Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 88.7%

Time: 18.8s

Alternatives: 5

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d\_m}\\ t_1 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(-2 \cdot \log \left(\frac{1}{D\_m}\right) + \left(\log \left(-0.25 \cdot \left(\frac{h}{\ell} \cdot {M\_m}^{2}\right)\right) + -2 \cdot \log d\_m\right)\right)}\right)}^{3}\\ \mathbf{elif}\;t\_1 \leq 0.01:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{t\_0 \cdot t\_0}{4}, \frac{h}{-\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(1 + \left(-1 - D\_m \cdot \left(0.25 \cdot \frac{D\_m \cdot \left(h \cdot {M\_m}^{2}\right)}{\ell \cdot {d\_m}^{2}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d_m)))
        (t_1 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l))))
   (if (<= t_1 (- INFINITY))
     (pow
      (*
       (cbrt w0)
       (exp
        (*
         0.16666666666666666
         (+
          (* -2.0 (log (/ 1.0 D_m)))
          (+ (log (* -0.25 (* (/ h l) (pow M_m 2.0)))) (* -2.0 (log d_m)))))))
      3.0)
     (if (<= t_1 0.01)
       (* w0 (sqrt (fma (/ (* t_0 t_0) 4.0) (/ h (- l)) 1.0)))
       (*
        w0
        (sqrt
         (+
          1.0
          (+
           1.0
           (-
            -1.0
            (*
             D_m
             (*
              0.25
              (/ (* D_m (* h (pow M_m 2.0))) (* l (pow d_m 2.0))))))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = D_m * (M_m / d_m);
	double t_1 = pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = pow((cbrt(w0) * exp((0.16666666666666666 * ((-2.0 * log((1.0 / D_m))) + (log((-0.25 * ((h / l) * pow(M_m, 2.0)))) + (-2.0 * log(d_m))))))), 3.0);
	} else if (t_1 <= 0.01) {
		tmp = w0 * sqrt(fma(((t_0 * t_0) / 4.0), (h / -l), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 + (1.0 + (-1.0 - (D_m * (0.25 * ((D_m * (h * pow(M_m, 2.0))) / (l * pow(d_m, 2.0)))))))));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(D_m * Float64(M_m / d_m))
	t_1 = Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cbrt(w0) * exp(Float64(0.16666666666666666 * Float64(Float64(-2.0 * log(Float64(1.0 / D_m))) + Float64(log(Float64(-0.25 * Float64(Float64(h / l) * (M_m ^ 2.0)))) + Float64(-2.0 * log(d_m))))))) ^ 3.0;
	elseif (t_1 <= 0.01)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(t_0 * t_0) / 4.0), Float64(h / Float64(-l)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(1.0 + Float64(-1.0 - Float64(D_m * Float64(0.25 * Float64(Float64(D_m * Float64(h * (M_m ^ 2.0))) / Float64(l * (d_m ^ 2.0))))))))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(N[(-2.0 * N[Log[N[(1.0 / D$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Log[N[(-0.25 * N[(N[(h / l), $MachinePrecision] * N[Power[M$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[t$95$1, 0.01], N[(w0 * N[Sqrt[N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] * N[(h / (-l)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(1.0 + N[(-1.0 - N[(D$95$m * N[(0.25 * N[(N[(D$95$m * N[(h * N[Power[M$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 56.5%

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

   3. Simplified 60.0%

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

   6. Applied egg-rr 60.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}}\right)}^{3}} \]

   7. Taylor expanded in D around inf 41.2%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}}^{3} \]

   8. Taylor expanded in d around 0 15.4%

      \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\color{blue}{\left(\log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{\ell}\right) + -2 \cdot \log d\right)} + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]
   9. Step-by-step derivation

      1. distribute-lft-neg-in 15.4%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\left(\log \color{blue}{\left(\left(-0.25\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} + -2 \cdot \log d\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      2. metadata-eval 15.4%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\left(\log \left(\color{blue}{-0.25} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) + -2 \cdot \log d\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      3. associate-/l* 15.4%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\left(\log \left(-0.25 \cdot \color{blue}{\left({M}^{2} \cdot \frac{h}{\ell}\right)}\right) + -2 \cdot \log d\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

   10. Simplified 15.4%

       \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\color{blue}{\left(\log \left(-0.25 \cdot \left({M}^{2} \cdot \frac{h}{\ell}\right)\right) + -2 \cdot \log d\right)} + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 0.0100000000000000002

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. cancel-sign-sub-inv 99.9%

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

      3. distribute-frac-neg2 99.9%

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

      4. *-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

      7. *-commutative 99.9%

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

      8. times-frac 98.8%

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

   3. Simplified 98.8%

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

      1. unpow2 98.8%

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

      2. associate-*l/ 98.8%

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

      3. associate-*l/ 98.8%

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

      4. frac-times 98.8%

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

      5. metadata-eval 98.8%

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

   6. Applied egg-rr 98.8%

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

   if 0.0100000000000000002 < (*.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. Step-by-step derivation

      1. *-commutative 0.0%

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

      2. cancel-sign-sub-inv 0.0%

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

      3. distribute-frac-neg2 0.0%

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

      4. *-commutative 0.0%

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

      5. +-commutative 0.0%

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

      6. fma-define 0.0%

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

      7. *-commutative 0.0%

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

      8. times-frac 3.8%

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

   3. Simplified 3.8%

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

      1. unpow2 3.8%

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

      2. associate-*r/ 0.0%

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

      3. associate-*l/ 0.0%

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

      4. frac-times 0.0%

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

      5. div-inv 0.0%

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

      6. metadata-eval 0.0%

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

      7. *-commutative 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. fma-undefine 0.0%

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

      2. associate-/l* 0.0%

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

      3. associate-*l* 0.0%

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

      4. distribute-frac-neg2 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. +-commutative 0.0%

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

      2. distribute-rgt-neg-out 0.0%

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

      3. unsub-neg 0.0%

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

      4. associate-*l* 3.8%

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

      5. *-commutative 3.8%

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

      6. times-frac 3.8%

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

   10. Simplified 3.8%

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

       1. expm1-log1p-u 3.7%

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

       2. expm1-undefine 3.7%

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

   12. Applied egg-rr 0.1%

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

   13. Taylor expanded in M around 0 52.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(-2 \cdot \log \left(\frac{1}{D}\right) + \left(\log \left(-0.25 \cdot \left(\frac{h}{\ell} \cdot {M}^{2}\right)\right) + -2 \cdot \log d\right)\right)}\right)}^{3}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.01:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{4}, \frac{h}{-\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(1 + \left(-1 - D \cdot \left(0.25 \cdot \frac{D \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+283}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\left(\log \left(-0.25 \cdot \frac{\frac{h}{\ell}}{{d\_m}^{2}}\right) + 2 \cdot \log M\_m\right) + -2 \cdot \log \left(\frac{1}{D\_m}\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(D\_m \cdot \frac{M\_m}{2 \cdot d\_m}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -4e+283)
   (pow
    (*
     (cbrt w0)
     (exp
      (*
       0.16666666666666666
       (+
        (+ (log (* -0.25 (/ (/ h l) (pow d_m 2.0)))) (* 2.0 (log M_m)))
        (* -2.0 (log (/ 1.0 D_m)))))))
    3.0)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* D_m (/ M_m (* 2.0 d_m))) 2.0)) l))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -4e+283) {
		tmp = pow((cbrt(w0) * exp((0.16666666666666666 * ((log((-0.25 * ((h / l) / pow(d_m, 2.0)))) + (2.0 * log(M_m))) + (-2.0 * log((1.0 / D_m))))))), 3.0);
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow((D_m * (M_m / (2.0 * d_m))), 2.0)) / l)));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -4e+283) {
		tmp = Math.pow((Math.cbrt(w0) * Math.exp((0.16666666666666666 * ((Math.log((-0.25 * ((h / l) / Math.pow(d_m, 2.0)))) + (2.0 * Math.log(M_m))) + (-2.0 * Math.log((1.0 / D_m))))))), 3.0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((D_m * (M_m / (2.0 * d_m))), 2.0)) / l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -4e+283)
		tmp = Float64(cbrt(w0) * exp(Float64(0.16666666666666666 * Float64(Float64(log(Float64(-0.25 * Float64(Float64(h / l) / (d_m ^ 2.0)))) + Float64(2.0 * log(M_m))) + Float64(-2.0 * log(Float64(1.0 / D_m))))))) ^ 3.0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(D_m * Float64(M_m / Float64(2.0 * d_m))) ^ 2.0)) / l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e+283], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(N[(N[Log[N[(-0.25 * N[(N[(h / l), $MachinePrecision] / N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(2.0 * N[Log[M$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[Log[N[(1.0 / D$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(D$95$m * N[(M$95$m / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $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)) < -3.99999999999999982e283

   1. Initial program 57.2%

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

   3. Simplified 59.0%

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

   6. Applied egg-rr 58.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}}\right)}^{3}} \]

   7. Taylor expanded in D around inf 40.5%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}}^{3} \]

   8. Taylor expanded in M around 0 25.4%

      \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\color{blue}{\left(\log \left(-0.25 \cdot \frac{h}{{d}^{2} \cdot \ell}\right) + 2 \cdot \log M\right)} + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]
   9. Step-by-step derivation

      1. distribute-lft-neg-in 25.4%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\left(\log \color{blue}{\left(\left(-0.25\right) \cdot \frac{h}{{d}^{2} \cdot \ell}\right)} + 2 \cdot \log M\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      2. metadata-eval 25.4%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\left(\log \left(\color{blue}{-0.25} \cdot \frac{h}{{d}^{2} \cdot \ell}\right) + 2 \cdot \log M\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      3. *-commutative 25.4%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\left(\log \left(-0.25 \cdot \frac{h}{\color{blue}{\ell \cdot {d}^{2}}}\right) + 2 \cdot \log M\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      4. associate-/r* 25.7%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\left(\log \left(-0.25 \cdot \color{blue}{\frac{\frac{h}{\ell}}{{d}^{2}}}\right) + 2 \cdot \log M\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

   10. Simplified 25.7%

       \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\color{blue}{\left(\log \left(-0.25 \cdot \frac{\frac{h}{\ell}}{{d}^{2}}\right) + 2 \cdot \log M\right)} + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

   if -3.99999999999999982e283 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.5%

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

   3. Simplified 86.5%

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

      1. associate-*r/ 94.4%

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

      2. add-sqr-sqrt 94.4%

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

      3. pow2 94.4%

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

      4. sqrt-pow1 94.4%

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

      5. metadata-eval 94.4%

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

      6. pow1 94.4%

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

      7. associate-/l/ 94.4%

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

      8. *-commutative 94.4%

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

   6. Applied egg-rr 94.4%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+283}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\left(\log \left(-0.25 \cdot \frac{\frac{h}{\ell}}{{d}^{2}}\right) + 2 \cdot \log M\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d\_m}\\ \mathbf{if}\;1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{+211}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{t\_0 \cdot t\_0}{4}, \frac{h}{-\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(1 + \left(-1 - D\_m \cdot \frac{h \cdot \left(\left(M\_m \cdot 0.5\right) \cdot \left(\left(D\_m \cdot 0.5\right) \cdot \left(M\_m \cdot {d\_m}^{-2}\right)\right)\right)}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d_m))))
   (if (<= (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l))) 2e+211)
     (* w0 (sqrt (fma (/ (* t_0 t_0) 4.0) (/ h (- l)) 1.0)))
     (*
      w0
      (sqrt
       (+
        1.0
        (+
         1.0
         (-
          -1.0
          (*
           D_m
           (/
            (* h (* (* M_m 0.5) (* (* D_m 0.5) (* M_m (pow d_m -2.0)))))
            l))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = D_m * (M_m / d_m);
	double tmp;
	if ((1.0 - (pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))) <= 2e+211) {
		tmp = w0 * sqrt(fma(((t_0 * t_0) / 4.0), (h / -l), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 + (1.0 + (-1.0 - (D_m * ((h * ((M_m * 0.5) * ((D_m * 0.5) * (M_m * pow(d_m, -2.0))))) / l))))));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(D_m * Float64(M_m / d_m))
	tmp = 0.0
	if (Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l))) <= 2e+211)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(t_0 * t_0) / 4.0), Float64(h / Float64(-l)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(1.0 + Float64(-1.0 - Float64(D_m * Float64(Float64(h * Float64(Float64(M_m * 0.5) * Float64(Float64(D_m * 0.5) * Float64(M_m * (d_m ^ -2.0))))) / l)))))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+211], N[(w0 * N[Sqrt[N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] * N[(h / (-l)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(1.0 + N[(-1.0 - N[(D$95$m * N[(N[(h * N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(N[(D$95$m * 0.5), $MachinePrecision] * N[(M$95$m * N[Power[d$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 1.9999999999999999e211

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. cancel-sign-sub-inv 99.9%

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

      3. distribute-frac-neg2 99.9%

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

      4. *-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

      7. *-commutative 99.9%

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

      8. times-frac 99.4%

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

   3. Simplified 99.4%

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

      1. unpow2 99.4%

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

      2. associate-*l/ 99.4%

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

      3. associate-*l/ 99.4%

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

      4. frac-times 99.4%

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

      5. metadata-eval 99.4%

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

   6. Applied egg-rr 99.4%

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

   if 1.9999999999999999e211 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 41.2%

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

      1. *-commutative 41.2%

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

      2. cancel-sign-sub-inv 41.2%

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

      3. distribute-frac-neg2 41.2%

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

      4. *-commutative 41.2%

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

      5. +-commutative 41.2%

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

      6. fma-define 41.2%

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

      7. *-commutative 41.2%

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

      8. times-frac 43.5%

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

   3. Simplified 43.5%

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

      1. unpow2 43.5%

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

      2. associate-*r/ 40.1%

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

      3. associate-*l/ 40.1%

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

      4. frac-times 40.1%

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

      5. div-inv 40.1%

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

      6. metadata-eval 40.1%

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

      7. *-commutative 40.1%

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

   6. Applied egg-rr 40.1%

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

      1. fma-undefine 40.1%

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

      2. associate-/l* 40.1%

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

      3. associate-*l* 40.1%

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

      4. distribute-frac-neg2 40.1%

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

   8. Applied egg-rr 40.1%

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

      1. +-commutative 40.1%

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

      2. distribute-rgt-neg-out 40.1%

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

      3. unsub-neg 40.1%

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

      4. associate-*l* 43.5%

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

      5. *-commutative 43.5%

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

      6. times-frac 42.5%

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

   10. Simplified 42.5%

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

       1. expm1-log1p-u 2.4%

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

       2. expm1-undefine 2.4%

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

   12. Applied egg-rr 41.1%

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

       1. associate-*r/ 60.0%

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

       2. associate-*l* 62.4%

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

       3. div-inv 62.4%

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

       4. pow-flip 62.4%

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

       5. metadata-eval 62.4%

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

   14. Applied egg-rr 62.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{+211}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{4}, \frac{h}{-\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(1 + \left(-1 - D \cdot \frac{h \cdot \left(\left(M \cdot 0.5\right) \cdot \left(\left(D \cdot 0.5\right) \cdot \left(M \cdot {d}^{-2}\right)\right)\right)}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d\_m}\\ w0 \cdot \sqrt{1 + h \cdot \left(-0.25 \cdot \frac{t\_0 \cdot t\_0}{\ell}\right)} \end{array} \end{array} \]

FPCore

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

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = D_m * (M_m / d_m);
	return w0 * sqrt((1.0 + (h * (-0.25 * ((t_0 * t_0) / l)))));
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: t_0
    t_0 = d_m * (m_m / d_m_1)
    code = w0 * sqrt((1.0d0 + (h * ((-0.25d0) * ((t_0 * t_0) / l)))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = D_m * (M_m / d_m);
	return w0 * Math.sqrt((1.0 + (h * (-0.25 * ((t_0 * t_0) / l)))));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = D_m * (M_m / d_m)
	return w0 * math.sqrt((1.0 + (h * (-0.25 * ((t_0 * t_0) / l)))))

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(D_m * Float64(M_m / d_m))
	return Float64(w0 * sqrt(Float64(1.0 + Float64(h * Float64(-0.25 * Float64(Float64(t_0 * t_0) / l))))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp = code(w0, M_m, D_m, h, l, d_m)
	t_0 = D_m * (M_m / d_m);
	tmp = w0 * sqrt((1.0 + (h * (-0.25 * ((t_0 * t_0) / l)))));
end

Wolfram

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

Derivation

1. Initial program 80.2%

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

3. Simplified 80.6%

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

5. Taylor expanded in h around inf 58.2%

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

   1. cancel-sign-sub-inv 58.2%

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

   2. metadata-eval 58.2%

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

   3. +-commutative 58.2%

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

   4. distribute-lft-in 58.2%

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

   5. fma-define 58.2%

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

7. Simplified 73.8%

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

   1. fma-undefine 73.8%

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

   2. associate-/l* 73.8%

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

   3. pow-prod-down 86.8%

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

9. Applied egg-rr 86.8%

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

    1. unpow2 86.8%

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

11. Applied egg-rr 86.8%

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

12. Final simplification 86.8%

    \[\leadsto w0 \cdot \sqrt{1 + h \cdot \left(-0.25 \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\ell}\right)} \]
13. Add Preprocessing

Alternative 5: 67.0% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

3. Simplified 80.6%

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

5. Taylor expanded in D around 0 69.8%

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

Reproduce

herbie shell --seed 2024163 
(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))))))