Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 90.9%

Time: 19.3s

Alternatives: 5

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.0% 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: 90.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+28}:\\ \;\;\;\;w0 \cdot \left(\sqrt{h \cdot \frac{\sqrt[3]{-0.015625}}{\ell}} \cdot \left(D\_m \cdot \frac{M\_m}{d\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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)) -5e+28)
   (* w0 (* (sqrt (* h (/ (cbrt -0.015625) l))) (* D_m (/ M_m d_m))))
   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) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -5e+28) {
		tmp = w0 * (sqrt((h * (cbrt(-0.015625) / l))) * (D_m * (M_m / d_m)));
	} else {
		tmp = w0;
	}
	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)) <= -5e+28) {
		tmp = w0 * (Math.sqrt((h * (Math.cbrt(-0.015625) / l))) * (D_m * (M_m / d_m)));
	} else {
		tmp = w0;
	}
	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)) <= -5e+28)
		tmp = Float64(w0 * Float64(sqrt(Float64(h * Float64(cbrt(-0.015625) / l))) * Float64(D_m * Float64(M_m / d_m))));
	else
		tmp = w0;
	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], -5e+28], N[(w0 * N[(N[Sqrt[N[(h * N[(N[Power[-0.015625, 1/3], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.99999999999999957e28

   1. Initial program 77.4%

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

   3. Simplified 76.2%

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

      1. associate-*r/ 72.2%

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

      2. add-sqr-sqrt 72.2%

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

      3. pow2 72.2%

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

      4. sqrt-pow1 72.2%

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

      5. metadata-eval 72.2%

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

      6. pow1 72.2%

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

      7. *-un-lft-identity 72.2%

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

      8. times-frac 72.2%

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

      9. metadata-eval 72.2%

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

   6. Applied egg-rr 72.2%

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

      1. add-cbrt-cube 62.3%

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

      2. pow3 62.3%

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

      3. clear-num 62.3%

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

      4. inv-pow 62.3%

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

      5. pow-pow 62.3%

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

   8. Applied egg-rr 62.3%

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

   9. Taylor expanded in h around -inf 0.0%

      \[\leadsto w0 \cdot \color{blue}{\left(-1 \cdot \left(\frac{D \cdot \left(M \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h \cdot \sqrt[3]{-0.015625}}{\ell}}\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 0.0%

          \[\leadsto w0 \cdot \color{blue}{\left(-\frac{D \cdot \left(M \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h \cdot \sqrt[3]{-0.015625}}{\ell}}\right)} \]

       2. distribute-rgt-neg-in 0.0%

          \[\leadsto w0 \cdot \color{blue}{\left(\frac{D \cdot \left(M \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \left(-\sqrt{\frac{h \cdot \sqrt[3]{-0.015625}}{\ell}}\right)\right)} \]

       3. associate-/l* 0.0%

          \[\leadsto w0 \cdot \left(\color{blue}{\left(D \cdot \frac{M \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)} \cdot \left(-\sqrt{\frac{h \cdot \sqrt[3]{-0.015625}}{\ell}}\right)\right) \]

       4. *-commutative 0.0%

          \[\leadsto w0 \cdot \left(\left(D \cdot \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot M}}{d}\right) \cdot \left(-\sqrt{\frac{h \cdot \sqrt[3]{-0.015625}}{\ell}}\right)\right) \]

       5. unpow2 0.0%

          \[\leadsto w0 \cdot \left(\left(D \cdot \frac{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot M}{d}\right) \cdot \left(-\sqrt{\frac{h \cdot \sqrt[3]{-0.015625}}{\ell}}\right)\right) \]

       6. rem-square-sqrt 47.7%

          \[\leadsto w0 \cdot \left(\left(D \cdot \frac{\color{blue}{-1} \cdot M}{d}\right) \cdot \left(-\sqrt{\frac{h \cdot \sqrt[3]{-0.015625}}{\ell}}\right)\right) \]

       7. mul-1-neg 47.7%

          \[\leadsto w0 \cdot \left(\left(D \cdot \frac{\color{blue}{-M}}{d}\right) \cdot \left(-\sqrt{\frac{h \cdot \sqrt[3]{-0.015625}}{\ell}}\right)\right) \]

       8. associate-/l* 47.7%

          \[\leadsto w0 \cdot \left(\left(D \cdot \frac{-M}{d}\right) \cdot \left(-\sqrt{\color{blue}{h \cdot \frac{\sqrt[3]{-0.015625}}{\ell}}}\right)\right) \]

   11. Simplified 47.7%

       \[\leadsto w0 \cdot \color{blue}{\left(\left(D \cdot \frac{-M}{d}\right) \cdot \left(-\sqrt{h \cdot \frac{\sqrt[3]{-0.015625}}{\ell}}\right)\right)} \]

   if -4.99999999999999957e28 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in M around 0 96.2%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+28}:\\ \;\;\;\;w0 \cdot \left(\sqrt{h \cdot \frac{\sqrt[3]{-0.015625}}{\ell}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-273}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{M\_m}{d\_m}\right)}^{2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= (/ h l) -5e-273)
   (* w0 (fma -0.125 (* (/ h l) (pow (* D_m (/ M_m d_m)) 2.0)) 1.0))
   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) {
	double tmp;
	if ((h / l) <= -5e-273) {
		tmp = w0 * fma(-0.125, ((h / l) * pow((D_m * (M_m / d_m)), 2.0)), 1.0);
	} else {
		tmp = w0;
	}
	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(h / l) <= -5e-273)
		tmp = Float64(w0 * fma(-0.125, Float64(Float64(h / l) * (Float64(D_m * Float64(M_m / d_m)) ^ 2.0)), 1.0));
	else
		tmp = w0;
	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[(h / l), $MachinePrecision], -5e-273], N[(w0 * N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -4.99999999999999965e-273

   1. Initial program 84.2%

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

   3. Simplified 83.5%

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

      1. associate-*r/ 84.2%

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

      2. associate-/r* 84.2%

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

      3. clear-num 84.2%

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

      4. associate-/l* 84.2%

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

      5. div-inv 84.2%

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

      6. metadata-eval 84.2%

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

   6. Applied egg-rr 84.2%

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

      1. associate-*r/ 85.0%

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

      2. inv-pow 85.0%

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

      3. pow-pow 85.0%

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

      4. associate-/r* 85.0%

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

      5. metadata-eval 85.0%

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

   8. Applied egg-rr 85.0%

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

   9. Taylor expanded in d around inf 55.2%

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

       1. +-commutative 55.2%

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

       2. fma-define 55.2%

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

       3. associate-*r* 56.8%

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

       4. times-frac 56.3%

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

       5. associate-/l* 56.4%

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

       6. unpow2 56.4%

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

       7. unpow2 56.4%

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

       8. unpow2 56.4%

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

       9. times-frac 61.1%

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

       10. swap-sqr 71.8%

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

       11. associate-/l* 71.8%

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

       12. associate-/l* 71.8%

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

       13. unpow2 71.8%

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

       14. associate-/l* 71.8%

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

   11. Simplified 71.8%

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

   if -4.99999999999999965e-273 < (/.f64 h l)

   1. Initial program 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in M around 0 95.8%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-273}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;d\_m \leq 950000000:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{{\left(D\_m \cdot \frac{M\_m}{d\_m}\right)}^{2}}{\ell} \cdot \left(h \cdot w0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= d_m 950000000.0)
   (+ w0 (* -0.125 (* (/ (pow (* D_m (/ M_m d_m)) 2.0) l) (* h w0))))
   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) {
	double tmp;
	if (d_m <= 950000000.0) {
		tmp = w0 + (-0.125 * ((pow((D_m * (M_m / d_m)), 2.0) / l) * (h * w0)));
	} else {
		tmp = w0;
	}
	return tmp;
}

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) :: tmp
    if (d_m_1 <= 950000000.0d0) then
        tmp = w0 + ((-0.125d0) * ((((d_m * (m_m / d_m_1)) ** 2.0d0) / l) * (h * w0)))
    else
        tmp = w0
    end if
    code = tmp
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 tmp;
	if (d_m <= 950000000.0) {
		tmp = w0 + (-0.125 * ((Math.pow((D_m * (M_m / d_m)), 2.0) / l) * (h * w0)));
	} else {
		tmp = w0;
	}
	return tmp;
}

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):
	tmp = 0
	if d_m <= 950000000.0:
		tmp = w0 + (-0.125 * ((math.pow((D_m * (M_m / d_m)), 2.0) / l) * (h * w0)))
	else:
		tmp = w0
	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 (d_m <= 950000000.0)
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64((Float64(D_m * Float64(M_m / d_m)) ^ 2.0) / l) * Float64(h * w0))));
	else
		tmp = w0;
	end
	return tmp
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_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (d_m <= 950000000.0)
		tmp = w0 + (-0.125 * ((((D_m * (M_m / d_m)) ^ 2.0) / l) * (h * w0)));
	else
		tmp = w0;
	end
	tmp_2 = 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[d$95$m, 950000000.0], N[(w0 + N[(-0.125 * N[(N[(N[Power[N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(h * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 9.5e8

   1. Initial program 85.3%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in M around 0 52.7%

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

      1. associate-/l* 52.1%

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

      2. times-frac 52.6%

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

   7. Simplified 52.6%

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

      1. unpow2 52.6%

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

      2. unpow2 52.6%

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

      3. times-frac 60.2%

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

   9. Applied egg-rr 60.2%

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

       1. associate-*r* 60.7%

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

       2. frac-times 52.6%

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

       3. unpow2 52.6%

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

       4. associate-*r/ 57.3%

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

       5. /-rgt-identity 57.3%

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

       6. associate-*r* 57.3%

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

       7. pow1 57.3%

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

       8. associate-*r* 57.3%

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

       9. /-rgt-identity 57.3%

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

       10. associate-*r/ 52.6%

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

       11. unpow2 52.6%

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

       12. frac-times 60.7%

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

       13. pow2 60.7%

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

       14. pow-prod-down 72.7%

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

       15. associate-*r/ 71.2%

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

   11. Applied egg-rr 71.2%

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

       1. unpow1 71.2%

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

   13. Simplified 71.2%

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

   14. Taylor expanded in D around 0 52.7%

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

       1. associate-*r* 52.6%

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

       2. times-frac 53.5%

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

       3. associate-/l* 52.6%

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

       4. unpow2 52.6%

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

       5. unpow2 52.6%

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

       6. unpow2 52.6%

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

       7. times-frac 60.7%

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

       8. swap-sqr 72.7%

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

       9. associate-/l* 72.7%

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

       10. associate-/l* 72.7%

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

       11. unpow2 72.7%

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

       12. *-lft-identity 72.7%

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

       13. associate-*l/ 72.7%

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

       14. associate-*r* 75.2%

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

       15. associate-*r/ 75.2%

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

       16. *-rgt-identity 75.2%

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

       17. associate-/l* 75.2%

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

   16. Simplified 75.2%

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

   if 9.5e8 < d

   1. Initial program 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in M around 0 92.7%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 950000000:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell} \cdot \left(h \cdot w0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.0% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{d\_m}\\ \mathbf{if}\;M\_m \leq 3.2 \cdot 10^{-138}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \end{array} \]

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 (/ (* M_m D_m) d_m)))
   (if (<= M_m 3.2e-138) w0 (+ w0 (* -0.125 (* (* t_0 t_0) (* h (/ w0 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 = (M_m * D_m) / d_m;
	double tmp;
	if (M_m <= 3.2e-138) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((t_0 * t_0) * (h * (w0 / l))));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = (m_m * d_m) / d_m_1
    if (m_m <= 3.2d-138) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((t_0 * t_0) * (h * (w0 / l))))
    end if
    code = tmp
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 = (M_m * D_m) / d_m;
	double tmp;
	if (M_m <= 3.2e-138) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((t_0 * t_0) * (h * (w0 / l))));
	}
	return tmp;
}

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 = (M_m * D_m) / d_m
	tmp = 0
	if M_m <= 3.2e-138:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((t_0 * t_0) * (h * (w0 / 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(Float64(M_m * D_m) / d_m)
	tmp = 0.0
	if (M_m <= 3.2e-138)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(t_0 * t_0) * Float64(h * Float64(w0 / l)))));
	end
	return tmp
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_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = (M_m * D_m) / d_m;
	tmp = 0.0;
	if (M_m <= 3.2e-138)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((t_0 * t_0) * (h * (w0 / l))));
	end
	tmp_2 = 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[(N[(M$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]}, If[LessEqual[M$95$m, 3.2e-138], w0, N[(w0 + N[(-0.125 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 3.2000000000000001e-138

   1. Initial program 89.0%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in M around 0 75.5%

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

   if 3.2000000000000001e-138 < M

   1. Initial program 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in M around 0 52.4%

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

      1. associate-/l* 52.4%

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

      2. times-frac 53.5%

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

   7. Simplified 53.5%

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

      1. unpow2 53.5%

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

      2. unpow2 53.5%

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

      3. times-frac 57.9%

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

   9. Applied egg-rr 57.9%

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

       1. associate-*r* 58.9%

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

       2. frac-times 54.6%

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

       3. unpow2 54.6%

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

       4. associate-*r/ 58.9%

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

       5. /-rgt-identity 58.9%

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

       6. associate-*r* 57.8%

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

       7. pow1 57.8%

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

       8. associate-*r* 58.9%

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

       9. /-rgt-identity 58.9%

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

       10. associate-*r/ 54.6%

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

       11. unpow2 54.6%

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

       12. frac-times 58.9%

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

       13. pow2 58.9%

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

       14. pow-prod-down 71.6%

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

       15. associate-*r/ 71.7%

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

   11. Applied egg-rr 71.7%

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

       1. unpow1 71.7%

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

   13. Simplified 71.7%

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

       1. unpow2 71.7%

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

       2. associate-*r/ 71.7%

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

       3. associate-*r/ 71.7%

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

   15. Applied egg-rr 71.7%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.2 \cdot 10^{-138}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right) \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% 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 87.4%

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

3. Simplified 87.1%

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

5. Taylor expanded in M around 0 69.1%

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

6. Final simplification 69.1%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(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))))))