then
        tmp = w0 * sqrt((((-0.25d0) * (h / (d * l))) * (m_m * ((d_m * (m_m * d_m)) / d))))
    else
        tmp = w0
    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 t_0 = Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -2e+163) {
		tmp = D_m * (w0 * Math.sqrt(((M_m * (-0.25 * (M_m * h))) / (d * (d * l)))));
	} else if (t_0 <= -40.0) {
		tmp = w0 * Math.sqrt(((-0.25 * (h / (d * l))) * (M_m * ((D_m * (M_m * D_m)) / d))));
	} else {
		tmp = w0;
	}
	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 = math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)
	tmp = 0
	if t_0 <= -2e+163:
		tmp = D_m * (w0 * math.sqrt(((M_m * (-0.25 * (M_m * h))) / (d * (d * l)))))
	elif t_0 <= -40.0:
		tmp = w0 * math.sqrt(((-0.25 * (h / (d * l))) * (M_m * ((D_m * (M_m * D_m)) / d))))
	else:
		tmp = w0
	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 <= -2e+163)
		tmp = Float64(D_m * Float64(w0 * sqrt(Float64(Float64(M_m * Float64(-0.25 * Float64(M_m * h))) / Float64(d * Float64(d * l))))));
	elseif (t_0 <= -40.0)
		tmp = Float64(w0 * sqrt(Float64(Float64(-0.25 * Float64(h / Float64(d * l))) * Float64(M_m * Float64(Float64(D_m * Float64(M_m * D_m)) / d)))));
	else
		tmp = w0;
	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 = (((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_0 <= -2e+163)
		tmp = D_m * (w0 * sqrt(((M_m * (-0.25 * (M_m * h))) / (d * (d * l)))));
	elseif (t_0 <= -40.0)
		tmp = w0 * sqrt(((-0.25 * (h / (d * l))) * (M_m * ((D_m * (M_m * D_m)) / d))));
	else
		tmp = w0;
	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[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, -2e+163], N[(D$95$m * N[(w0 * N[Sqrt[N[(N[(M$95$m * N[(-0.25 * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -40.0], N[(w0 * N[Sqrt[N[(N[(-0.25 * N[(h / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * N[(N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

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)) < -1.9999999999999999e163

   1. Initial program 60.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied egg-rr 69.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   6. Applied egg-rr 65.7%

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

   7. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 48.7

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

   9. Simplified 48.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-sqrt.f64 N/A

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

       10. *-commutative N/A

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

   11. Applied egg-rr 28.6%

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

   if -1.9999999999999999e163 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -40

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied egg-rr 90.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 19.8

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

   9. Simplified 19.8%

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

   11. Applied egg-rr 63.4%

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

   if -40 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 96.7%

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

      1. *-rgt-identity 96.7

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

   7. Applied egg-rr 96.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+163}:\\ \;\;\;\;D \cdot \left(w0 \cdot \sqrt{\frac{M \cdot \left(-0.25 \cdot \left(M \cdot h\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right)\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -40:\\ \;\;\;\;w0 \cdot \sqrt{\left(-0.25 \cdot \frac{h}{d \cdot \ell}\right) \cdot \left(M \cdot \frac{D \cdot \left(M \cdot D\right)}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{2 \cdot d}\\ \mathbf{if}\;{t\_0}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{h}{\ell} \cdot \frac{M\_m \cdot D\_m}{d \cdot -2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D\_m}{d} \cdot \left(\frac{D\_m}{d} \cdot \frac{h \cdot \left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right)}{\ell}\right)}\\ \end{array} \end{array} \]

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 (/ (* M_m D_m) (* 2.0 d))))
   (if (<= (* (pow t_0 2.0) (/ h l)) 2e-14)
     (* w0 (sqrt (fma t_0 (* (/ h l) (/ (* M_m D_m) (* d -2.0))) 1.0)))
     (*
      w0
      (sqrt
       (-
        1.0
        (* (/ D_m d) (* (/ D_m d) (/ (* h (* (* M_m M_m) 0.25)) l)))))))))

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 = (M_m * D_m) / (2.0 * d);
	double tmp;
	if ((pow(t_0, 2.0) * (h / l)) <= 2e-14) {
		tmp = w0 * sqrt(fma(t_0, ((h / l) * ((M_m * D_m) / (d * -2.0))), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 - ((D_m / d) * ((D_m / d) * ((h * ((M_m * M_m) * 0.25)) / l)))));
	}
	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(M_m * D_m) / Float64(2.0 * d))
	tmp = 0.0
	if (Float64((t_0 ^ 2.0) * Float64(h / l)) <= 2e-14)
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(Float64(h / l) * Float64(Float64(M_m * D_m) / Float64(d * -2.0))), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D_m / d) * Float64(Float64(D_m / d) * Float64(Float64(h * Float64(Float64(M_m * M_m) * 0.25)) / l))))));
	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[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 2e-14], N[(w0 * N[Sqrt[N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / l), $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)) < 2e-14

   1. Initial program 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. associate-*l* N/A

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

   4. Applied egg-rr 90.3%

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

   if 2e-14 < (*.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. times-frac N/A

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

      2. unpow-prod-down N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. unpow-prod-down N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 0.0

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

   4. Applied egg-rr 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*r/ N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 38.1

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

   6. Applied egg-rr 38.1%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{h}{\ell} \cdot \frac{M \cdot D}{d \cdot -2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{h \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{-5}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D\_m \cdot \frac{M\_m}{2 \cdot d}, h \cdot \frac{M\_m \cdot D\_m}{\ell \cdot \left(d \cdot -2\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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)) -4e-5)
   (*
    w0
    (sqrt
     (fma
      (* D_m (/ M_m (* 2.0 d)))
      (* h (/ (* M_m D_m) (* l (* d -2.0))))
      1.0)))
   w0))

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)) <= -4e-5) {
		tmp = w0 * sqrt(fma((D_m * (M_m / (2.0 * d))), (h * ((M_m * D_m) / (l * (d * -2.0)))), 1.0));
	} else {
		tmp = w0;
	}
	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)) <= -4e-5)
		tmp = Float64(w0 * sqrt(fma(Float64(D_m * Float64(M_m / Float64(2.0 * d))), Float64(h * Float64(Float64(M_m * D_m) / Float64(l * Float64(d * -2.0)))), 1.0)));
	else
		tmp = w0;
	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], -4e-5], N[(w0 * N[Sqrt[N[(N[(D$95$m * N[(M$95$m / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(l * N[(d * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

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)) < -4.00000000000000033e-5

   1. Initial program 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied egg-rr 72.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   6. Applied egg-rr 70.1%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 67.9

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

   8. Applied egg-rr 67.9%

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

   if -4.00000000000000033e-5 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 97.0%

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

      1. *-rgt-identity 97.0

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

   7. Applied egg-rr 97.0%

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

Alternative 6: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{-5}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{-0.25 \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right)}{d}, \frac{h}{d \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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)) -4e-5)
   (*
    w0
    (sqrt (fma (/ (* -0.25 (* (* M_m D_m) (* M_m D_m))) d) (/ h (* d l)) 1.0)))
   w0))

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)) <= -4e-5) {
		tmp = w0 * sqrt(fma(((-0.25 * ((M_m * D_m) * (M_m * D_m))) / d), (h / (d * l)), 1.0));
	} else {
		tmp = w0;
	}
	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)) <= -4e-5)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(-0.25 * Float64(Float64(M_m * D_m) * Float64(M_m * D_m))) / d), Float64(h / Float64(d * l)), 1.0)));
	else
		tmp = w0;
	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], -4e-5], N[(w0 * N[Sqrt[N[(N[(N[(-0.25 * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(h / N[(d * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

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)) < -4.00000000000000033e-5

   1. Initial program 65.8%

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

   3. Taylor expanded in w0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

   5. Simplified 44.9%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 49.2

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

   7. Applied egg-rr 49.2%

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

   9. Applied egg-rr 63.9%

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

   if -4.00000000000000033e-5 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 97.0%

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

      1. *-rgt-identity 97.0

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

   7. Applied egg-rr 97.0%

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

Alternative 7: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -40:\\ \;\;\;\;w0 \cdot \sqrt{\frac{-0.25 \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right)}{d} \cdot \frac{h}{d \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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)) -40.0)
   (* w0 (sqrt (* (/ (* -0.25 (* (* M_m D_m) (* M_m D_m))) d) (/ h (* d l)))))
   w0))

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)) <= -40.0) {
		tmp = w0 * sqrt((((-0.25 * ((M_m * D_m) * (M_m * D_m))) / d) * (h / (d * l))));
	} else {
		tmp = w0;
	}
	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)) <= (-40.0d0)) then
        tmp = w0 * sqrt(((((-0.25d0) * ((m_m * d_m) * (m_m * d_m))) / d) * (h / (d * l))))
    else
        tmp = w0
    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)) <= -40.0) {
		tmp = w0 * Math.sqrt((((-0.25 * ((M_m * D_m) * (M_m * D_m))) / d) * (h / (d * l))));
	} else {
		tmp = w0;
	}
	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)) <= -40.0:
		tmp = w0 * math.sqrt((((-0.25 * ((M_m * D_m) * (M_m * D_m))) / d) * (h / (d * l))))
	else:
		tmp = w0
	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)) <= -40.0)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(-0.25 * Float64(Float64(M_m * D_m) * Float64(M_m * D_m))) / d) * Float64(h / Float64(d * l)))));
	else
		tmp = w0;
	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)) <= -40.0)
		tmp = w0 * sqrt((((-0.25 * ((M_m * D_m) * (M_m * D_m))) / d) * (h / (d * l))));
	else
		tmp = w0;
	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], -40.0], N[(w0 * N[Sqrt[N[(N[(N[(-0.25 * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(h / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

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)) < -40

   1. Initial program 65.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 42.7

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

   5. Simplified 42.7%

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

   7. Applied egg-rr 61.5%

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

   if -40 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 96.7%

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

      1. *-rgt-identity 96.7

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

   7. Applied egg-rr 96.7%

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

Alternative 8: 84.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -40:\\ \;\;\;\;D\_m \cdot \left(w0 \cdot \sqrt{\frac{M\_m \cdot \left(-0.25 \cdot \left(M\_m \cdot h\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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)) -40.0)
   (* D_m (* w0 (sqrt (/ (* M_m (* -0.25 (* M_m h))) (* d (* d l))))))
   w0))

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)) <= -40.0) {
		tmp = D_m * (w0 * sqrt(((M_m * (-0.25 * (M_m * h))) / (d * (d * l)))));
	} else {
		tmp = w0;
	}
	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)) <= (-40.0d0)) then
        tmp = d_m * (w0 * sqrt(((m_m * ((-0.25d0) * (m_m * h))) / (d * (d * l)))))
    else
        tmp = w0
    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)) <= -40.0) {
		tmp = D_m * (w0 * Math.sqrt(((M_m * (-0.25 * (M_m * h))) / (d * (d * l)))));
	} else {
		tmp = w0;
	}
	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)) <= -40.0:
		tmp = D_m * (w0 * math.sqrt(((M_m * (-0.25 * (M_m * h))) / (d * (d * l)))))
	else:
		tmp = w0
	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)) <= -40.0)
		tmp = Float64(D_m * Float64(w0 * sqrt(Float64(Float64(M_m * Float64(-0.25 * Float64(M_m * h))) / Float64(d * Float64(d * l))))));
	else
		tmp = w0;
	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)) <= -40.0)
		tmp = D_m * (w0 * sqrt(((M_m * (-0.25 * (M_m * h))) / (d * (d * l)))));
	else
		tmp = w0;
	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], -40.0], N[(D$95$m * N[(w0 * N[Sqrt[N[(N[(M$95$m * N[(-0.25 * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

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)) < -40

   1. Initial program 65.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied egg-rr 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   6. Applied egg-rr 69.8%

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

   7. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 45.2

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

   9. Simplified 45.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-sqrt.f64 N/A

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

       10. *-commutative N/A

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

   11. Applied egg-rr 26.8%

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

   if -40 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 96.7%

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

      1. *-rgt-identity 96.7

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

   7. Applied egg-rr 96.7%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -40:\\ \;\;\;\;D \cdot \left(w0 \cdot \sqrt{\frac{M \cdot \left(-0.25 \cdot \left(M \cdot h\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+182}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m \cdot D\_m, \left(\left(M\_m \cdot M\_m\right) \cdot -0.125\right) \cdot \frac{h}{d \cdot \left(d \cdot \ell\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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+182)
   (* w0 (fma (* D_m D_m) (* (* (* M_m M_m) -0.125) (/ h (* d (* d l)))) 1.0))
   w0))

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+182) {
		tmp = w0 * fma((D_m * D_m), (((M_m * M_m) * -0.125) * (h / (d * (d * l)))), 1.0);
	} else {
		tmp = w0;
	}
	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+182)
		tmp = Float64(w0 * fma(Float64(D_m * D_m), Float64(Float64(Float64(M_m * M_m) * -0.125) * Float64(h / Float64(d * Float64(d * l)))), 1.0));
	else
		tmp = w0;
	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+182], N[(w0 * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], w0]

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)) < -1.0000000000000001e182

   1. Initial program 60.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied egg-rr 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   6. Applied egg-rr 65.2%

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

   7. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right)} \]

   9. Simplified 45.8%

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

       1. lift-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right) \]

       2. lift-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot h\right)}}{d \cdot \left(d \cdot \ell\right)}, 1\right) \]

       3. lift-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}, 1\right) \]

       4. lift-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot h\right)}}{d \cdot \left(d \cdot \ell\right)}, 1\right) \]

       5. associate-*r* N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\color{blue}{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}}{d \cdot \left(d \cdot \ell\right)}, 1\right) \]

       6. lift-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}, 1\right) \]

       7. associate-/l* N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \color{blue}{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot \left(d \cdot \ell\right)}}, 1\right) \]

       8. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \color{blue}{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot \left(d \cdot \ell\right)}}, 1\right) \]

       9. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \color{blue}{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right)} \cdot \frac{h}{d \cdot \left(d \cdot \ell\right)}, 1\right) \]

       10. lower-/.f64 47.3

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

   11. Applied egg-rr 47.3%

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

   if -1.0000000000000001e182 < (*.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. Simplified 91.5%

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

      1. *-rgt-identity 91.5

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

   7. Applied egg-rr 91.5%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+182}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D \cdot D, \left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{h}{d \cdot \left(d \cdot \ell\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+182}:\\ \;\;\;\;\left(D\_m \cdot D\_m\right) \cdot \frac{-0.125 \cdot \left(h \cdot \left(w0 \cdot \left(M\_m \cdot M\_m\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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+182)
   (* (* D_m D_m) (/ (* -0.125 (* h (* w0 (* M_m M_m)))) (* l (* d d))))
   w0))

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+182) {
		tmp = (D_m * D_m) * ((-0.125 * (h * (w0 * (M_m * M_m)))) / (l * (d * d)));
	} else {
		tmp = w0;
	}
	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+182)) then
        tmp = (d_m * d_m) * (((-0.125d0) * (h * (w0 * (m_m * m_m)))) / (l * (d * d)))
    else
        tmp = w0
    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+182) {
		tmp = (D_m * D_m) * ((-0.125 * (h * (w0 * (M_m * M_m)))) / (l * (d * d)));
	} else {
		tmp = w0;
	}
	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+182:
		tmp = (D_m * D_m) * ((-0.125 * (h * (w0 * (M_m * M_m)))) / (l * (d * d)))
	else:
		tmp = w0
	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+182)
		tmp = Float64(Float64(D_m * D_m) * Float64(Float64(-0.125 * Float64(h * Float64(w0 * Float64(M_m * M_m)))) / Float64(l * Float64(d * d))));
	else
		tmp = w0;
	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+182)
		tmp = (D_m * D_m) * ((-0.125 * (h * (w0 * (M_m * M_m)))) / (l * (d * d)));
	else
		tmp = w0;
	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+182], N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(-0.125 * N[(h * N[(w0 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

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)) < -1.0000000000000001e182

   1. Initial program 60.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied egg-rr 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   6. Applied egg-rr 65.2%

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

   7. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right)} \]

   9. Simplified 45.8%

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

   10. Taylor expanded in D around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}} \]

       2. associate-/l* N/A

          \[\leadsto \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} \cdot \frac{-1}{8} \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)} \]

       4. *-commutative N/A

          \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} \]

       6. unpow2 N/A

          \[\leadsto \color{blue}{\left(D \cdot D\right)} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(D \cdot D\right)} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right) \]

       8. associate-*r/ N/A

          \[\leadsto \left(D \cdot D\right) \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]

       9. lower-/.f64 N/A

          \[\leadsto \left(D \cdot D\right) \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]

       10. lower-*.f64 N/A

           \[\leadsto \left(D \cdot D\right) \cdot \frac{\color{blue}{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}}{{d}^{2} \cdot \ell} \]

       11. *-commutative N/A

           \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \color{blue}{\left(w0 \cdot h\right)}\right)}{{d}^{2} \cdot \ell} \]

       12. associate-*r* N/A

           \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot w0\right) \cdot h\right)}}{{d}^{2} \cdot \ell} \]

       13. lower-*.f64 N/A

           \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot w0\right) \cdot h\right)}}{{d}^{2} \cdot \ell} \]

       14. lower-*.f64 N/A

           \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \left(\color{blue}{\left({M}^{2} \cdot w0\right)} \cdot h\right)}{{d}^{2} \cdot \ell} \]

       15. unpow2 N/A

           \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot w0\right) \cdot h\right)}{{d}^{2} \cdot \ell} \]

       16. lower-*.f64 N/A

           \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot w0\right) \cdot h\right)}{{d}^{2} \cdot \ell} \]

       17. lower-*.f64 N/A

           \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \left(\left(\left(M \cdot M\right) \cdot w0\right) \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

       18. unpow2 N/A

           \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \left(\left(\left(M \cdot M\right) \cdot w0\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} \]

       19. lower-*.f64 45.6

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

   12. Simplified 45.6%

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

   if -1.0000000000000001e182 < (*.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. Simplified 91.5%

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

      1. *-rgt-identity 91.5

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

   7. Applied egg-rr 91.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+182}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+182}:\\ \;\;\;\;w0 \cdot \left(\left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right) \cdot \frac{h \cdot \left(M\_m \cdot M\_m\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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+182)
   (* w0 (* (* (* D_m D_m) -0.125) (/ (* h (* M_m M_m)) (* l (* d d)))))
   w0))

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+182) {
		tmp = w0 * (((D_m * D_m) * -0.125) * ((h * (M_m * M_m)) / (l * (d * d))));
	} else {
		tmp = w0;
	}
	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+182)) then
        tmp = w0 * (((d_m * d_m) * (-0.125d0)) * ((h * (m_m * m_m)) / (l * (d * d))))
    else
        tmp = w0
    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+182) {
		tmp = w0 * (((D_m * D_m) * -0.125) * ((h * (M_m * M_m)) / (l * (d * d))));
	} else {
		tmp = w0;
	}
	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+182:
		tmp = w0 * (((D_m * D_m) * -0.125) * ((h * (M_m * M_m)) / (l * (d * d))))
	else:
		tmp = w0
	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+182)
		tmp = Float64(w0 * Float64(Float64(Float64(D_m * D_m) * -0.125) * Float64(Float64(h * Float64(M_m * M_m)) / Float64(l * Float64(d * d)))));
	else
		tmp = w0;
	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+182)
		tmp = w0 * (((D_m * D_m) * -0.125) * ((h * (M_m * M_m)) / (l * (d * d))));
	else
		tmp = w0;
	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+182], N[(w0 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

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)) < -1.0000000000000001e182

   1. Initial program 60.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied egg-rr 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   6. Applied egg-rr 65.2%

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

   7. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right)} \]

   9. Simplified 45.8%

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

   10. Taylor expanded in D around inf

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

       1. *-commutative N/A

          \[\leadsto w0 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)} \]

       2. associate-/l* N/A

          \[\leadsto w0 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)} \cdot \frac{-1}{8}\right) \]

       3. associate-*r* N/A

          \[\leadsto w0 \cdot \color{blue}{\left({D}^{2} \cdot \left(\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)\right)} \]

       4. *-commutative N/A

          \[\leadsto w0 \cdot \left({D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right) \]

       5. associate-*r* N/A

          \[\leadsto w0 \cdot \color{blue}{\left(\left({D}^{2} \cdot \frac{-1}{8}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto w0 \cdot \color{blue}{\left(\left({D}^{2} \cdot \frac{-1}{8}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto w0 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{-1}{8}\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \]

       8. unpow2 N/A

          \[\leadsto w0 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{-1}{8}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto w0 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{-1}{8}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \]

       10. lower-/.f64 N/A

           \[\leadsto w0 \cdot \left(\left(\left(D \cdot D\right) \cdot \frac{-1}{8}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto w0 \cdot \left(\left(\left(D \cdot D\right) \cdot \frac{-1}{8}\right) \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2} \cdot \ell}\right) \]

       12. unpow2 N/A

           \[\leadsto w0 \cdot \left(\left(\left(D \cdot D\right) \cdot \frac{-1}{8}\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2} \cdot \ell}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto w0 \cdot \left(\left(\left(D \cdot D\right) \cdot \frac{-1}{8}\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2} \cdot \ell}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto w0 \cdot \left(\left(\left(D \cdot D\right) \cdot \frac{-1}{8}\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{{d}^{2} \cdot \ell}}\right) \]

       15. unpow2 N/A

           \[\leadsto w0 \cdot \left(\left(\left(D \cdot D\right) \cdot \frac{-1}{8}\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

       16. lower-*.f64 45.7

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

   12. Simplified 45.7%

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

   if -1.0000000000000001e182 < (*.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. Simplified 91.5%

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

      1. *-rgt-identity 91.5

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

   7. Applied egg-rr 91.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+182}:\\ \;\;\;\;w0 \cdot \left(\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \left(M\_m \cdot M\_m\right) \cdot 0.25\\ \mathbf{if}\;M\_m \cdot D\_m \leq 5 \cdot 10^{-307}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D\_m}{d} \cdot \left(\frac{D\_m}{d} \cdot \frac{h \cdot t\_0}{\ell}\right)}\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 10^{+111}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D\_m \cdot \frac{M\_m}{2 \cdot d}, h \cdot \frac{M\_m \cdot D\_m}{\ell \cdot \left(d \cdot -2\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\frac{D\_m}{d} \cdot t\_0\right) \cdot \frac{h}{-\ell}, \frac{D\_m}{d}, 1\right)}\\ \end{array} \end{array} \]

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 (* (* M_m M_m) 0.25)))
   (if (<= (* M_m D_m) 5e-307)
     (* w0 (sqrt (- 1.0 (* (/ D_m d) (* (/ D_m d) (/ (* h t_0) l))))))
     (if (<= (* M_m D_m) 1e+111)
       (*
        w0
        (sqrt
         (fma
          (* D_m (/ M_m (* 2.0 d)))
          (* h (/ (* M_m D_m) (* l (* d -2.0))))
          1.0)))
       (* w0 (sqrt (fma (* (* (/ D_m d) t_0) (/ h (- l))) (/ D_m d) 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 = (M_m * M_m) * 0.25;
	double tmp;
	if ((M_m * D_m) <= 5e-307) {
		tmp = w0 * sqrt((1.0 - ((D_m / d) * ((D_m / d) * ((h * t_0) / l)))));
	} else if ((M_m * D_m) <= 1e+111) {
		tmp = w0 * sqrt(fma((D_m * (M_m / (2.0 * d))), (h * ((M_m * D_m) / (l * (d * -2.0)))), 1.0));
	} else {
		tmp = w0 * sqrt(fma((((D_m / d) * t_0) * (h / -l)), (D_m / d), 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(M_m * M_m) * 0.25)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 5e-307)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D_m / d) * Float64(Float64(D_m / d) * Float64(Float64(h * t_0) / l))))));
	elseif (Float64(M_m * D_m) <= 1e+111)
		tmp = Float64(w0 * sqrt(fma(Float64(D_m * Float64(M_m / Float64(2.0 * d))), Float64(h * Float64(Float64(M_m * D_m) / Float64(l * Float64(d * -2.0)))), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(D_m / d) * t_0) * Float64(h / Float64(-l))), Float64(D_m / d), 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[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision]}, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e-307], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e+111], N[(w0 * N[Sqrt[N[(N[(D$95$m * N[(M$95$m / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(l * N[(d * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(h / (-l)), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 5.00000000000000014e-307

   1. Initial program 84.6%

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

      1. times-frac N/A

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

      2. unpow-prod-down N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. unpow-prod-down N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 71.1

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

   4. Applied egg-rr 71.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*r/ N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 77.0

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

   6. Applied egg-rr 77.0%

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

   if 5.00000000000000014e-307 < (*.f64 M D) < 9.99999999999999957e110

   1. Initial program 86.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied egg-rr 94.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   6. Applied egg-rr 92.9%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 91.7

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

   8. Applied egg-rr 91.7%

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

   if 9.99999999999999957e110 < (*.f64 M D)

   1. Initial program 60.5%

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

   4. Applied egg-rr 55.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 5 \cdot 10^{-307}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{h \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\ell}\right)}\\ \mathbf{elif}\;M \cdot D \leq 10^{+111}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D \cdot \frac{M}{2 \cdot d}, h \cdot \frac{M \cdot D}{\ell \cdot \left(d \cdot -2\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \frac{h}{-\ell}, \frac{D}{d}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := w0 \cdot \sqrt{1 - \frac{D\_m}{d} \cdot \left(\frac{D\_m}{d} \cdot \frac{h \cdot \left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right)}{\ell}\right)}\\ \mathbf{if}\;M\_m \cdot D\_m \leq 5 \cdot 10^{-307}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 5 \cdot 10^{+230}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D\_m \cdot \frac{M\_m}{2 \cdot d}, h \cdot \frac{M\_m \cdot D\_m}{\ell \cdot \left(d \cdot -2\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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
         (*
          w0
          (sqrt
           (-
            1.0
            (* (/ D_m d) (* (/ D_m d) (/ (* h (* (* M_m M_m) 0.25)) l))))))))
   (if (<= (* M_m D_m) 5e-307)
     t_0
     (if (<= (* M_m D_m) 5e+230)
       (*
        w0
        (sqrt
         (fma
          (* D_m (/ M_m (* 2.0 d)))
          (* h (/ (* M_m D_m) (* l (* d -2.0))))
          1.0)))
       t_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 = w0 * sqrt((1.0 - ((D_m / d) * ((D_m / d) * ((h * ((M_m * M_m) * 0.25)) / l)))));
	double tmp;
	if ((M_m * D_m) <= 5e-307) {
		tmp = t_0;
	} else if ((M_m * D_m) <= 5e+230) {
		tmp = w0 * sqrt(fma((D_m * (M_m / (2.0 * d))), (h * ((M_m * D_m) / (l * (d * -2.0)))), 1.0));
	} else {
		tmp = t_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(w0 * sqrt(Float64(1.0 - Float64(Float64(D_m / d) * Float64(Float64(D_m / d) * Float64(Float64(h * Float64(Float64(M_m * M_m) * 0.25)) / l))))))
	tmp = 0.0
	if (Float64(M_m * D_m) <= 5e-307)
		tmp = t_0;
	elseif (Float64(M_m * D_m) <= 5e+230)
		tmp = Float64(w0 * sqrt(fma(Float64(D_m * Float64(M_m / Float64(2.0 * d))), Float64(h * Float64(Float64(M_m * D_m) / Float64(l * Float64(d * -2.0)))), 1.0)));
	else
		tmp = t_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[(w0 * N[Sqrt[N[(1.0 - N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e-307], t$95$0, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e+230], N[(w0 * N[Sqrt[N[(N[(D$95$m * N[(M$95$m / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(l * N[(d * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 5.00000000000000014e-307 or 5.0000000000000003e230 < (*.f64 M D)

   1. Initial program 80.7%

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

      1. times-frac N/A

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

      2. unpow-prod-down N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. unpow-prod-down N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 68.7

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

   4. Applied egg-rr 68.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*r/ N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 75.1

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

   6. Applied egg-rr 75.1%

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

   if 5.00000000000000014e-307 < (*.f64 M D) < 5.0000000000000003e230

   1. Initial program 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied egg-rr 88.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   6. Applied egg-rr 88.3%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 87.4

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

   8. Applied egg-rr 87.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 5 \cdot 10^{-307}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{h \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\ell}\right)}\\ \mathbf{elif}\;M \cdot D \leq 5 \cdot 10^{+230}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D \cdot \frac{M}{2 \cdot d}, h \cdot \frac{M \cdot D}{\ell \cdot \left(d \cdot -2\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{h \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 87.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;2 \cdot d \leq 10^{+21}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M\_m \cdot D\_m}{2 \cdot d}, \frac{\frac{\left(M\_m \cdot D\_m\right) \cdot h}{2 \cdot d}}{-\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D\_m}{d} \cdot \left(\frac{D\_m}{d} \cdot \frac{h \cdot \left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right)}{\ell}\right)}\\ \end{array} \end{array} \]

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 (<= (* 2.0 d) 1e+21)
   (*
    w0
    (sqrt
     (fma
      (/ (* M_m D_m) (* 2.0 d))
      (/ (/ (* (* M_m D_m) h) (* 2.0 d)) (- l))
      1.0)))
   (*
    w0
    (sqrt
     (- 1.0 (* (/ D_m d) (* (/ D_m d) (/ (* h (* (* M_m M_m) 0.25)) l))))))))

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 ((2.0 * d) <= 1e+21) {
		tmp = w0 * sqrt(fma(((M_m * D_m) / (2.0 * d)), ((((M_m * D_m) * h) / (2.0 * d)) / -l), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 - ((D_m / d) * ((D_m / d) * ((h * ((M_m * M_m) * 0.25)) / l)))));
	}
	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(2.0 * d) <= 1e+21)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M_m * D_m) / Float64(2.0 * d)), Float64(Float64(Float64(Float64(M_m * D_m) * h) / Float64(2.0 * d)) / Float64(-l)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D_m / d) * Float64(Float64(D_m / d) * Float64(Float64(h * Float64(Float64(M_m * M_m) * 0.25)) / l))))));
	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[(2.0 * d), $MachinePrecision], 1e+21], N[(w0 * N[Sqrt[N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) d) < 1e21

   1. Initial program 79.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied egg-rr 86.4%

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

   if 1e21 < (*.f64 #s(literal 2 binary64) d)

   1. Initial program 86.6%

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

      1. times-frac N/A

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

      2. unpow-prod-down N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. unpow-prod-down N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 71.4

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

   4. Applied egg-rr 71.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*r/ N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 76.5

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

   6. Applied egg-rr 76.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot d \leq 10^{+21}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{\frac{\left(M \cdot D\right) \cdot h}{2 \cdot d}}{-\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{h \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.7% accurate, 157.0× speedup

Math

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

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)

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;
}

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
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;
}

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

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 w0
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;
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_] := w0

Derivation

1. Initial program 81.4%

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

3. Taylor expanded in M around 0

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

5. Simplified 67.2%

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

   1. *-rgt-identity 67.2

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

7. Applied egg-rr 67.2%

   \[\leadsto \color{blue}{w0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))))))