Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 91.2%

Time: 17.0s

Alternatives: 5

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.2% accurate, 0.3× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) (- INFINITY))
   (*
    w0
    (pow
     (exp
      (*
       0.25
       (+
        (log (* (/ h l) -0.25))
        (+ (* -2.0 (log d)) (+ (* -2.0 (log (/ 1.0 D))) (* 2.0 (log M)))))))
     2.0))
   (* w0 (sqrt (- 1.0 (/ h (/ l (pow (* D (/ M (* 2.0 d))) 2.0))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -((double) INFINITY)) {
		tmp = w0 * pow(exp((0.25 * (log(((h / l) * -0.25)) + ((-2.0 * log(d)) + ((-2.0 * log((1.0 / D))) + (2.0 * log(M))))))), 2.0);
	} else {
		tmp = w0 * sqrt((1.0 - (h / (l / pow((D * (M / (2.0 * d))), 2.0)))));
	}
	return tmp;
}

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.pow(Math.exp((0.25 * (Math.log(((h / l) * -0.25)) + ((-2.0 * Math.log(d)) + ((-2.0 * Math.log((1.0 / D))) + (2.0 * Math.log(M))))))), 2.0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h / (l / Math.pow((D * (M / (2.0 * d))), 2.0)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -math.inf:
		tmp = w0 * math.pow(math.exp((0.25 * (math.log(((h / l) * -0.25)) + ((-2.0 * math.log(d)) + ((-2.0 * math.log((1.0 / D))) + (2.0 * math.log(M))))))), 2.0)
	else:
		tmp = w0 * math.sqrt((1.0 - (h / (l / math.pow((D * (M / (2.0 * d))), 2.0)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= Float64(-Inf))
		tmp = Float64(w0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(h / l) * -0.25)) + Float64(Float64(-2.0 * log(d)) + Float64(Float64(-2.0 * log(Float64(1.0 / D))) + Float64(2.0 * log(M))))))) ^ 2.0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h / Float64(l / (Float64(D * Float64(M / Float64(2.0 * d))) ^ 2.0))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -Inf)
		tmp = w0 * (exp((0.25 * (log(((h / l) * -0.25)) + ((-2.0 * log(d)) + ((-2.0 * log((1.0 / D))) + (2.0 * log(M))))))) ^ 2.0);
	else
		tmp = w0 * sqrt((1.0 - (h / (l / ((D * (M / (2.0 * d))) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(w0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(N[(h / l), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] + N[(N[(-2.0 * N[Log[d], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[Log[N[(1.0 / D), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Log[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h / N[(l / N[Power[N[(D * N[(M / N[(2.0 * 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)) < -inf.0

   1. Initial program 56.3%

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

   3. Simplified 56.2%

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

      1. *-commutative 56.2%

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

      2. frac-times 56.3%

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

      3. *-commutative 56.3%

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

      4. *-commutative 56.3%

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

      5. add-sqr-sqrt 56.3%

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

      6. pow2 56.3%

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

   5. Applied egg-rr 56.3%

      \[\leadsto w0 \cdot \color{blue}{{\left({\left(1 - \frac{h}{\ell} \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}\right)}^{0.25}\right)}^{2}} \]

   6. Taylor expanded in D around inf 20.2%

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

   7. Taylor expanded in d around 0 8.5%

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

      1. +-commutative 8.5%

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

      2. distribute-lft-neg-in 8.5%

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

      3. metadata-eval 8.5%

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

      4. associate-*r/ 8.4%

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

   9. Simplified 8.4%

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

   10. Taylor expanded in M around 0 5.7%

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

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

   1. Initial program 84.3%

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

   3. Simplified 84.9%

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

      1. associate-*r/ 94.3%

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

      2. frac-times 93.7%

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

      3. *-commutative 93.7%

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

      4. clear-num 93.7%

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

      5. *-commutative 93.7%

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

      6. *-commutative 93.7%

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

      7. frac-times 94.3%

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

      8. frac-times 93.7%

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

      9. *-commutative 93.7%

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

      10. div-inv 93.7%

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

      11. *-commutative 93.7%

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

      12. associate-/r* 93.7%

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

      13. metadata-eval 93.7%

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

   5. Applied egg-rr 93.7%

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

      1. associate-/r/ 93.7%

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

      2. *-commutative 93.7%

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

      3. associate-*r* 94.3%

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

      4. associate-*r/ 94.3%

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

      5. /-rgt-identity 94.3%

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

      6. associate-/l* 94.3%

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

      7. metadata-eval 94.3%

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

      8. associate-/r* 94.3%

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

      9. associate-*r/ 93.7%

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

      10. associate-*l/ 94.3%

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

      11. *-commutative 94.3%

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

   7. Simplified 94.3%

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

      1. expm1-log1p-u 94.1%

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

      2. expm1-udef 94.1%

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

   9. Applied egg-rr 94.1%

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

       1. expm1-def 94.1%

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

       2. expm1-log1p 94.3%

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

       3. associate-/l* 94.8%

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

       4. associate-/l/ 94.8%

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

       5. *-commutative 94.8%

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

   11. Simplified 94.8%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot {\left(e^{0.25 \cdot \left(\log \left(\frac{h}{\ell} \cdot -0.25\right) + \left(-2 \cdot \log d + \left(-2 \cdot \log \left(\frac{1}{D}\right) + 2 \cdot \log M\right)\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\frac{\ell}{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}}}\\ \end{array} \]

Alternative 2: 90.0% accurate, 0.3× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) (- INFINITY))
   (*
    w0
    (pow
     (pow
      (exp 0.25)
      (+ (log (* (/ h l) -0.25)) (* -2.0 (- (log d) (log (* M D))))))
     2.0))
   (* w0 (sqrt (- 1.0 (/ h (/ l (pow (* D (/ M (* 2.0 d))) 2.0))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -((double) INFINITY)) {
		tmp = w0 * pow(pow(exp(0.25), (log(((h / l) * -0.25)) + (-2.0 * (log(d) - log((M * D)))))), 2.0);
	} else {
		tmp = w0 * sqrt((1.0 - (h / (l / pow((D * (M / (2.0 * d))), 2.0)))));
	}
	return tmp;
}

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.pow(Math.pow(Math.exp(0.25), (Math.log(((h / l) * -0.25)) + (-2.0 * (Math.log(d) - Math.log((M * D)))))), 2.0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h / (l / Math.pow((D * (M / (2.0 * d))), 2.0)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -math.inf:
		tmp = w0 * math.pow(math.pow(math.exp(0.25), (math.log(((h / l) * -0.25)) + (-2.0 * (math.log(d) - math.log((M * D)))))), 2.0)
	else:
		tmp = w0 * math.sqrt((1.0 - (h / (l / math.pow((D * (M / (2.0 * d))), 2.0)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= Float64(-Inf))
		tmp = Float64(w0 * ((exp(0.25) ^ Float64(log(Float64(Float64(h / l) * -0.25)) + Float64(-2.0 * Float64(log(d) - log(Float64(M * D)))))) ^ 2.0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h / Float64(l / (Float64(D * Float64(M / Float64(2.0 * d))) ^ 2.0))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -Inf)
		tmp = w0 * ((exp(0.25) ^ (log(((h / l) * -0.25)) + (-2.0 * (log(d) - log((M * D)))))) ^ 2.0);
	else
		tmp = w0 * sqrt((1.0 - (h / (l / ((D * (M / (2.0 * d))) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(w0 * N[Power[N[Power[N[Exp[0.25], $MachinePrecision], N[(N[Log[N[(N[(h / l), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[(N[Log[d], $MachinePrecision] - N[Log[N[(M * D), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h / N[(l / N[Power[N[(D * N[(M / N[(2.0 * 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)) < -inf.0

   1. Initial program 56.3%

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

   3. Simplified 56.2%

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

      1. *-commutative 56.2%

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

      2. frac-times 56.3%

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

      3. *-commutative 56.3%

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

      4. *-commutative 56.3%

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

      5. add-sqr-sqrt 56.3%

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

      6. pow2 56.3%

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

   5. Applied egg-rr 56.3%

      \[\leadsto w0 \cdot \color{blue}{{\left({\left(1 - \frac{h}{\ell} \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}\right)}^{0.25}\right)}^{2}} \]

   6. Taylor expanded in D around inf 20.2%

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

   7. Taylor expanded in d around 0 8.5%

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

      1. +-commutative 8.5%

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

      2. distribute-lft-neg-in 8.5%

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

      3. metadata-eval 8.5%

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

      4. associate-*r/ 8.4%

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

   9. Simplified 8.4%

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

   10. Taylor expanded in M around inf 5.7%

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

       1. exp-prod 5.5%

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

       2. distribute-lft-out 5.5%

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

       3. distribute-lft-out 5.5%

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

       4. log-rec 5.5%

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

       5. log-rec 5.5%

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

       6. distribute-neg-out 5.5%

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

       7. log-prod 15.6%

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

   12. Simplified 15.6%

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

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

   1. Initial program 84.3%

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

   3. Simplified 84.9%

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

      1. associate-*r/ 94.3%

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

      2. frac-times 93.7%

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

      3. *-commutative 93.7%

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

      4. clear-num 93.7%

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

      5. *-commutative 93.7%

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

      6. *-commutative 93.7%

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

      7. frac-times 94.3%

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

      8. frac-times 93.7%

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

      9. *-commutative 93.7%

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

      10. div-inv 93.7%

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

      11. *-commutative 93.7%

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

      12. associate-/r* 93.7%

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

      13. metadata-eval 93.7%

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

   5. Applied egg-rr 93.7%

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

      1. associate-/r/ 93.7%

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

      2. *-commutative 93.7%

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

      3. associate-*r* 94.3%

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

      4. associate-*r/ 94.3%

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

      5. /-rgt-identity 94.3%

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

      6. associate-/l* 94.3%

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

      7. metadata-eval 94.3%

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

      8. associate-/r* 94.3%

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

      9. associate-*r/ 93.7%

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

      10. associate-*l/ 94.3%

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

      11. *-commutative 94.3%

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

   7. Simplified 94.3%

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

      1. expm1-log1p-u 94.1%

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

      2. expm1-udef 94.1%

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

   9. Applied egg-rr 94.1%

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

       1. expm1-def 94.1%

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

       2. expm1-log1p 94.3%

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

       3. associate-/l* 94.8%

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

       4. associate-/l/ 94.8%

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

       5. *-commutative 94.8%

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

   11. Simplified 94.8%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\log \left(\frac{h}{\ell} \cdot -0.25\right) + -2 \cdot \left(\log d - \log \left(M \cdot D\right)\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\frac{\ell}{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}}}\\ \end{array} \]

Alternative 3: 86.2% accurate, 1.0× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* h (/ (pow (/ M (* d (/ 2.0 D))) 2.0) l))))))

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 - (h * (((m / (d_1 * (2.0d0 / d))) ** 2.0d0) / l))))
end function

Java

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

Python

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

Julia

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

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - (h * (((M / (d * (2.0 / D))) ^ 2.0) / l))));
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(M / N[(d * N[(2.0 / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

3. Simplified 77.1%

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

   1. frac-times 76.7%

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

   2. *-commutative 76.7%

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

   3. clear-num 76.7%

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

   4. un-div-inv 78.6%

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

   5. *-commutative 78.6%

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

   6. frac-times 79.1%

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

   7. frac-times 78.6%

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

   8. *-commutative 78.6%

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

   9. div-inv 78.6%

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

   10. *-commutative 78.6%

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

   11. associate-/r* 78.6%

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

   12. metadata-eval 78.6%

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

5. Applied egg-rr 78.6%

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

   1. associate-/r/ 85.3%

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

   2. *-commutative 85.3%

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

   3. *-commutative 85.3%

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

   4. associate-*r* 85.7%

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

   5. associate-*r/ 85.7%

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

   6. associate-*r/ 85.3%

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

   7. /-rgt-identity 85.3%

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

   8. associate-/l* 85.3%

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

   9. metadata-eval 85.3%

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

   10. associate-*r/ 85.3%

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

   11. associate-/l* 85.3%

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

   12. associate-/l/ 86.1%

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

7. Simplified 86.1%

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

8. Final simplification 86.1%

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

Alternative 4: 72.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4.8 \cdot 10^{-74}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 4.8e-74)
   w0
   (+ w0 (* -0.125 (/ (* (* (* M D) (* M D)) (* h w0)) (* l (pow d 2.0)))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.8e-74) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((((M * D) * (M * D)) * (h * w0)) / (l * pow(d, 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (m <= 4.8d-74) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((((m * d) * (m * d)) * (h * w0)) / (l * (d_1 ** 2.0d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.8e-74) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((((M * D) * (M * D)) * (h * w0)) / (l * Math.pow(d, 2.0))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4.8e-74:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((((M * D) * (M * D)) * (h * w0)) / (l * math.pow(d, 2.0))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4.8e-74)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(h * w0)) / Float64(l * (d ^ 2.0)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4.8e-74)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((((M * D) * (M * D)) * (h * w0)) / (l * (d ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 4.8e-74], w0, N[(w0 + N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(h * w0), $MachinePrecision]), $MachinePrecision] / N[(l * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.7999999999999998e-74

   1. Initial program 76.4%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in D around 0 71.0%

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

   if 4.7999999999999998e-74 < M

   1. Initial program 77.6%

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

   3. Simplified 78.2%

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

   4. Taylor expanded in D around 0 41.5%

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

      1. expm1-log1p-u 32.2%

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

      2. expm1-udef 32.2%

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

      3. associate-*r* 32.2%

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

      4. pow-prod-down 37.0%

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

   6. Applied egg-rr 37.0%

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

      1. expm1-def 37.0%

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

      2. expm1-log1p 46.3%

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

   8. Simplified 46.3%

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

      1. *-commutative 46.3%

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

      2. pow2 46.3%

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

   10. Applied egg-rr 46.3%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.8 \cdot 10^{-74}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(h \cdot w0\right)}{\ell \cdot {d}^{2}}\\ \end{array} \]

Alternative 5: 68.6% accurate, 216.0× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d) :precision binary64 w0)

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
end function

Java

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

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0

Julia

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

MATLAB

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

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 76.7%

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

3. Simplified 77.1%

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

4. Taylor expanded in D around 0 65.7%

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

5. Final simplification 65.7%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023312 
(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))))))