Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.4% → 92.0%

Time: 20.1s

Alternatives: 4

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: 80.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \mathsf{hypot}\left(1, D\_m \cdot \left(0.5 \cdot \left(\frac{M\_m}{d} \cdot \sqrt{\frac{h}{-\ell}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\ell} \cdot \left(h \cdot {\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -5e-8)
   (* w0 (hypot 1.0 (* D_m (* 0.5 (* (/ M_m d) (sqrt (/ h (- l))))))))
   (*
    w0
    (sqrt (- 1.0 (* (/ 1.0 l) (* h (pow (* D_m (* 0.5 (/ M_m d))) 2.0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e-8) {
		tmp = w0 * hypot(1.0, (D_m * (0.5 * ((M_m / d) * sqrt((h / -l))))));
	} else {
		tmp = w0 * sqrt((1.0 - ((1.0 / l) * (h * pow((D_m * (0.5 * (M_m / d))), 2.0)))));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e-8) {
		tmp = w0 * Math.hypot(1.0, (D_m * (0.5 * ((M_m / d) * Math.sqrt((h / -l))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((1.0 / l) * (h * Math.pow((D_m * (0.5 * (M_m / d))), 2.0)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e-8:
		tmp = w0 * math.hypot(1.0, (D_m * (0.5 * ((M_m / d) * math.sqrt((h / -l))))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((1.0 / l) * (h * math.pow((D_m * (0.5 * (M_m / d))), 2.0)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e-8)
		tmp = Float64(w0 * hypot(1.0, Float64(D_m * Float64(0.5 * Float64(Float64(M_m / d) * sqrt(Float64(h / Float64(-l))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(1.0 / l) * Float64(h * (Float64(D_m * Float64(0.5 * Float64(M_m / d))) ^ 2.0))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e-8)
		tmp = w0 * hypot(1.0, (D_m * (0.5 * ((M_m / d) * sqrt((h / -l))))));
	else
		tmp = w0 * sqrt((1.0 - ((1.0 / l) * (h * ((D_m * (0.5 * (M_m / d))) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e-8], N[(w0 * N[Sqrt[1.0 ^ 2 + N[(D$95$m * N[(0.5 * N[(N[(M$95$m / d), $MachinePrecision] * N[Sqrt[N[(h / (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(1.0 / l), $MachinePrecision] * N[(h * N[Power[N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $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)) < -4.9999999999999998e-8

   1. Initial program 62.5%

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

   3. Simplified 64.7%

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

      1. associate-*r/ 63.6%

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

      2. add-sqr-sqrt 63.6%

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

      3. pow2 63.6%

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

      4. unpow2 63.6%

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

      5. sqrt-prod 30.3%

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

      6. add-sqr-sqrt 63.6%

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

      7. *-un-lft-identity 63.6%

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

      8. times-frac 63.6%

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

      9. metadata-eval 63.6%

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

   6. Applied egg-rr 63.6%

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

      1. *-un-lft-identity 63.6%

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

      2. sub-neg 63.6%

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

      3. associate-/l* 64.7%

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

      4. distribute-rgt-neg-out 64.7%

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

      5. add-sqr-sqrt 64.7%

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

      6. hypot-1-def 64.7%

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

      7. sqrt-prod 70.5%

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

      8. unpow2 70.5%

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

      9. sqrt-prod 34.9%

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

      10. add-sqr-sqrt 78.3%

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

      11. associate-*r/ 78.3%

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

   8. Applied egg-rr 78.3%

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

      1. *-lft-identity 78.3%

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

      2. associate-*r/ 77.1%

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

      3. *-commutative 77.1%

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

      4. associate-*r* 77.1%

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

      5. *-commutative 77.1%

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

      6. associate-*r* 77.1%

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

      7. *-commutative 77.1%

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

      8. associate-*r* 77.1%

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

      9. *-commutative 77.1%

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

      10. associate-*r* 77.1%

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

      11. associate-*r/ 77.1%

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

      12. *-commutative 77.1%

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

      13. associate-/l* 77.1%

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

      14. associate-*l* 77.1%

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

   10. Simplified 77.1%

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

       1. pow1 77.1%

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

       2. associate-*l* 82.8%

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

       3. *-commutative 82.8%

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

   12. Applied egg-rr 82.8%

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

       1. unpow1 82.8%

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

       2. associate-*l* 82.8%

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

   14. Simplified 82.8%

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

   if -4.9999999999999998e-8 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 83.8%

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

   3. Simplified 83.7%

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

      1. associate-*r/ 94.6%

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

      2. clear-num 94.6%

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

      3. add-sqr-sqrt 94.6%

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

      4. pow2 94.6%

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

      5. unpow2 94.6%

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

      6. sqrt-prod 62.6%

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

      7. add-sqr-sqrt 94.6%

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

      8. *-un-lft-identity 94.6%

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

      9. times-frac 94.6%

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

      10. metadata-eval 94.6%

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

   6. Applied egg-rr 94.6%

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

      1. associate-/r/ 94.6%

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

      2. *-commutative 94.6%

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

      3. associate-*r/ 94.6%

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

      4. *-commutative 94.6%

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

      5. associate-/l* 94.6%

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

      6. associate-*l/ 94.0%

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

      7. *-commutative 94.0%

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

      8. associate-*l* 94.0%

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

   8. Simplified 94.0%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \mathsf{hypot}\left(1, D \cdot \left(0.5 \cdot \left(\frac{M}{d} \cdot \sqrt{\frac{h}{-\ell}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\ell} \cdot \left(h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.9% accurate, 0.7× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -4e-10)
   (* w0 (hypot 1.0 (* D_m (* 0.5 (* (/ M_m d) (sqrt (/ h (- l))))))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -4e-10) {
		tmp = w0 * hypot(1.0, (D_m * (0.5 * ((M_m / d) * sqrt((h / -l))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -4e-10) {
		tmp = w0 * Math.hypot(1.0, (D_m * (0.5 * ((M_m / d) * Math.sqrt((h / -l))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -4e-10:
		tmp = w0 * math.hypot(1.0, (D_m * (0.5 * ((M_m / d) * math.sqrt((h / -l))))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -4e-10)
		tmp = Float64(w0 * hypot(1.0, Float64(D_m * Float64(0.5 * Float64(Float64(M_m / d) * sqrt(Float64(h / Float64(-l))))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -4e-10)
		tmp = w0 * hypot(1.0, (D_m * (0.5 * ((M_m / d) * sqrt((h / -l))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e-10], N[(w0 * N[Sqrt[1.0 ^ 2 + N[(D$95$m * N[(0.5 * N[(N[(M$95$m / d), $MachinePrecision] * N[Sqrt[N[(h / (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $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)) < -4.00000000000000015e-10

   1. Initial program 62.9%

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

   3. Simplified 65.1%

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

      1. associate-*r/ 64.0%

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

      2. add-sqr-sqrt 64.0%

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

      3. pow2 64.0%

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

      4. unpow2 64.0%

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

      5. sqrt-prod 30.0%

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

      6. add-sqr-sqrt 64.0%

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

      7. *-un-lft-identity 64.0%

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

      8. times-frac 64.0%

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

      9. metadata-eval 64.0%

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

   6. Applied egg-rr 64.0%

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

      1. *-un-lft-identity 64.0%

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

      2. sub-neg 64.0%

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

      3. associate-/l* 65.1%

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

      4. distribute-rgt-neg-out 65.1%

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

      5. add-sqr-sqrt 65.1%

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

      6. hypot-1-def 65.2%

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

      7. sqrt-prod 70.8%

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

      8. unpow2 70.8%

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

      9. sqrt-prod 34.5%

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

      10. add-sqr-sqrt 78.6%

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

      11. associate-*r/ 78.6%

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

   8. Applied egg-rr 78.6%

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

      1. *-lft-identity 78.6%

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

      2. associate-*r/ 77.3%

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

      3. *-commutative 77.3%

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

      4. associate-*r* 77.3%

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

      5. *-commutative 77.3%

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

      6. associate-*r* 77.3%

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

      7. *-commutative 77.3%

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

      8. associate-*r* 77.3%

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

      9. *-commutative 77.3%

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

      10. associate-*r* 77.3%

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

      11. associate-*r/ 77.3%

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

      12. *-commutative 77.3%

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

      13. associate-/l* 77.4%

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

      14. associate-*l* 77.4%

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

   10. Simplified 77.4%

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

       1. pow1 77.4%

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

       2. associate-*l* 83.0%

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

       3. *-commutative 83.0%

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

   12. Applied egg-rr 83.0%

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

       1. unpow1 83.0%

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

       2. associate-*l* 83.0%

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

   14. Simplified 83.0%

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

   if -4.00000000000000015e-10 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 83.7%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in M around 0 94.6%

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

4. Final simplification 90.7%

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

Alternative 3: 71.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 7 \cdot 10^{-16}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right) \cdot \frac{h \cdot w0}{\ell \cdot {d}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 7e-16)
   w0
   (+
    w0
    (*
     -0.125
     (* (* (* M_m D_m) (* M_m D_m)) (/ (* h w0) (* l (pow d 2.0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 7e-16) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((M_m * D_m) * (M_m * D_m)) * ((h * w0) / (l * pow(d, 2.0)))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 7d-16) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((m_m * d_m) * (m_m * d_m)) * ((h * w0) / (l * (d ** 2.0d0)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 7e-16) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((M_m * D_m) * (M_m * D_m)) * ((h * w0) / (l * Math.pow(d, 2.0)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 7e-16:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (((M_m * D_m) * (M_m * D_m)) * ((h * w0) / (l * math.pow(d, 2.0)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 7e-16)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) * Float64(Float64(h * w0) / Float64(l * (d ^ 2.0))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 7e-16)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((M_m * D_m) * (M_m * D_m)) * ((h * w0) / (l * (d ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 7e-16], w0, N[(w0 + N[(-0.125 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(h * w0), $MachinePrecision] / N[(l * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.00000000000000035e-16

   1. Initial program 76.1%

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

   3. Simplified 76.6%

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

   5. Taylor expanded in M around 0 69.2%

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

   if 7.00000000000000035e-16 < M

   1. Initial program 78.8%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in M around 0 44.4%

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

      1. *-un-lft-identity 44.4%

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

      2. associate-*r* 45.9%

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

      3. pow-prod-down 61.8%

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

      4. *-commutative 61.8%

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

   7. Applied egg-rr 61.8%

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

      1. *-lft-identity 61.8%

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

      2. associate-/l* 60.4%

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

      3. *-commutative 60.4%

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

   9. Simplified 60.4%

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

       1. unpow2 60.4%

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

   11. Applied egg-rr 60.4%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7 \cdot 10^{-16}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w0}{\ell \cdot {d}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.2% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

3. Simplified 77.5%

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

5. Taylor expanded in M around 0 65.5%

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

6. Final simplification 65.5%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024052 
(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))))))