Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 88.0%

Time: 22.1s

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: 81.4% 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: 88.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D, h, l, d_m])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;{\left(\sqrt{w0\_m} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{h \cdot {\left(M\_m \cdot D\right)}^{2}}{\ell}\right) + -2 \cdot \log d\_m\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \left(M\_m \cdot \frac{0.5}{d\_m}\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D) (* 2.0 d_m)) 2.0) (/ h l)) (- INFINITY))
    (pow
     (*
      (sqrt w0_m)
      (pow
       (exp 0.25)
       (+ (log (* -0.25 (/ (* h (pow (* M_m D) 2.0)) l))) (* -2.0 (log d_m)))))
     2.0)
    (* w0_m (sqrt (- 1.0 (* h (/ (pow (* D (* M_m (/ 0.5 d_m))) 2.0) l))))))))

C

M_m = fabs(M);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D && D < h && h < l && l < d_m);
double code(double w0_s, double w0_m, double M_m, double D, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -((double) INFINITY)) {
		tmp = pow((sqrt(w0_m) * pow(exp(0.25), (log((-0.25 * ((h * pow((M_m * D), 2.0)) / l))) + (-2.0 * log(d_m))))), 2.0);
	} else {
		tmp = w0_m * sqrt((1.0 - (h * (pow((D * (M_m * (0.5 / d_m))), 2.0) / l))));
	}
	return w0_s * tmp;
}

Java

M_m = Math.abs(M);
d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M_m && M_m < D && D < h && h < l && l < d_m;
public static double code(double w0_s, double w0_m, double M_m, double D, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M_m * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.sqrt(w0_m) * Math.pow(Math.exp(0.25), (Math.log((-0.25 * ((h * Math.pow((M_m * D), 2.0)) / l))) + (-2.0 * Math.log(d_m))))), 2.0);
	} else {
		tmp = w0_m * Math.sqrt((1.0 - (h * (Math.pow((D * (M_m * (0.5 / d_m))), 2.0) / l))));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D, h, l, d_m] = sort([w0_m, M_m, D, h, l, d_m])
def code(w0_s, w0_m, M_m, D, h, l, d_m):
	tmp = 0
	if (math.pow(((M_m * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -math.inf:
		tmp = math.pow((math.sqrt(w0_m) * math.pow(math.exp(0.25), (math.log((-0.25 * ((h * math.pow((M_m * D), 2.0)) / l))) + (-2.0 * math.log(d_m))))), 2.0)
	else:
		tmp = w0_m * math.sqrt((1.0 - (h * (math.pow((D * (M_m * (0.5 / d_m))), 2.0) / l))))
	return w0_s * tmp

Julia

M_m = abs(M)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D, h, l, d_m = sort([w0_m, M_m, D, h, l, d_m])
function code(w0_s, w0_m, M_m, D, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= Float64(-Inf))
		tmp = Float64(sqrt(w0_m) * (exp(0.25) ^ Float64(log(Float64(-0.25 * Float64(Float64(h * (Float64(M_m * D) ^ 2.0)) / l))) + Float64(-2.0 * log(d_m))))) ^ 2.0;
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(h * Float64((Float64(D * Float64(M_m * Float64(0.5 / d_m))) ^ 2.0) / l)))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D, h, l, d_m = num2cell(sort([w0_m, M_m, D, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D, h, l, d_m)
	tmp = 0.0;
	if (((((M_m * D) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -Inf)
		tmp = (sqrt(w0_m) * (exp(0.25) ^ (log((-0.25 * ((h * ((M_m * D) ^ 2.0)) / l))) + (-2.0 * log(d_m))))) ^ 2.0;
	else
		tmp = w0_m * sqrt((1.0 - (h * (((D * (M_m * (0.5 / d_m))) ^ 2.0) / l))));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[Power[N[(N[Sqrt[w0$95$m], $MachinePrecision] * N[Power[N[Exp[0.25], $MachinePrecision], N[(N[Log[N[(-0.25 * N[(N[(h * N[Power[N[(M$95$m * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(D * N[(M$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.6%

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

   3. Simplified 52.7%

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

      1. clear-num 52.7%

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

      2. un-div-inv 53.7%

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

      3. add-sqr-sqrt 53.7%

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

      4. pow2 53.7%

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

      5. sqrt-pow1 53.7%

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

      6. metadata-eval 53.7%

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

      7. pow1 53.7%

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

      8. associate-/r* 53.7%

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

      9. div-inv 53.7%

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

      10. associate-/r* 53.7%

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

      11. metadata-eval 53.7%

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

   6. Applied egg-rr 53.7%

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

      1. associate-/r/ 55.0%

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

   8. Simplified 55.0%

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

      1. add-sqr-sqrt 26.4%

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

      2. pow2 26.4%

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

      3. *-commutative 26.4%

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

      4. associate-*l/ 26.4%

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

      5. associate-*r* 26.4%

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

   10. Applied egg-rr 26.4%

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

   11. Taylor expanded in d around 0 7.1%

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

       1. exp-prod 7.1%

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

       2. distribute-lft-neg-in 7.1%

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

       3. metadata-eval 7.1%

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

       4. associate-*r* 7.1%

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

       5. unpow2 7.1%

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

       6. unpow2 7.1%

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

       7. swap-sqr 11.6%

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

       8. unpow2 11.6%

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

   13. Simplified 11.6%

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

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

   1. Initial program 89.3%

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

   3. Simplified 88.8%

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

      1. clear-num 88.8%

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

      2. un-div-inv 88.8%

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

      3. add-sqr-sqrt 88.8%

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

      4. pow2 88.8%

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

      5. sqrt-pow1 88.8%

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

      6. metadata-eval 88.8%

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

      7. pow1 88.8%

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

      8. associate-/r* 88.8%

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

      9. div-inv 88.8%

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

      10. associate-/r* 88.8%

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

      11. metadata-eval 88.8%

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

   6. Applied egg-rr 88.8%

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

      1. associate-/r/ 95.3%

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

   8. Simplified 95.3%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;{\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{h \cdot {\left(M \cdot D\right)}^{2}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D, h, l, d_m])\\ \\ w0\_s \cdot \left(w0\_m \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \left(M\_m \cdot \frac{0.5}{d\_m}\right)\right)}^{2}}{\ell}}\right) \end{array} \]

FPCore

M_m = (fabs.f64 M)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D h l d_m)
 :precision binary64
 (*
  w0_s
  (* w0_m (sqrt (- 1.0 (* h (/ (pow (* D (* M_m (/ 0.5 d_m))) 2.0) l)))))))

C

M_m = fabs(M);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D && D < h && h < l && l < d_m);
double code(double w0_s, double w0_m, double M_m, double D, double h, double l, double d_m) {
	return w0_s * (w0_m * sqrt((1.0 - (h * (pow((D * (M_m * (0.5 / d_m))), 2.0) / l)))));
}

Fortran

M_m = abs(m)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
NOTE: w0_m, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0_s, w0_m, m_m, d, h, l, d_m)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m
    code = w0_s * (w0_m * sqrt((1.0d0 - (h * (((d * (m_m * (0.5d0 / d_m))) ** 2.0d0) / l)))))
end function

Java

M_m = Math.abs(M);
d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M_m && M_m < D && D < h && h < l && l < d_m;
public static double code(double w0_s, double w0_m, double M_m, double D, double h, double l, double d_m) {
	return w0_s * (w0_m * Math.sqrt((1.0 - (h * (Math.pow((D * (M_m * (0.5 / d_m))), 2.0) / l)))));
}

Python

M_m = math.fabs(M)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D, h, l, d_m] = sort([w0_m, M_m, D, h, l, d_m])
def code(w0_s, w0_m, M_m, D, h, l, d_m):
	return w0_s * (w0_m * math.sqrt((1.0 - (h * (math.pow((D * (M_m * (0.5 / d_m))), 2.0) / l)))))

Julia

M_m = abs(M)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D, h, l, d_m = sort([w0_m, M_m, D, h, l, d_m])
function code(w0_s, w0_m, M_m, D, h, l, d_m)
	return Float64(w0_s * Float64(w0_m * sqrt(Float64(1.0 - Float64(h * Float64((Float64(D * Float64(M_m * Float64(0.5 / d_m))) ^ 2.0) / l))))))
end

MATLAB

M_m = abs(M);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D, h, l, d_m = num2cell(sort([w0_m, M_m, D, h, l, d_m])){:}
function tmp = code(w0_s, w0_m, M_m, D, h, l, d_m)
	tmp = w0_s * (w0_m * sqrt((1.0 - (h * (((D * (M_m * (0.5 / d_m))) ^ 2.0) / l)))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D_, h_, l_, d$95$m_] := N[(w0$95$s * N[(w0$95$m * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(D * N[(M$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $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 76.7%

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

   1. clear-num 76.7%

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

   2. un-div-inv 77.0%

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

   3. add-sqr-sqrt 77.0%

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

   4. pow2 77.0%

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

   5. sqrt-pow1 77.0%

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

   6. metadata-eval 77.0%

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

   7. pow1 77.0%

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

   8. associate-/r* 77.0%

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

   9. div-inv 77.0%

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

   10. associate-/r* 77.0%

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

   11. metadata-eval 77.0%

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

6. Applied egg-rr 77.0%

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

   1. associate-/r/ 81.7%

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

8. Simplified 81.7%

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

9. Final simplification 81.7%

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

Alternative 3: 80.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D, h, l, d_m])\\ \\ w0\_s \cdot \left(w0\_m + -0.125 \cdot \left(h \cdot \left(w0\_m \cdot \frac{{\left(D \cdot \frac{M\_m}{d\_m}\right)}^{2}}{\ell}\right)\right)\right) \end{array} \]

FPCore

M_m = (fabs.f64 M)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D h l d_m)
 :precision binary64
 (* w0_s (+ w0_m (* -0.125 (* h (* w0_m (/ (pow (* D (/ M_m d_m)) 2.0) l)))))))

C

M_m = fabs(M);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D && D < h && h < l && l < d_m);
double code(double w0_s, double w0_m, double M_m, double D, double h, double l, double d_m) {
	return w0_s * (w0_m + (-0.125 * (h * (w0_m * (pow((D * (M_m / d_m)), 2.0) / l)))));
}

Fortran

M_m = abs(m)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
NOTE: w0_m, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0_s, w0_m, m_m, d, h, l, d_m)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m
    code = w0_s * (w0_m + ((-0.125d0) * (h * (w0_m * (((d * (m_m / d_m)) ** 2.0d0) / l)))))
end function

Java

M_m = Math.abs(M);
d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M_m && M_m < D && D < h && h < l && l < d_m;
public static double code(double w0_s, double w0_m, double M_m, double D, double h, double l, double d_m) {
	return w0_s * (w0_m + (-0.125 * (h * (w0_m * (Math.pow((D * (M_m / d_m)), 2.0) / l)))));
}

Python

M_m = math.fabs(M)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D, h, l, d_m] = sort([w0_m, M_m, D, h, l, d_m])
def code(w0_s, w0_m, M_m, D, h, l, d_m):
	return w0_s * (w0_m + (-0.125 * (h * (w0_m * (math.pow((D * (M_m / d_m)), 2.0) / l)))))

Julia

M_m = abs(M)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D, h, l, d_m = sort([w0_m, M_m, D, h, l, d_m])
function code(w0_s, w0_m, M_m, D, h, l, d_m)
	return Float64(w0_s * Float64(w0_m + Float64(-0.125 * Float64(h * Float64(w0_m * Float64((Float64(D * Float64(M_m / d_m)) ^ 2.0) / l))))))
end

MATLAB

M_m = abs(M);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D, h, l, d_m = num2cell(sort([w0_m, M_m, D, h, l, d_m])){:}
function tmp = code(w0_s, w0_m, M_m, D, h, l, d_m)
	tmp = w0_s * (w0_m + (-0.125 * (h * (w0_m * (((D * (M_m / d_m)) ^ 2.0) / l)))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D_, h_, l_, d$95$m_] := N[(w0$95$s * N[(w0$95$m + N[(-0.125 * N[(h * N[(w0$95$m * N[(N[Power[N[(D * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $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 76.7%

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

   1. clear-num 76.7%

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

   2. un-div-inv 77.0%

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

   3. add-sqr-sqrt 77.0%

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

   4. pow2 77.0%

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

   5. sqrt-pow1 77.0%

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

   6. metadata-eval 77.0%

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

   7. pow1 77.0%

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

   8. associate-/r* 77.0%

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

   9. div-inv 77.0%

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

   10. associate-/r* 77.0%

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

   11. metadata-eval 77.0%

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

6. Applied egg-rr 77.0%

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

   1. associate-/r/ 81.7%

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

8. Simplified 81.7%

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

   1. add-exp-log 33.5%

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

   2. *-commutative 33.5%

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

   3. associate-*l/ 33.5%

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

   4. associate-*r* 33.9%

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

10. Applied egg-rr 33.9%

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

11. Taylor expanded in D around 0 49.6%

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

13. Simplified 70.9%

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

14. Taylor expanded in w0 around 0 52.0%

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

    1. associate-*r* 53.6%

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

    2. times-frac 52.0%

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

    3. associate-/l* 52.0%

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

    4. unpow2 52.0%

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

    5. unpow2 52.0%

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

    6. unpow2 52.0%

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

    7. times-frac 63.1%

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

    8. swap-sqr 70.9%

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

    9. unpow2 70.9%

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

    10. associate-*r/ 74.5%

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

    11. *-commutative 74.5%

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

    12. associate-/l* 75.3%

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

16. Simplified 75.3%

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

Alternative 4: 76.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D, h, l, d_m])\\ \\ \begin{array}{l} t_0 := D \cdot \frac{M\_m}{d\_m}\\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;D \leq 3.5 \cdot 10^{+75}:\\ \;\;\;\;w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m + -0.125 \cdot \left(h \cdot \left(\frac{w0\_m}{\ell} \cdot \left(t\_0 \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D h l d_m)
 :precision binary64
 (let* ((t_0 (* D (/ M_m d_m))))
   (*
    w0_s
    (if (<= D 3.5e+75)
      w0_m
      (+ w0_m (* -0.125 (* h (* (/ w0_m l) (* t_0 t_0)))))))))

C

M_m = fabs(M);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D && D < h && h < l && l < d_m);
double code(double w0_s, double w0_m, double M_m, double D, double h, double l, double d_m) {
	double t_0 = D * (M_m / d_m);
	double tmp;
	if (D <= 3.5e+75) {
		tmp = w0_m;
	} else {
		tmp = w0_m + (-0.125 * (h * ((w0_m / l) * (t_0 * t_0))));
	}
	return w0_s * tmp;
}

Fortran

M_m = abs(m)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
NOTE: w0_m, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0_s, w0_m, m_m, d, h, l, d_m)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * (m_m / d_m)
    if (d <= 3.5d+75) then
        tmp = w0_m
    else
        tmp = w0_m + ((-0.125d0) * (h * ((w0_m / l) * (t_0 * t_0))))
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M_m && M_m < D && D < h && h < l && l < d_m;
public static double code(double w0_s, double w0_m, double M_m, double D, double h, double l, double d_m) {
	double t_0 = D * (M_m / d_m);
	double tmp;
	if (D <= 3.5e+75) {
		tmp = w0_m;
	} else {
		tmp = w0_m + (-0.125 * (h * ((w0_m / l) * (t_0 * t_0))));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D, h, l, d_m] = sort([w0_m, M_m, D, h, l, d_m])
def code(w0_s, w0_m, M_m, D, h, l, d_m):
	t_0 = D * (M_m / d_m)
	tmp = 0
	if D <= 3.5e+75:
		tmp = w0_m
	else:
		tmp = w0_m + (-0.125 * (h * ((w0_m / l) * (t_0 * t_0))))
	return w0_s * tmp

Julia

M_m = abs(M)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D, h, l, d_m = sort([w0_m, M_m, D, h, l, d_m])
function code(w0_s, w0_m, M_m, D, h, l, d_m)
	t_0 = Float64(D * Float64(M_m / d_m))
	tmp = 0.0
	if (D <= 3.5e+75)
		tmp = w0_m;
	else
		tmp = Float64(w0_m + Float64(-0.125 * Float64(h * Float64(Float64(w0_m / l) * Float64(t_0 * t_0)))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D, h, l, d_m = num2cell(sort([w0_m, M_m, D, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D, h, l, d_m)
	t_0 = D * (M_m / d_m);
	tmp = 0.0;
	if (D <= 3.5e+75)
		tmp = w0_m;
	else
		tmp = w0_m + (-0.125 * (h * ((w0_m / l) * (t_0 * t_0))));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(D * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[D, 3.5e+75], w0$95$m, N[(w0$95$m + N[(-0.125 * N[(h * N[(N[(w0$95$m / l), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 3.4999999999999998e75

   1. Initial program 78.3%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in D around 0 64.7%

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

   if 3.4999999999999998e75 < D

   1. Initial program 66.7%

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

   3. Simplified 69.4%

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

      1. clear-num 69.4%

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

      2. un-div-inv 69.4%

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

      3. add-sqr-sqrt 69.4%

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

      4. pow2 69.4%

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

      5. sqrt-pow1 69.4%

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

      6. metadata-eval 69.4%

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

      7. pow1 69.4%

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

      8. associate-/r* 69.4%

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

      9. div-inv 69.4%

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

      10. associate-/r* 69.4%

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

      11. metadata-eval 69.4%

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

   6. Applied egg-rr 69.4%

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

      1. associate-/r/ 69.4%

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

   8. Simplified 69.4%

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

      1. add-exp-log 25.5%

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

      2. *-commutative 25.5%

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

      3. associate-*l/ 25.5%

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

      4. associate-*r* 23.1%

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

   10. Applied egg-rr 23.1%

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

   11. Taylor expanded in D around 0 39.2%

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

   13. Simplified 61.6%

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

       1. unpow2 61.6%

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

   15. Applied egg-rr 61.6%

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

Alternative 5: 68.3% accurate, 216.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D, h, l, d_m])\\ \\ w0\_s \cdot w0\_m \end{array} \]

FPCore

M_m = (fabs.f64 M)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D h l d_m) :precision binary64 (* w0_s w0_m))

C

M_m = fabs(M);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D && D < h && h < l && l < d_m);
double code(double w0_s, double w0_m, double M_m, double D, double h, double l, double d_m) {
	return w0_s * w0_m;
}

Fortran

M_m = abs(m)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
NOTE: w0_m, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0_s, w0_m, m_m, d, h, l, d_m)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m
    code = w0_s * w0_m
end function

Java

M_m = Math.abs(M);
d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M_m && M_m < D && D < h && h < l && l < d_m;
public static double code(double w0_s, double w0_m, double M_m, double D, double h, double l, double d_m) {
	return w0_s * w0_m;
}

Python

M_m = math.fabs(M)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D, h, l, d_m] = sort([w0_m, M_m, D, h, l, d_m])
def code(w0_s, w0_m, M_m, D, h, l, d_m):
	return w0_s * w0_m

Julia

M_m = abs(M)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D, h, l, d_m = sort([w0_m, M_m, D, h, l, d_m])
function code(w0_s, w0_m, M_m, D, h, l, d_m)
	return Float64(w0_s * w0_m)
end

MATLAB

M_m = abs(M);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D, h, l, d_m = num2cell(sort([w0_m, M_m, D, h, l, d_m])){:}
function tmp = code(w0_s, w0_m, M_m, D, h, l, d_m)
	tmp = w0_s * w0_m;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D_, h_, l_, d$95$m_] := N[(w0$95$s * w0$95$m), $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 76.7%

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

5. Taylor expanded in D around 0 60.1%

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

Reproduce

herbie shell --seed 2024090 
(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))))))