Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 89.0%

Time: 17.1s

Alternatives: 5

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: 81.0% 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, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{2 \cdot d}\\ \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{+71}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M\_m}{\frac{2 \cdot d}{D\_m}}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m (* 2.0 d)))))
   (if (<= (/ h l) -5e+71)
     (* w0 (sqrt (- 1.0 (/ (* h (pow (/ M_m (/ (* 2.0 d) D_m)) 2.0)) l))))
     (* w0 (sqrt (- 1.0 (* h (* t_0 (/ t_0 l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M_m / (2.0 * d));
	double tmp;
	if ((h / l) <= -5e+71) {
		tmp = w0 * sqrt((1.0 - ((h * pow((M_m / ((2.0 * d) / D_m)), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_m * (m_m / (2.0d0 * d))
    if ((h / l) <= (-5d+71)) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((m_m / ((2.0d0 * d) / d_m)) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (h * (t_0 * (t_0 / l)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M_m / (2.0 * d));
	double tmp;
	if ((h / l) <= -5e+71) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M_m / ((2.0 * d) / D_m)), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = D_m * (M_m / (2.0 * d))
	tmp = 0
	if (h / l) <= -5e+71:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((M_m / ((2.0 * d) / D_m)), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (h * (t_0 * (t_0 / l)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m * Float64(M_m / Float64(2.0 * d)))
	tmp = 0.0
	if (Float64(h / l) <= -5e+71)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M_m / Float64(Float64(2.0 * d) / D_m)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(t_0 * Float64(t_0 / l))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = D_m * (M_m / (2.0 * d));
	tmp = 0.0;
	if ((h / l) <= -5e+71)
		tmp = w0 * sqrt((1.0 - ((h * ((M_m / ((2.0 * d) / D_m)) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], -5e+71], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M$95$m / N[(N[(2.0 * d), $MachinePrecision] / D$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -4.99999999999999972e71

   1. Initial program 62.0%

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

   3. Simplified 63.3%

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

      1. clear-num 63.3%

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

      2. un-div-inv 64.2%

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

      3. associate-/r* 64.2%

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

   6. Applied egg-rr 64.2%

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

      1. associate-/r/ 75.6%

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

      2. associate-/r* 75.6%

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

   8. Simplified 75.6%

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

      1. associate-/r* 75.6%

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

      2. associate-*r/ 75.7%

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

      3. associate-*l/ 74.3%

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

      4. clear-num 74.3%

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

      5. associate-*l/ 74.4%

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

      6. *-un-lft-identity 74.4%

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

      7. *-un-lft-identity 74.4%

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

      8. times-frac 74.4%

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

      9. metadata-eval 74.4%

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

   10. Applied egg-rr 74.4%

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

       1. associate-*l/ 77.0%

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

       2. associate-*r/ 77.0%

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

   12. Applied egg-rr 77.0%

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

   if -4.99999999999999972e71 < (/.f64 h l)

   1. Initial program 87.3%

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

   3. Simplified 87.9%

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

      1. clear-num 87.9%

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

      2. un-div-inv 87.9%

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

      3. associate-/r* 87.9%

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

   6. Applied egg-rr 87.9%

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

      1. associate-/r/ 91.2%

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

      2. associate-/r* 91.2%

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

   8. Simplified 91.2%

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

      1. unpow2 91.2%

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

      2. associate-/l/ 91.2%

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

      3. associate-/l/ 91.2%

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

   10. Applied egg-rr 91.2%

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

       1. associate-/l* 93.3%

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

       2. *-commutative 93.3%

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

       3. *-commutative 93.3%

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

   12. Applied egg-rr 93.3%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{+71}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(D \cdot \frac{M}{2 \cdot d}\right) \cdot \frac{D \cdot \frac{M}{2 \cdot d}}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.4 \cdot 10^{+23}:\\ \;\;\;\;w0 + \frac{-0.125}{d} \cdot \frac{\frac{\left(h \cdot w0\right) \cdot {\left(M\_m \cdot D\_m\right)}^{2}}{\ell}}{d}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= d 2.4e+23)
   (+ w0 (* (/ -0.125 d) (/ (/ (* (* h w0) (pow (* M_m D_m) 2.0)) l) d)))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 2.4e+23) {
		tmp = w0 + ((-0.125 / d) * ((((h * w0) * pow((M_m * D_m), 2.0)) / l) / d));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 2.4d+23) then
        tmp = w0 + (((-0.125d0) / d) * ((((h * w0) * ((m_m * d_m) ** 2.0d0)) / l) / d))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 2.4e+23) {
		tmp = w0 + ((-0.125 / d) * ((((h * w0) * Math.pow((M_m * D_m), 2.0)) / l) / d));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if d <= 2.4e+23:
		tmp = w0 + ((-0.125 / d) * ((((h * w0) * math.pow((M_m * D_m), 2.0)) / l) / d))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (d <= 2.4e+23)
		tmp = Float64(w0 + Float64(Float64(-0.125 / d) * Float64(Float64(Float64(Float64(h * w0) * (Float64(M_m * D_m) ^ 2.0)) / l) / d)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (d <= 2.4e+23)
		tmp = w0 + ((-0.125 / d) * ((((h * w0) * ((M_m * D_m) ^ 2.0)) / l) / d));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[d, 2.4e+23], N[(w0 + N[(N[(-0.125 / d), $MachinePrecision] * N[(N[(N[(N[(h * w0), $MachinePrecision] * N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 2.4e23

   1. Initial program 79.8%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in D around 0 50.1%

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

      1. expm1-log1p-u 43.0%

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

      2. expm1-undefine 43.0%

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

      3. associate-*r* 44.0%

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

      4. unpow-prod-down 51.0%

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

      5. *-commutative 51.0%

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

   7. Applied egg-rr 51.0%

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

      1. expm1-define 51.0%

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

      2. associate-*r/ 51.0%

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

      3. times-frac 55.2%

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

      4. *-commutative 55.2%

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

   9. Simplified 55.2%

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

       1. expm1-log1p-u 63.0%

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

       2. *-commutative 63.0%

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

       3. times-frac 58.8%

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

       4. associate-/r* 63.0%

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

       5. *-commutative 63.0%

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

       6. associate-*l* 65.7%

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

       7. *-commutative 65.7%

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

   11. Applied egg-rr 65.7%

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

       1. associate-/l* 65.7%

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

       2. unpow2 65.7%

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

       3. times-frac 76.0%

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

       4. pow2 76.0%

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

       5. associate-*r* 72.8%

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

       6. pow2 72.8%

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

       7. *-commutative 72.8%

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

   13. Applied egg-rr 72.8%

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

   if 2.4e23 < d

   1. Initial program 81.6%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in D around 0 83.7%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.4 \cdot 10^{+23}:\\ \;\;\;\;w0 + \frac{-0.125}{d} \cdot \frac{\frac{\left(h \cdot w0\right) \cdot {\left(M \cdot D\right)}^{2}}{\ell}}{d}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{2 \cdot d}\\ w0 \cdot \sqrt{1 - h \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)} \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m (* 2.0 d)))))
   (* w0 (sqrt (- 1.0 (* h (* t_0 (/ t_0 l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M_m / (2.0 * d));
	return w0 * sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    t_0 = d_m * (m_m / (2.0d0 * d))
    code = w0 * sqrt((1.0d0 - (h * (t_0 * (t_0 / l)))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M_m / (2.0 * d));
	return w0 * Math.sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = D_m * (M_m / (2.0 * d))
	return w0 * math.sqrt((1.0 - (h * (t_0 * (t_0 / l)))))

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m * Float64(M_m / Float64(2.0 * d)))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(t_0 * Float64(t_0 / l))))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp = code(w0, M_m, D_m, h, l, d)
	t_0 = D_m * (M_m / (2.0 * d));
	tmp = w0 * sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.3%

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

3. Simplified 81.1%

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

   1. clear-num 81.1%

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

   2. un-div-inv 81.3%

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

   3. associate-/r* 81.3%

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

6. Applied egg-rr 81.3%

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

   1. associate-/r/ 86.9%

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

   2. associate-/r* 86.9%

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

8. Simplified 86.9%

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

   1. unpow2 86.9%

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

   2. associate-/l/ 86.9%

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

   3. associate-/l/ 86.9%

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

10. Applied egg-rr 86.9%

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

    1. associate-/l* 88.4%

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

    2. *-commutative 88.4%

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

    3. *-commutative 88.4%

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

12. Applied egg-rr 88.4%

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

13. Final simplification 88.4%

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

Alternative 4: 72.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2.3 \cdot 10^{-58}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + \frac{-0.125 \cdot \left(\left(h \cdot w0\right) \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 2.3e-58)
   w0
   (+
    w0
    (/ (* -0.125 (* (* h w0) (* (* M_m D_m) (* M_m D_m)))) (* l (* d d))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.3e-58) {
		tmp = w0;
	} else {
		tmp = w0 + ((-0.125 * ((h * w0) * ((M_m * D_m) * (M_m * D_m)))) / (l * (d * d)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 2.3d-58) then
        tmp = w0
    else
        tmp = w0 + (((-0.125d0) * ((h * w0) * ((m_m * d_m) * (m_m * d_m)))) / (l * (d * d)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.3e-58) {
		tmp = w0;
	} else {
		tmp = w0 + ((-0.125 * ((h * w0) * ((M_m * D_m) * (M_m * D_m)))) / (l * (d * d)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 2.3e-58:
		tmp = w0
	else:
		tmp = w0 + ((-0.125 * ((h * w0) * ((M_m * D_m) * (M_m * D_m)))) / (l * (d * d)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 2.3e-58)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(Float64(-0.125 * Float64(Float64(h * w0) * Float64(Float64(M_m * D_m) * Float64(M_m * D_m)))) / Float64(l * Float64(d * d))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 2.3e-58)
		tmp = w0;
	else
		tmp = w0 + ((-0.125 * ((h * w0) * ((M_m * D_m) * (M_m * D_m)))) / (l * (d * d)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 2.3e-58], w0, N[(w0 + N[(N[(-0.125 * N[(N[(h * w0), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.2999999999999999e-58

   1. Initial program 83.1%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in D around 0 79.0%

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

   if 2.2999999999999999e-58 < M

   1. Initial program 72.2%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in D around 0 56.5%

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

      1. associate-*r/ 56.5%

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

      2. associate-*r* 58.1%

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

      3. unpow-prod-down 64.6%

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

      4. *-commutative 64.6%

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

   7. Applied egg-rr 64.6%

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

      1. unpow2 64.6%

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

      2. *-commutative 64.6%

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

      3. *-commutative 64.6%

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

   9. Applied egg-rr 64.6%

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

       1. unpow2 64.6%

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

   11. Applied egg-rr 64.6%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.3 \cdot 10^{-58}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + \frac{-0.125 \cdot \left(\left(h \cdot w0\right) \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.4% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.3%

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

3. Simplified 81.1%

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

5. Taylor expanded in D around 0 71.3%

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

Reproduce

herbie shell --seed 2024181 
(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))))))