Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 87.6%

Time: 25.4s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d_m}\\ \mathbf{if}\;{t_0}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+282}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d_m, \log \left(-0.25 \cdot \frac{h \cdot {\left(M \cdot D\right)}^{2}}{\ell}\right)\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left({\left(\sqrt[3]{t_0}\right)}^{3}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* 2.0 d_m))))
   (if (<= (* (pow t_0 2.0) (/ h l)) -5e+282)
     (pow
      (*
       (cbrt w0)
       (pow
        (exp 0.16666666666666666)
        (fma -2.0 (log d_m) (log (* -0.25 (/ (* h (pow (* M D) 2.0)) l))))))
      3.0)
     (* w0 (sqrt (- 1.0 (/ (* h (pow (pow (cbrt t_0) 3.0) 2.0)) l)))))))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double t_0 = (M * D) / (2.0 * d_m);
	double tmp;
	if ((pow(t_0, 2.0) * (h / l)) <= -5e+282) {
		tmp = pow((cbrt(w0) * pow(exp(0.16666666666666666), fma(-2.0, log(d_m), log((-0.25 * ((h * pow((M * D), 2.0)) / l)))))), 3.0);
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow(pow(cbrt(t_0), 3.0), 2.0)) / l)));
	}
	return tmp;
}

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	t_0 = Float64(Float64(M * D) / Float64(2.0 * d_m))
	tmp = 0.0
	if (Float64((t_0 ^ 2.0) * Float64(h / l)) <= -5e+282)
		tmp = Float64(cbrt(w0) * (exp(0.16666666666666666) ^ fma(-2.0, log(d_m), log(Float64(-0.25 * Float64(Float64(h * (Float64(M * D) ^ 2.0)) / l)))))) ^ 3.0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * ((cbrt(t_0) ^ 3.0) ^ 2.0)) / l))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+282], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Power[N[Exp[0.16666666666666666], $MachinePrecision], N[(-2.0 * N[Log[d$95$m], $MachinePrecision] + N[Log[N[(-0.25 * N[(N[(h * N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $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 2 d)) 2) (/.f64 h l)) < -4.99999999999999978e282

   1. Initial program 61.1%

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

   3. Simplified 62.7%

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

      1. associate-/l* 61.1%

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

      2. *-commutative 61.1%

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

      3. add-sqr-sqrt 32.7%

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

      4. sqrt-unprod 55.0%

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

      5. *-commutative 55.0%

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

      6. *-commutative 55.0%

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

      7. swap-sqr 55.0%

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

      8. metadata-eval 55.0%

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

      9. metadata-eval 55.0%

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

      10. swap-sqr 55.0%

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

      11. sqrt-unprod 28.5%

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

      12. add-sqr-sqrt 61.1%

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

      13. times-frac 62.7%

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

   6. Applied egg-rr 62.7%

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

      1. associate-*l/ 61.1%

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

      2. associate-*r/ 61.1%

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

      3. associate-*l/ 61.1%

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

      4. associate-/l* 64.1%

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

   8. Simplified 64.1%

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

      1. add-cube-cbrt 64.0%

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

      2. pow3 64.0%

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

      3. *-commutative 64.0%

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

      4. associate-/r/ 62.7%

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

      5. div-inv 62.7%

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

      6. metadata-eval 62.7%

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

   10. Applied egg-rr 62.7%

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

   11. Taylor expanded in d around 0 10.6%

       \[\leadsto {\color{blue}{\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right) + -2 \cdot \log d\right)}\right)}}^{3} \]
   12. Step-by-step derivation

       1. unpow1/3 30.4%

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

       2. *-lft-identity 30.4%

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

       3. exp-prod 30.4%

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

       4. +-commutative 30.4%

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

       5. fma-def 30.4%

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

       6. distribute-lft-neg-in 30.4%

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

       7. metadata-eval 30.4%

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

       8. associate-*r* 30.5%

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

       9. unpow2 30.5%

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

       10. unpow2 30.5%

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

       11. swap-sqr 39.5%

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

       12. unpow2 39.5%

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

       13. *-commutative 39.5%

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

   13. Simplified 39.5%

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

   if -4.99999999999999978e282 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 86.2%

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

   3. Simplified 85.7%

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

   6. Applied egg-rr 94.0%

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

      1. add-cube-cbrt 93.9%

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

      2. pow3 93.9%

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

      3. associate-*r/ 94.4%

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

   8. Applied egg-rr 94.4%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+282}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(-0.25 \cdot \frac{h \cdot {\left(M \cdot D\right)}^{2}}{\ell}\right)\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left({\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{3}\right)}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{\frac{2 \cdot d_m}{D}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d_m)) 2.0) (/ h l)) -2e-13)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ M (/ (* 2.0 d_m) D)) 2.0)))))
   w0))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e-13) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((M / ((2.0 * d_m) / D)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_m)) ** 2.0d0) * (h / l)) <= (-2d-13)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((m / ((2.0d0 * d_m) / d)) ** 2.0d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e-13) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((M / ((2.0 * d_m) / D)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e-13:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((M / ((2.0 * d_m) / D)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e-13)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(M / Float64(Float64(2.0 * d_m) / D)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -2e-13)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M / ((2.0 * d_m) / D)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-13], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(M / N[(N[(2.0 * d$95$m), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -2.0000000000000001e-13

   1. Initial program 70.0%

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

   3. Simplified 70.2%

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

   if -2.0000000000000001e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in M around 0 94.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{\frac{D}{-2}}{\frac{d_m}{M}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d_m)) 2.0) (/ h l)) -2e-13)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ (/ D -2.0) (/ d_m M)) 2.0)))))
   w0))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e-13) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D / -2.0) / (d_m / M)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_m)) ** 2.0d0) * (h / l)) <= (-2d-13)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d / (-2.0d0)) / (d_m / m)) ** 2.0d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e-13) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D / -2.0) / (d_m / M)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e-13:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D / -2.0) / (d_m / M)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e-13)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D / -2.0) / Float64(d_m / M)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -2e-13)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D / -2.0) / (d_m / M)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-13], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / -2.0), $MachinePrecision] / N[(d$95$m / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -2.0000000000000001e-13

   1. Initial program 70.0%

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

   3. Simplified 70.2%

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

      1. associate-/l* 70.0%

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

      2. *-commutative 70.0%

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

      3. add-sqr-sqrt 35.4%

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

      4. sqrt-unprod 59.9%

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

      5. *-commutative 59.9%

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

      6. *-commutative 59.9%

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

      7. swap-sqr 59.9%

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

      8. metadata-eval 59.9%

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

      9. metadata-eval 59.9%

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

      10. swap-sqr 59.9%

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

      11. sqrt-unprod 34.5%

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

      12. add-sqr-sqrt 70.0%

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

      13. times-frac 70.2%

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

   6. Applied egg-rr 70.2%

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

      1. associate-*l/ 70.0%

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

      2. associate-*r/ 70.0%

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

      3. associate-*l/ 70.0%

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

      4. associate-/l* 70.2%

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

   8. Simplified 70.2%

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

   if -2.0000000000000001e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in M around 0 94.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{\frac{D}{-2}}{\frac{d}{M}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;D \leq 5.4 \cdot 10^{+51}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{d_m}{M \cdot \left(D \cdot -0.5\right)}\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= D 5.4e+51)
   w0
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ d_m (* M (* D -0.5))) -2.0)))))))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (D <= 5.4e+51) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((d_m / (M * (D * -0.5))), -2.0))));
	}
	return tmp;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if (d <= 5.4d+51) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d_m / (m * (d * (-0.5d0)))) ** (-2.0d0)))))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (D <= 5.4e+51) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((d_m / (M * (D * -0.5))), -2.0))));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if D <= 5.4e+51:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((d_m / (M * (D * -0.5))), -2.0))))
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (D <= 5.4e+51)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(d_m / Float64(M * Float64(D * -0.5))) ^ -2.0)))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if (D <= 5.4e+51)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((h / l) * ((d_m / (M * (D * -0.5))) ^ -2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[D, 5.4e+51], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(d$95$m / N[(M * N[(D * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 5.39999999999999983e51

   1. Initial program 78.9%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in M around 0 71.7%

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

   if 5.39999999999999983e51 < D

   1. Initial program 83.1%

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

   3. Simplified 85.0%

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

      1. associate-/l* 83.1%

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

      2. *-commutative 83.1%

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

      3. add-sqr-sqrt 36.7%

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

      4. sqrt-unprod 73.7%

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

      5. *-commutative 73.7%

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

      6. *-commutative 73.7%

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

      7. swap-sqr 73.7%

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

      8. metadata-eval 73.7%

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

      9. metadata-eval 73.7%

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

      10. swap-sqr 73.7%

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

      11. sqrt-unprod 46.4%

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

      12. add-sqr-sqrt 83.1%

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

      13. times-frac 85.0%

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

   6. Applied egg-rr 85.0%

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

      1. associate-*l/ 83.1%

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

      2. associate-*r/ 83.1%

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

      3. associate-*l/ 83.1%

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

      4. associate-/l* 85.0%

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

   8. Simplified 85.0%

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

      1. clear-num 85.0%

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

      2. inv-pow 85.0%

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

      3. div-inv 85.0%

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

      4. metadata-eval 85.0%

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

   10. Applied egg-rr 85.0%

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

       1. unpow-1 85.0%

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

       2. associate-/l/ 83.1%

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

   12. Simplified 83.1%

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

       1. associate-*r/ 81.5%

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

       2. inv-pow 81.5%

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

       3. pow-pow 81.5%

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

       4. associate-/r* 83.4%

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

       5. metadata-eval 83.4%

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

   14. Applied egg-rr 83.4%

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

       1. associate-*r/ 85.0%

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

       2. *-commutative 85.0%

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

       3. associate-/l/ 83.1%

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

   16. Simplified 83.1%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 5.4 \cdot 10^{+51}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{d}{M \cdot \left(D \cdot -0.5\right)}\right)}^{-2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;M \leq 4.7 \cdot 10^{+39}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(w0 \cdot \left(\frac{h}{{d_m}^{2}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= M 4.7e+39)
   w0
   (* -0.125 (* w0 (* (/ h (pow d_m 2.0)) (/ (pow (* M D) 2.0) l))))))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (M <= 4.7e+39) {
		tmp = w0;
	} else {
		tmp = -0.125 * (w0 * ((h / pow(d_m, 2.0)) * (pow((M * D), 2.0) / l)));
	}
	return tmp;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if (m <= 4.7d+39) then
        tmp = w0
    else
        tmp = (-0.125d0) * (w0 * ((h / (d_m ** 2.0d0)) * (((m * d) ** 2.0d0) / l)))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (M <= 4.7e+39) {
		tmp = w0;
	} else {
		tmp = -0.125 * (w0 * ((h / Math.pow(d_m, 2.0)) * (Math.pow((M * D), 2.0) / l)));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if M <= 4.7e+39:
		tmp = w0
	else:
		tmp = -0.125 * (w0 * ((h / math.pow(d_m, 2.0)) * (math.pow((M * D), 2.0) / l)))
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (M <= 4.7e+39)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(w0 * Float64(Float64(h / (d_m ^ 2.0)) * Float64((Float64(M * D) ^ 2.0) / l))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if (M <= 4.7e+39)
		tmp = w0;
	else
		tmp = -0.125 * (w0 * ((h / (d_m ^ 2.0)) * (((M * D) ^ 2.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[M, 4.7e+39], w0, N[(-0.125 * N[(w0 * N[(N[(h / N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.6999999999999999e39

   1. Initial program 81.8%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in M around 0 68.5%

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

   if 4.6999999999999999e39 < M

   1. Initial program 70.5%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in M around 0 42.1%

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

      1. *-commutative 42.1%

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

      2. times-frac 37.8%

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

   7. Simplified 37.8%

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

      1. frac-times 42.1%

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

      2. associate-/r* 42.2%

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

      3. associate-*r* 46.8%

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

      4. pow-prod-down 64.7%

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

   9. Applied egg-rr 64.7%

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

   10. Taylor expanded in D around inf 23.5%

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

   12. Simplified 22.8%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.7 \cdot 10^{+39}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(w0 \cdot \left(\frac{h}{{d}^{2}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;M \leq 3.3 \cdot 10^{+37}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(w0 \cdot \frac{\left(h \cdot {\left(M \cdot D\right)}^{2}\right) \cdot {d_m}^{-2}}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= M 3.3e+37)
   w0
   (* -0.125 (* w0 (/ (* (* h (pow (* M D) 2.0)) (pow d_m -2.0)) l)))))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (M <= 3.3e+37) {
		tmp = w0;
	} else {
		tmp = -0.125 * (w0 * (((h * pow((M * D), 2.0)) * pow(d_m, -2.0)) / l));
	}
	return tmp;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if (m <= 3.3d+37) then
        tmp = w0
    else
        tmp = (-0.125d0) * (w0 * (((h * ((m * d) ** 2.0d0)) * (d_m ** (-2.0d0))) / l))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (M <= 3.3e+37) {
		tmp = w0;
	} else {
		tmp = -0.125 * (w0 * (((h * Math.pow((M * D), 2.0)) * Math.pow(d_m, -2.0)) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if M <= 3.3e+37:
		tmp = w0
	else:
		tmp = -0.125 * (w0 * (((h * math.pow((M * D), 2.0)) * math.pow(d_m, -2.0)) / l))
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (M <= 3.3e+37)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(w0 * Float64(Float64(Float64(h * (Float64(M * D) ^ 2.0)) * (d_m ^ -2.0)) / l)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if (M <= 3.3e+37)
		tmp = w0;
	else
		tmp = -0.125 * (w0 * (((h * ((M * D) ^ 2.0)) * (d_m ^ -2.0)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[M, 3.3e+37], w0, N[(-0.125 * N[(w0 * N[(N[(N[(h * N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[d$95$m, -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.3000000000000001e37

   1. Initial program 81.7%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in M around 0 68.3%

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

   if 3.3000000000000001e37 < M

   1. Initial program 71.1%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in M around 0 43.3%

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

      1. *-commutative 43.3%

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

      2. times-frac 39.2%

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

   7. Simplified 39.2%

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

      1. frac-times 43.3%

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

      2. associate-/r* 43.4%

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

      3. associate-*r* 47.9%

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

      4. pow-prod-down 65.5%

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

   9. Applied egg-rr 65.5%

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

   10. Taylor expanded in D around inf 23.1%

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

   12. Simplified 22.4%

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

       1. expm1-log1p-u 1.7%

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

       2. expm1-udef 1.6%

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

       3. div-inv 1.6%

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

       4. pow-flip 1.6%

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

       5. metadata-eval 1.6%

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

   14. Applied egg-rr 1.6%

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

       1. expm1-def 1.7%

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

       2. expm1-log1p 22.4%

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

       3. associate-*r/ 22.7%

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

       4. *-commutative 22.7%

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

       5. associate-*r* 24.9%

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

       6. *-commutative 24.9%

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

   16. Simplified 24.9%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.3 \cdot 10^{+37}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(w0 \cdot \frac{\left(h \cdot {\left(M \cdot D\right)}^{2}\right) \cdot {d}^{-2}}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.2% accurate, 1.0× speedup

Math

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

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (* w0 (sqrt (- 1.0 (/ (* h (pow (* M (/ D (* 2.0 d_m))) 2.0)) l)))))

C

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

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    code = w0 * sqrt((1.0d0 - ((h * ((m * (d / (2.0d0 * d_m))) ** 2.0d0)) / l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M * N[(D / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

3. Simplified 79.8%

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

6. Applied egg-rr 86.3%

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

7. Final simplification 86.3%

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

Alternative 8: 86.3% accurate, 1.0× speedup

Math

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

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (* w0 (sqrt (- 1.0 (/ (* h (pow (/ (/ d_m (* D -0.5)) M) -2.0)) l)))))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	return w0 * sqrt((1.0 - ((h * pow(((d_m / (D * -0.5)) / M), -2.0)) / l)));
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    code = w0 * sqrt((1.0d0 - ((h * (((d_m / (d * (-0.5d0))) / m) ** (-2.0d0))) / l)))
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	return w0 * Math.sqrt((1.0 - ((h * Math.pow(((d_m / (D * -0.5)) / M), -2.0)) / l)));
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	return w0 * math.sqrt((1.0 - ((h * math.pow(((d_m / (D * -0.5)) / M), -2.0)) / l)))

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(d_m / Float64(D * -0.5)) / M) ^ -2.0)) / l))))
end

MATLAB

d_m = abs(d);
function tmp = code(w0, M, D, h, l, d_m)
	tmp = w0 * sqrt((1.0 - ((h * (((d_m / (D * -0.5)) / M) ^ -2.0)) / l)));
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(d$95$m / N[(D * -0.5), $MachinePrecision]), $MachinePrecision] / M), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

3. Simplified 79.8%

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

   1. associate-/l* 79.8%

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

   2. *-commutative 79.8%

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

   3. add-sqr-sqrt 39.2%

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

   4. sqrt-unprod 72.5%

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

   5. *-commutative 72.5%

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

   6. *-commutative 72.5%

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

   7. swap-sqr 72.5%

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

   8. metadata-eval 72.5%

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

   9. metadata-eval 72.5%

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

   10. swap-sqr 72.5%

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

   11. sqrt-unprod 40.5%

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

   12. add-sqr-sqrt 79.8%

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

   13. times-frac 79.8%

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

6. Applied egg-rr 79.8%

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

   1. associate-*l/ 79.8%

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

   2. associate-*r/ 79.8%

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

   3. associate-*l/ 79.8%

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

   4. associate-/l* 79.4%

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

8. Simplified 79.4%

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

   1. clear-num 79.4%

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

   2. inv-pow 79.4%

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

   3. div-inv 79.4%

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

   4. metadata-eval 79.4%

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

10. Applied egg-rr 79.4%

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

    1. unpow-1 79.4%

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

    2. associate-/l/ 79.8%

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

12. Simplified 79.8%

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

    1. associate-*r/ 85.9%

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

    2. inv-pow 85.9%

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

    3. pow-pow 85.9%

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

    4. associate-/r* 86.3%

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

    5. metadata-eval 86.3%

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

14. Applied egg-rr 86.3%

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

15. Final simplification 86.3%

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

Alternative 9: 68.2% accurate, 216.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ w0 \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m) :precision binary64 w0)

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	return w0;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    code = w0
end function

Java

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

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	return w0

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	return w0
end

MATLAB

d_m = abs(d);
function tmp = code(w0, M, D, h, l, d_m)
	tmp = w0;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := w0

Derivation

1. Initial program 79.8%

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

3. Simplified 79.8%

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

5. Taylor expanded in M around 0 64.8%

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

6. Final simplification 64.8%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(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))))))