Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.6% → 88.6%

Time: 15.4s

Alternatives: 7

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: 80.6% 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: 88.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;w0\_m \cdot \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 2 \cdot 10^{+221}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - h \cdot \left(\left(0.5 \cdot \frac{M\_m \cdot D\_m}{d}\right) \cdot \frac{\frac{D\_m \cdot \left(M\_m \cdot 0.5\right)}{d}}{\ell}\right)}\\ \end{array} \end{array} \]

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)))))
       2e+221)
    (* w0_m (sqrt (- 1.0 (* (/ h l) (pow (* D_m (/ (/ M_m 2.0) d)) 2.0)))))
    (*
     w0_m
     (sqrt
      (-
       1.0
       (*
        h
        (* (* 0.5 (/ (* M_m D_m) d)) (/ (/ (* D_m (* M_m 0.5)) d) 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) {
	double tmp;
	if ((w0_m * sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 2e+221) {
		tmp = w0_m * sqrt((1.0 - ((h / l) * pow((D_m * ((M_m / 2.0) / d)), 2.0))));
	} else {
		tmp = w0_m * sqrt((1.0 - (h * ((0.5 * ((M_m * D_m) / d)) * (((D_m * (M_m * 0.5)) / d) / l)))));
	}
	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))))) <= 2d+221) then
        tmp = w0_m * sqrt((1.0d0 - ((h / l) * ((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0))))
    else
        tmp = w0_m * sqrt((1.0d0 - (h * ((0.5d0 * ((m_m * d_m) / d)) * (((d_m * (m_m * 0.5d0)) / d) / l)))))
    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))))) <= 2e+221) {
		tmp = w0_m * Math.sqrt((1.0 - ((h / l) * Math.pow((D_m * ((M_m / 2.0) / d)), 2.0))));
	} else {
		tmp = w0_m * Math.sqrt((1.0 - (h * ((0.5 * ((M_m * D_m) / d)) * (((D_m * (M_m * 0.5)) / d) / 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_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))))) <= 2e+221:
		tmp = w0_m * math.sqrt((1.0 - ((h / l) * math.pow((D_m * ((M_m / 2.0) / d)), 2.0))))
	else:
		tmp = w0_m * math.sqrt((1.0 - (h * ((0.5 * ((M_m * D_m) / d)) * (((D_m * (M_m * 0.5)) / d) / 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_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))))) <= 2e+221)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0)))));
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(h * Float64(Float64(0.5 * Float64(Float64(M_m * D_m) / d)) * Float64(Float64(Float64(D_m * Float64(M_m * 0.5)) / d) / 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_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))))) <= 2e+221)
		tmp = w0_m * sqrt((1.0 - ((h / l) * ((D_m * ((M_m / 2.0) / d)) ^ 2.0))));
	else
		tmp = w0_m * sqrt((1.0 - (h * ((0.5 * ((M_m * D_m) / d)) * (((D_m * (M_m * 0.5)) / d) / 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_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], 2e+221], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(h * N[(N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / l), $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))))) < 2.0000000000000001e221

   1. Initial program 93.2%

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

   3. Simplified 91.2%

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

   if 2.0000000000000001e221 < (*.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 46.0%

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

   3. Simplified 47.5%

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

      1. associate-*r/ 68.3%

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

      2. clear-num 68.3%

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

      3. *-commutative 68.3%

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

      4. associate-/r* 68.3%

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

   6. Applied egg-rr 68.3%

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

      1. clear-num 68.3%

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

      2. associate-/l* 70.9%

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

      3. clear-num 70.9%

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

      4. *-commutative 70.9%

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

      5. associate-*r/ 70.9%

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

      6. div-inv 70.9%

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

      7. *-un-lft-identity 70.9%

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

      8. *-commutative 70.9%

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

      9. times-frac 70.9%

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

      10. clear-num 70.9%

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

      11. div-inv 70.9%

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

      12. metadata-eval 70.9%

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

   8. Applied egg-rr 70.9%

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

      1. unpow2 70.9%

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

      2. *-un-lft-identity 70.9%

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

      3. times-frac 76.3%

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

      4. associate-*r/ 72.3%

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

      5. associate-*r/ 72.3%

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

   10. Applied egg-rr 72.3%

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

   11. Taylor expanded in M around 0 72.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 2 \cdot 10^{+221}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \frac{\frac{D \cdot \left(M \cdot 0.5\right)}{d}}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.1% accurate, 1.8× speedup

Math

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

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 (* (* 0.5 (/ (* M_m D_m) d)) (/ (/ (* D_m (* M_m 0.5)) d) 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 * ((0.5 * ((M_m * D_m) / d)) * (((D_m * (M_m * 0.5)) / d) / 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 * ((0.5d0 * ((m_m * d_m) / d)) * (((d_m * (m_m * 0.5d0)) / d) / 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 * ((0.5 * ((M_m * D_m) / d)) * (((D_m * (M_m * 0.5)) / d) / 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 * ((0.5 * ((M_m * D_m) / d)) * (((D_m * (M_m * 0.5)) / d) / 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(h * Float64(Float64(0.5 * Float64(Float64(M_m * D_m) / d)) * Float64(Float64(Float64(D_m * Float64(M_m * 0.5)) / d) / 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 * ((0.5 * ((M_m * D_m) / d)) * (((D_m * (M_m * 0.5)) / d) / 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[(h * N[(N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 78.9%

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

   1. associate-*r/ 84.4%

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

   2. clear-num 84.4%

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

   3. *-commutative 84.4%

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

   4. associate-/r* 84.4%

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

6. Applied egg-rr 84.4%

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

   1. clear-num 84.4%

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

   2. associate-/l* 84.3%

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

   3. clear-num 84.3%

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

   4. *-commutative 84.3%

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

   5. associate-*r/ 84.3%

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

   6. div-inv 84.3%

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

   7. *-un-lft-identity 84.3%

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

   8. *-commutative 84.3%

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

   9. times-frac 84.0%

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

   10. clear-num 84.0%

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

   11. div-inv 84.0%

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

   12. metadata-eval 84.0%

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

8. Applied egg-rr 84.0%

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

   1. unpow2 84.0%

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

   2. *-un-lft-identity 84.0%

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

   3. times-frac 85.5%

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

   4. associate-*r/ 84.4%

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

   5. associate-*r/ 85.5%

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

10. Applied egg-rr 85.5%

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

11. Taylor expanded in M around 0 85.5%

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

12. Final simplification 85.5%

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

Alternative 3: 76.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;D\_m \leq 3.2 \cdot 10^{+93}:\\ \;\;\;\;w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m + -0.125 \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \left(w0\_m \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \end{array} \]

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 (<= D_m 3.2e+93)
    w0_m
    (+ w0_m (* -0.125 (* (pow (* D_m (/ M_m d)) 2.0) (* w0_m (/ h 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) {
	double tmp;
	if (D_m <= 3.2e+93) {
		tmp = w0_m;
	} else {
		tmp = w0_m + (-0.125 * (pow((D_m * (M_m / d)), 2.0) * (w0_m * (h / l))));
	}
	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 (d_m <= 3.2d+93) then
        tmp = w0_m
    else
        tmp = w0_m + ((-0.125d0) * (((d_m * (m_m / d)) ** 2.0d0) * (w0_m * (h / l))))
    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 (D_m <= 3.2e+93) {
		tmp = w0_m;
	} else {
		tmp = w0_m + (-0.125 * (Math.pow((D_m * (M_m / d)), 2.0) * (w0_m * (h / 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_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 D_m <= 3.2e+93:
		tmp = w0_m
	else:
		tmp = w0_m + (-0.125 * (math.pow((D_m * (M_m / d)), 2.0) * (w0_m * (h / 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_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 (D_m <= 3.2e+93)
		tmp = w0_m;
	else
		tmp = Float64(w0_m + Float64(-0.125 * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(w0_m * Float64(h / 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_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 (D_m <= 3.2e+93)
		tmp = w0_m;
	else
		tmp = w0_m + (-0.125 * (((D_m * (M_m / d)) ^ 2.0) * (w0_m * (h / 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_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[D$95$m, 3.2e+93], w0$95$m, N[(w0$95$m + N[(-0.125 * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(w0$95$m * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if D < 3.2000000000000001e93

   1. Initial program 79.5%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in D around 0 68.6%

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

   if 3.2000000000000001e93 < D

   1. Initial program 82.7%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in D around 0 31.1%

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

      1. associate-/l* 31.0%

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

      2. times-frac 36.1%

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

      3. *-commutative 36.1%

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

   7. Simplified 36.1%

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

      1. expm1-log1p-u 36.1%

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

      2. expm1-undefine 33.5%

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

      3. add-sqr-sqrt 33.5%

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

      4. pow2 33.5%

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

      5. sqrt-div 33.5%

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

      6. sqrt-pow1 36.2%

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

      7. metadata-eval 36.2%

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

      8. pow1 36.2%

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

      9. sqrt-pow1 41.8%

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

      10. metadata-eval 41.8%

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

      11. pow1 41.8%

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

   9. Applied egg-rr 41.8%

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

       1. expm1-define 44.4%

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

   11. Simplified 44.4%

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

       1. pow-to-exp 44.4%

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

       2. *-commutative 44.4%

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

       3. expm1-log1p-u 44.4%

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

       4. pow2 44.4%

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

       5. frac-times 36.1%

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

       6. unpow2 36.1%

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

       7. unpow2 36.1%

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

       8. pow1 36.1%

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

   13. Applied egg-rr 75.9%

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

       1. unpow1 75.9%

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

   15. Simplified 75.9%

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

Alternative 4: 73.2% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;M\_m \leq 118000:\\ \;\;\;\;w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m + -0.125 \cdot \left(\left(D\_m \cdot D\_m\right) \cdot \frac{\left(M\_m \cdot \frac{M\_m}{d}\right) \cdot \left(w0\_m \cdot h\right)}{d \cdot \ell}\right)\\ \end{array} \end{array} \]

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 (<= M_m 118000.0)
    w0_m
    (+
     w0_m
     (*
      -0.125
      (* (* D_m D_m) (/ (* (* M_m (/ M_m d)) (* w0_m h)) (* d 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) {
	double tmp;
	if (M_m <= 118000.0) {
		tmp = w0_m;
	} else {
		tmp = w0_m + (-0.125 * ((D_m * D_m) * (((M_m * (M_m / d)) * (w0_m * h)) / (d * l))));
	}
	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 (m_m <= 118000.0d0) then
        tmp = w0_m
    else
        tmp = w0_m + ((-0.125d0) * ((d_m * d_m) * (((m_m * (m_m / d)) * (w0_m * h)) / (d * l))))
    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 (M_m <= 118000.0) {
		tmp = w0_m;
	} else {
		tmp = w0_m + (-0.125 * ((D_m * D_m) * (((M_m * (M_m / d)) * (w0_m * h)) / (d * 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_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 M_m <= 118000.0:
		tmp = w0_m
	else:
		tmp = w0_m + (-0.125 * ((D_m * D_m) * (((M_m * (M_m / d)) * (w0_m * h)) / (d * 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_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 (M_m <= 118000.0)
		tmp = w0_m;
	else
		tmp = Float64(w0_m + Float64(-0.125 * Float64(Float64(D_m * D_m) * Float64(Float64(Float64(M_m * Float64(M_m / d)) * Float64(w0_m * h)) / Float64(d * 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_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 (M_m <= 118000.0)
		tmp = w0_m;
	else
		tmp = w0_m + (-0.125 * ((D_m * D_m) * (((M_m * (M_m / d)) * (w0_m * h)) / (d * 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_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[M$95$m, 118000.0], w0$95$m, N[(w0$95$m + N[(-0.125 * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(w0$95$m * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if M < 118000

   1. Initial program 81.3%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in D around 0 70.6%

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

   if 118000 < M

   1. Initial program 74.9%

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

   3. Simplified 76.6%

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

   5. Taylor expanded in D around 0 46.6%

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

      1. associate-/l* 46.6%

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

      2. times-frac 48.4%

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

      3. *-commutative 48.4%

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

   7. Simplified 48.4%

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

      1. expm1-log1p-u 48.4%

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

      2. expm1-undefine 46.5%

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

      3. add-sqr-sqrt 46.5%

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

      4. pow2 46.5%

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

      5. sqrt-div 46.5%

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

      6. sqrt-pow1 59.5%

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

      7. metadata-eval 59.5%

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

      8. pow1 59.5%

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

      9. sqrt-pow1 59.6%

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

      10. metadata-eval 59.6%

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

      11. pow1 59.6%

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

   9. Applied egg-rr 59.6%

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

       1. expm1-define 61.5%

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

   11. Simplified 61.5%

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

       1. expm1-log1p-u 61.5%

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

       2. pow2 61.5%

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

       3. associate-*r/ 61.5%

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

       4. frac-times 59.6%

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

       5. *-commutative 59.6%

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

   13. Applied egg-rr 59.6%

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

       1. unpow2 59.6%

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

   15. Applied egg-rr 59.6%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 118000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot \frac{M}{d}\right) \cdot \left(w0 \cdot h\right)}{d \cdot \ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;M\_m \leq 2050000:\\ \;\;\;\;w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m + -0.125 \cdot \left(\left(D\_m \cdot D\_m\right) \cdot \left(\left(\frac{M\_m}{d} \cdot \frac{M\_m}{d}\right) \cdot \frac{w0\_m \cdot h}{\ell}\right)\right)\\ \end{array} \end{array} \]

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 (<= M_m 2050000.0)
    w0_m
    (+
     w0_m
     (*
      -0.125
      (* (* D_m D_m) (* (* (/ M_m d) (/ M_m d)) (/ (* w0_m h) 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) {
	double tmp;
	if (M_m <= 2050000.0) {
		tmp = w0_m;
	} else {
		tmp = w0_m + (-0.125 * ((D_m * D_m) * (((M_m / d) * (M_m / d)) * ((w0_m * h) / l))));
	}
	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 (m_m <= 2050000.0d0) then
        tmp = w0_m
    else
        tmp = w0_m + ((-0.125d0) * ((d_m * d_m) * (((m_m / d) * (m_m / d)) * ((w0_m * h) / l))))
    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 (M_m <= 2050000.0) {
		tmp = w0_m;
	} else {
		tmp = w0_m + (-0.125 * ((D_m * D_m) * (((M_m / d) * (M_m / d)) * ((w0_m * h) / 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_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 M_m <= 2050000.0:
		tmp = w0_m
	else:
		tmp = w0_m + (-0.125 * ((D_m * D_m) * (((M_m / d) * (M_m / d)) * ((w0_m * h) / 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_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 (M_m <= 2050000.0)
		tmp = w0_m;
	else
		tmp = Float64(w0_m + Float64(-0.125 * Float64(Float64(D_m * D_m) * Float64(Float64(Float64(M_m / d) * Float64(M_m / d)) * Float64(Float64(w0_m * h) / 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_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 (M_m <= 2050000.0)
		tmp = w0_m;
	else
		tmp = w0_m + (-0.125 * ((D_m * D_m) * (((M_m / d) * (M_m / d)) * ((w0_m * h) / 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_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[M$95$m, 2050000.0], w0$95$m, N[(w0$95$m + N[(-0.125 * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(w0$95$m * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if M < 2.05e6

   1. Initial program 81.4%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in D around 0 70.3%

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

   if 2.05e6 < M

   1. Initial program 74.4%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in D around 0 47.5%

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

      1. associate-/l* 47.5%

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

      2. times-frac 49.3%

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

      3. *-commutative 49.3%

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

   7. Simplified 49.3%

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

      1. unpow2 49.3%

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

      2. unpow2 49.3%

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

      3. times-frac 62.6%

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

   9. Applied egg-rr 62.6%

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

       1. unpow2 60.8%

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

   11. Applied egg-rr 62.6%

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

Alternative 6: 72.9% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;M\_m \leq 2500000:\\ \;\;\;\;w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m + -0.125 \cdot \left(\left(D\_m \cdot D\_m\right) \cdot \left(\left(w0\_m \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M\_m}{d} \cdot \frac{M\_m}{d}\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= M_m 2500000.0)
    w0_m
    (+
     w0_m
     (*
      -0.125
      (* (* D_m D_m) (* (* w0_m (/ h l)) (* (/ M_m d) (/ M_m 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 (M_m <= 2500000.0) {
		tmp = w0_m;
	} else {
		tmp = w0_m + (-0.125 * ((D_m * D_m) * ((w0_m * (h / l)) * ((M_m / d) * (M_m / 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 (m_m <= 2500000.0d0) then
        tmp = w0_m
    else
        tmp = w0_m + ((-0.125d0) * ((d_m * d_m) * ((w0_m * (h / l)) * ((m_m / d) * (m_m / 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 (M_m <= 2500000.0) {
		tmp = w0_m;
	} else {
		tmp = w0_m + (-0.125 * ((D_m * D_m) * ((w0_m * (h / l)) * ((M_m / d) * (M_m / 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 M_m <= 2500000.0:
		tmp = w0_m
	else:
		tmp = w0_m + (-0.125 * ((D_m * D_m) * ((w0_m * (h / l)) * ((M_m / d) * (M_m / 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 (M_m <= 2500000.0)
		tmp = w0_m;
	else
		tmp = Float64(w0_m + Float64(-0.125 * Float64(Float64(D_m * D_m) * Float64(Float64(w0_m * Float64(h / l)) * Float64(Float64(M_m / d) * Float64(M_m / 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 (M_m <= 2500000.0)
		tmp = w0_m;
	else
		tmp = w0_m + (-0.125 * ((D_m * D_m) * ((w0_m * (h / l)) * ((M_m / d) * (M_m / 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[M$95$m, 2500000.0], w0$95$m, N[(w0$95$m + N[(-0.125 * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(w0$95$m * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if M < 2.5e6

   1. Initial program 81.4%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in D around 0 70.3%

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

   if 2.5e6 < M

   1. Initial program 74.4%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in D around 0 47.5%

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

      1. associate-/l* 47.5%

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

      2. times-frac 49.3%

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

      3. *-commutative 49.3%

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

   7. Simplified 49.3%

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

      1. unpow2 49.3%

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

      2. unpow2 49.3%

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

      3. times-frac 62.6%

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

   9. Applied egg-rr 62.6%

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

       1. unpow2 60.8%

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

   11. Applied egg-rr 62.6%

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

   12. Taylor expanded in w0 around 0 62.6%

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

       1. *-commutative 62.6%

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

       2. associate-*r/ 62.6%

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

   14. Simplified 62.6%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2500000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(w0 \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 80.0%

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

3. Simplified 78.9%

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

5. Taylor expanded in D around 0 65.6%

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

Reproduce

herbie shell --seed 2024177 
(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))))))