Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 86.4%

Time: 12.8s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.1% 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: 86.4% accurate, 1.0× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* h (/ (pow (* D (* M_m (/ 0.5 d))) 2.0) l))))))

C

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

Fortran

M_m = abs(M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - (h * (((d * (m_m * (0.5d0 / d_1))) ** 2.0d0) / l))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(D * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.1%

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

3. Simplified 85.2%

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

   1. frac-times 84.1%

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

   2. *-commutative 84.1%

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

   3. expm1-log1p-u 62.1%

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

   4. expm1-udef 62.1%

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

6. Applied egg-rr 84.1%

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

   1. +-commutative 84.1%

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

   2. *-commutative 84.1%

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

   3. fma-udef 84.1%

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

   4. rem-exp-log 62.1%

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

   5. expm1-def 62.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\log \left(\mathsf{fma}\left({\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)\right)\right)}} \]

   6. fma-udef 62.1%

      \[\leadsto w0 \cdot \sqrt{1 - \mathsf{expm1}\left(\log \color{blue}{\left({\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2} \cdot \frac{h}{\ell} + 1\right)}\right)} \]

   7. *-commutative 62.1%

      \[\leadsto w0 \cdot \sqrt{1 - \mathsf{expm1}\left(\log \left(\color{blue}{\frac{h}{\ell} \cdot {\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2}} + 1\right)\right)} \]

   8. +-commutative 62.1%

      \[\leadsto w0 \cdot \sqrt{1 - \mathsf{expm1}\left(\log \color{blue}{\left(1 + \frac{h}{\ell} \cdot {\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2}\right)}\right)} \]

   9. log1p-def 62.1%

      \[\leadsto w0 \cdot \sqrt{1 - \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{h}{\ell} \cdot {\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2}\right)}\right)} \]

   10. expm1-log1p 84.1%

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

   11. associate-*l/ 85.4%

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

   12. *-commutative 85.4%

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

   13. associate-*l/ 85.8%

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

   14. *-commutative 85.8%

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

8. Simplified 86.5%

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

9. Final simplification 86.5%

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

Alternative 2: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 3.1 \cdot 10^{+52}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(h \cdot {\left(M_m \cdot \frac{D}{d}\right)}^{2}\right) \cdot \frac{w0}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= D 3.1e+52)
   w0
   (+ w0 (* -0.125 (* (* h (pow (* M_m (/ D d)) 2.0)) (/ w0 l))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (D <= 3.1e+52) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((h * pow((M_m * (D / d)), 2.0)) * (w0 / l)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 3.1d+52) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((h * ((m_m * (d / d_1)) ** 2.0d0)) * (w0 / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (D <= 3.1e+52) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((h * Math.pow((M_m * (D / d)), 2.0)) * (w0 / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if D <= 3.1e+52:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((h * math.pow((M_m * (D / d)), 2.0)) * (w0 / l)))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (D <= 3.1e+52)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(h * (Float64(M_m * Float64(D / d)) ^ 2.0)) * Float64(w0 / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (D <= 3.1e+52)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((h * ((M_m * (D / d)) ^ 2.0)) * (w0 / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[D, 3.1e+52], w0, N[(w0 + N[(-0.125 * N[(N[(h * N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 3.1e52

   1. Initial program 87.8%

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

   3. Simplified 87.8%

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

   5. Taylor expanded in D around 0 71.7%

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

   if 3.1e52 < D

   1. Initial program 68.5%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in D around 0 31.0%

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

      1. expm1-log1p-u 22.8%

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

      2. expm1-udef 22.8%

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

      3. associate-*r* 20.9%

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

      4. unpow-prod-down 37.6%

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

      5. *-commutative 37.6%

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

   7. Applied egg-rr 37.6%

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

      1. expm1-def 37.6%

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

      2. expm1-log1p 46.1%

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

      3. associate-*r* 44.1%

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

      4. *-commutative 44.1%

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

      5. *-commutative 44.1%

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

      6. times-frac 46.2%

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

   9. Simplified 61.0%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 3.1 \cdot 10^{+52}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(h \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right) \cdot \frac{w0}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 1.8× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (+ w0 (* -0.125 (/ 1.0 (/ (/ l (* h (pow (* M_m (/ D d)) 2.0))) w0)))))

C

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

Fortran

M_m = abs(M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 + ((-0.125d0) * (1.0d0 / ((l / (h * ((m_m * (d / d_1)) ** 2.0d0))) / w0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := N[(w0 + N[(-0.125 * N[(1.0 / N[(N[(l / N[(h * N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.1%

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

3. Simplified 85.2%

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

5. Taylor expanded in D around 0 52.4%

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

   1. clear-num 52.4%

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

   2. inv-pow 52.4%

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

   3. *-commutative 52.4%

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

   4. associate-*r* 52.0%

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

   5. unpow-prod-down 61.6%

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

7. Applied egg-rr 61.6%

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

   1. unpow-1 61.6%

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

   2. associate-*r* 64.3%

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

   3. *-commutative 64.3%

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

   4. associate-/r* 66.3%

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

9. Simplified 82.2%

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

10. Final simplification 82.2%

    \[\leadsto w0 + -0.125 \cdot \frac{1}{\frac{\frac{\ell}{h \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}}{w0}} \]
11. Add Preprocessing

Alternative 4: 68.7% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

3. Simplified 85.2%

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

5. Taylor expanded in D around 0 67.5%

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

6. Final simplification 67.5%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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))))))