Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 89.0%

Time: 19.4s

Alternatives: 4

Speedup: 1.8×

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: 89.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;w0\_m \cdot \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 5 \cdot 10^{+92}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\left(M\_m \cdot D\_m\right) \cdot \frac{0.5}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{M\_m \cdot \frac{0.5}{\frac{d}{D\_m}}}{\ell} \cdot \left(h \cdot \left(M\_m \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\right)\right)}\\ \end{array} \end{array} \]

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_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (<=
       (* w0_m (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)))))
       5e+92)
    (* w0_m (sqrt (- 1.0 (* (/ h l) (pow (* (* M_m D_m) (/ 0.5 d)) 2.0)))))
    (*
     w0_m
     (sqrt
      (-
       1.0
       (*
        (/ (* M_m (/ 0.5 (/ d D_m))) l)
        (* h (* M_m (* D_m (/ 0.5 d)))))))))))

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_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((w0_m * sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 5e+92) {
		tmp = w0_m * sqrt((1.0 - ((h / l) * pow(((M_m * D_m) * (0.5 / d)), 2.0))));
	} else {
		tmp = w0_m * sqrt((1.0 - (((M_m * (0.5 / (d / D_m))) / l) * (h * (M_m * (D_m * (0.5 / d)))))));
	}
	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_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((w0_m * sqrt((1.0d0 - ((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l))))) <= 5d+92) then
        tmp = w0_m * sqrt((1.0d0 - ((h / l) * (((m_m * d_m) * (0.5d0 / d)) ** 2.0d0))))
    else
        tmp = w0_m * sqrt((1.0d0 - (((m_m * (0.5d0 / (d / d_m))) / l) * (h * (m_m * (d_m * (0.5d0 / d)))))))
    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_m && D_m < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((w0_m * Math.sqrt((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 5e+92) {
		tmp = w0_m * Math.sqrt((1.0 - ((h / l) * Math.pow(((M_m * D_m) * (0.5 / d)), 2.0))));
	} else {
		tmp = w0_m * Math.sqrt((1.0 - (((M_m * (0.5 / (d / D_m))) / l) * (h * (M_m * (D_m * (0.5 / d)))))));
	}
	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_m, h, l, d] = sort([w0_m, M_m, D_m, h, l, d])
def code(w0_s, w0_m, M_m, D_m, h, l, d):
	tmp = 0
	if (w0_m * math.sqrt((1.0 - (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 5e+92:
		tmp = w0_m * math.sqrt((1.0 - ((h / l) * math.pow(((M_m * D_m) * (0.5 / d)), 2.0))))
	else:
		tmp = w0_m * math.sqrt((1.0 - (((M_m * (0.5 / (d / D_m))) / l) * (h * (M_m * (D_m * (0.5 / d)))))))
	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_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(w0_m * sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= 5e+92)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) * Float64(0.5 / d)) ^ 2.0)))));
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(Float64(M_m * Float64(0.5 / Float64(d / D_m))) / l) * Float64(h * Float64(M_m * Float64(D_m * Float64(0.5 / d))))))));
	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_m, h, l, d = num2cell(sort([w0_m, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((w0_m * sqrt((1.0 - ((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l))))) <= 5e+92)
		tmp = w0_m * sqrt((1.0 - ((h / l) * (((M_m * D_m) * (0.5 / d)) ^ 2.0))));
	else
		tmp = w0_m * sqrt((1.0 - (((M_m * (0.5 / (d / D_m))) / l) * (h * (M_m * (D_m * (0.5 / d)))))));
	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_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+92], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(N[(M$95$m * N[(0.5 / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(h * N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < 5.00000000000000022e92

   1. Initial program 95.6%

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

      1. clear-num 95.7%

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

      2. associate-/r/ 95.6%

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

      3. associate-/r* 95.6%

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

      4. metadata-eval 95.6%

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

   4. Applied egg-rr 95.6%

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

   if 5.00000000000000022e92 < (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))))

   1. Initial program 52.6%

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

   3. Simplified 55.1%

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

      1. associate-*r/ 70.3%

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

      2. add-sqr-sqrt 70.3%

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

      3. pow2 70.3%

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

      4. sqrt-pow1 70.3%

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

      5. metadata-eval 70.3%

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

      6. pow1 70.3%

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

      7. *-un-lft-identity 70.3%

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

      8. times-frac 70.3%

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

      9. metadata-eval 70.3%

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

   6. Applied egg-rr 70.3%

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

      1. associate-*l/ 70.3%

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

      2. associate-/r/ 56.4%

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

      3. unpow2 56.4%

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

      4. div-inv 56.4%

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

      5. times-frac 80.0%

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

      6. clear-num 80.0%

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

      7. un-div-inv 80.0%

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

      8. clear-num 80.0%

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

      9. un-div-inv 80.0%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{M \cdot \frac{0.5}{\frac{d}{D}}}{\ell} \cdot \frac{M \cdot \color{blue}{\frac{0.5}{\frac{d}{D}}}}{\frac{1}{h}}} \]

   8. Applied egg-rr 80.0%

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

      1. associate-/l* 77.5%

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

      2. associate-/r/ 77.5%

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

      3. associate-/r/ 77.5%

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

      4. /-rgt-identity 77.5%

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

      5. associate-/r/ 77.5%

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

   10. Simplified 77.5%

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

       1. associate-*r/ 80.0%

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

       2. associate-/r/ 80.0%

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

   12. Applied egg-rr 80.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 5 \cdot 10^{+92}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot \frac{0.5}{\frac{d}{D}}}{\ell} \cdot \left(h \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.7% accurate, 1.8× speedup

Math

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

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_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (*
   w0_m
   (sqrt
    (-
     1.0
     (* (/ (* M_m (/ 0.5 (/ d D_m))) l) (* h (* M_m (* D_m (/ 0.5 d))))))))))

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_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	return w0_s * (w0_m * sqrt((1.0 - (((M_m * (0.5 / (d / D_m))) / l) * (h * (M_m * (D_m * (0.5 / d))))))));
}

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_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0_s * (w0_m * sqrt((1.0d0 - (((m_m * (0.5d0 / (d / d_m))) / l) * (h * (m_m * (d_m * (0.5d0 / d))))))))
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_m && D_m < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	return w0_s * (w0_m * Math.sqrt((1.0 - (((M_m * (0.5 / (d / D_m))) / l) * (h * (M_m * (D_m * (0.5 / d))))))));
}

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_m, h, l, d] = sort([w0_m, M_m, D_m, h, l, d])
def code(w0_s, w0_m, M_m, D_m, h, l, d):
	return w0_s * (w0_m * math.sqrt((1.0 - (((M_m * (0.5 / (d / D_m))) / l) * (h * (M_m * (D_m * (0.5 / d))))))))

Julia

M_m = abs(M)
D_m = abs(D)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	return Float64(w0_s * Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(Float64(M_m * Float64(0.5 / Float64(d / D_m))) / l) * Float64(h * Float64(M_m * Float64(D_m * Float64(0.5 / d)))))))))
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_m, h, l, d = num2cell(sort([w0_m, M_m, D_m, h, l, d])){:}
function tmp = code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = w0_s * (w0_m * sqrt((1.0 - (((M_m * (0.5 / (d / D_m))) / l) * (h * (M_m * (D_m * (0.5 / d))))))));
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_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(N[(M$95$m * N[(0.5 / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(h * N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

3. Simplified 81.7%

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

   1. associate-*r/ 87.0%

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

   2. add-sqr-sqrt 87.0%

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

   3. pow2 87.0%

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

   4. sqrt-pow1 87.0%

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

   5. metadata-eval 87.0%

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

   6. pow1 87.0%

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

   7. *-un-lft-identity 87.0%

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

   8. times-frac 87.0%

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

   9. metadata-eval 87.0%

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

6. Applied egg-rr 87.0%

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

   1. associate-*l/ 85.9%

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

   2. associate-/r/ 82.1%

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

   3. unpow2 82.1%

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

   4. div-inv 82.1%

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

   5. times-frac 89.9%

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

   6. clear-num 89.8%

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

   7. un-div-inv 89.8%

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

   8. clear-num 89.8%

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

   9. un-div-inv 89.8%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{M \cdot \frac{0.5}{\frac{d}{D}}}{\ell} \cdot \frac{M \cdot \color{blue}{\frac{0.5}{\frac{d}{D}}}}{\frac{1}{h}}} \]

8. Applied egg-rr 89.8%

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

   1. associate-/l* 88.4%

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

   2. associate-/r/ 88.4%

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

   3. associate-/r/ 88.4%

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

   4. /-rgt-identity 88.4%

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

   5. associate-/r/ 88.4%

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

10. Simplified 88.4%

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

    1. associate-*r/ 89.8%

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

    2. associate-/r/ 89.9%

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

12. Applied egg-rr 89.9%

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

13. Final simplification 89.9%

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

Alternative 3: 86.6% accurate, 1.8× speedup

Math

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

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_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (*
   w0_m
   (sqrt
    (-
     1.0
     (* (* h (* M_m (* D_m (/ 0.5 d)))) (* 0.5 (* (/ D_m d) (/ M_m 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_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	return w0_s * (w0_m * sqrt((1.0 - ((h * (M_m * (D_m * (0.5 / d)))) * (0.5 * ((D_m / d) * (M_m / 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_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0_s * (w0_m * sqrt((1.0d0 - ((h * (m_m * (d_m * (0.5d0 / d)))) * (0.5d0 * ((d_m / d) * (m_m / 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_m && D_m < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	return w0_s * (w0_m * Math.sqrt((1.0 - ((h * (M_m * (D_m * (0.5 / d)))) * (0.5 * ((D_m / d) * (M_m / 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_m, h, l, d] = sort([w0_m, M_m, D_m, h, l, d])
def code(w0_s, w0_m, M_m, D_m, h, l, d):
	return w0_s * (w0_m * math.sqrt((1.0 - ((h * (M_m * (D_m * (0.5 / d)))) * (0.5 * ((D_m / d) * (M_m / 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_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	return Float64(w0_s * Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h * Float64(M_m * Float64(D_m * Float64(0.5 / d)))) * Float64(0.5 * Float64(Float64(D_m / d) * Float64(M_m / 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_m, h, l, d = num2cell(sort([w0_m, M_m, D_m, h, l, d])){:}
function tmp = code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = w0_s * (w0_m * sqrt((1.0 - ((h * (M_m * (D_m * (0.5 / d)))) * (0.5 * ((D_m / d) * (M_m / 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_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h * N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

3. Simplified 81.7%

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

   1. associate-*r/ 87.0%

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

   2. add-sqr-sqrt 87.0%

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

   3. pow2 87.0%

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

   4. sqrt-pow1 87.0%

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

   5. metadata-eval 87.0%

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

   6. pow1 87.0%

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

   7. *-un-lft-identity 87.0%

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

   8. times-frac 87.0%

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

   9. metadata-eval 87.0%

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

6. Applied egg-rr 87.0%

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

   1. associate-*l/ 85.9%

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

   2. associate-/r/ 82.1%

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

   3. unpow2 82.1%

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

   4. div-inv 82.1%

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

   5. times-frac 89.9%

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

   6. clear-num 89.8%

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

   7. un-div-inv 89.8%

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

   8. clear-num 89.8%

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

   9. un-div-inv 89.8%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{M \cdot \frac{0.5}{\frac{d}{D}}}{\ell} \cdot \frac{M \cdot \color{blue}{\frac{0.5}{\frac{d}{D}}}}{\frac{1}{h}}} \]

8. Applied egg-rr 89.8%

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

   1. associate-/l* 88.4%

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

   2. associate-/r/ 88.4%

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

   3. associate-/r/ 88.4%

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

   4. /-rgt-identity 88.4%

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

   5. associate-/r/ 88.4%

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

10. Simplified 88.4%

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

11. Taylor expanded in M around 0 85.7%

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

    1. times-frac 87.5%

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

13. Simplified 87.5%

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

14. Final simplification 87.5%

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

Alternative 4: 67.7% accurate, 216.0× speedup

Math

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

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_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d) :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_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	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_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0_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_m && D_m < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	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_m, h, l, d] = sort([w0_m, M_m, D_m, h, l, d])
def code(w0_s, w0_m, M_m, D_m, h, l, d):
	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_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	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_m, h, l, d = num2cell(sort([w0_m, M_m, D_m, h, l, d])){:}
function tmp = code(w0_s, w0_m, M_m, D_m, h, l, d)
	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_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * w0$95$m), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

3. Simplified 81.7%

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

5. Taylor expanded in M around 0 72.0%

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

Reproduce

herbie shell --seed 2024111 
(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))))))