Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 87.3%

Time: 15.7s

Alternatives: 6

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: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ h l)) -1e-9)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* D_m (/ (/ M_m 2.0) d)) 2.0)))))
   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 ((pow(((M_m * D_m) / (d * 2.0)), 2.0) * (h / l)) <= -1e-9) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D_m * ((M_m / 2.0) / d)), 2.0))));
	} 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 (((((m_m * d_m) / (d * 2.0d0)) ** 2.0d0) * (h / l)) <= (-1d-9)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0))))
    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 ((Math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (h / l)) <= -1e-9) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D_m * ((M_m / 2.0) / d)), 2.0))));
	} 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 (math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (h / l)) <= -1e-9:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((D_m * ((M_m / 2.0) / d)), 2.0))))
	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 (Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -1e-9)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0)))));
	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 (((((M_m * D_m) / (d * 2.0)) ^ 2.0) * (h / l)) <= -1e-9)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D_m * ((M_m / 2.0) / d)) ^ 2.0))));
	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[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e-9], N[(w0 * 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], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.00000000000000006e-9

   1. Initial program 69.4%

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

   3. Simplified 68.9%

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

   if -1.00000000000000006e-9 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 82.6%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in D around 0 95.8%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 1.8 \cdot 10^{+18}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}, 1\right)\\ \end{array} \end{array} \]

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 (<= D_m 1.8e+18)
   w0
   (* w0 (fma -0.125 (* (/ h l) (pow (* D_m (/ M_m d)) 2.0)) 1.0))))

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_m <= 1.8e+18) {
		tmp = w0;
	} else {
		tmp = w0 * fma(-0.125, ((h / l) * pow((D_m * (M_m / d)), 2.0)), 1.0);
	}
	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_m <= 1.8e+18)
		tmp = w0;
	else
		tmp = Float64(w0 * fma(-0.125, Float64(Float64(h / l) * (Float64(D_m * Float64(M_m / d)) ^ 2.0)), 1.0));
	end
	return 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$95$m, 1.8e+18], w0, N[(w0 * N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.8e18

   1. Initial program 78.9%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in D around 0 73.9%

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

   if 1.8e18 < D

   1. Initial program 77.0%

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

   3. Simplified 82.4%

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

      1. associate-*r/ 86.5%

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

      2. *-commutative 86.5%

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

      3. associate-*l/ 81.4%

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

      4. associate-*r/ 87.7%

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

      5. div-inv 87.7%

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

      6. metadata-eval 87.7%

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

   6. Applied egg-rr 87.7%

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

      1. add-sqr-sqrt 53.6%

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

      2. pow2 53.6%

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

      3. sqrt-prod 45.1%

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

      4. sqrt-pow1 45.1%

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

      5. metadata-eval 45.1%

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

      6. pow1 45.1%

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

      7. associate-*l* 45.1%

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

   8. Applied egg-rr 45.1%

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

   9. Taylor expanded in M around 0 45.1%

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

   10. Taylor expanded in D around 0 45.0%

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

       1. +-commutative 45.0%

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

       2. fma-define 45.0%

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

   12. Simplified 69.8%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.8 \cdot 10^{+18}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 1.0× speedup

Math

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

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 (sqrt (- 1.0 (/ (* (pow (/ (* M_m 0.5) (/ d D_m)) 2.0) h) 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) {
	return w0 * sqrt((1.0 - ((pow(((M_m * 0.5) / (d / D_m)), 2.0) * h) / 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
    code = w0 * sqrt((1.0d0 - (((((m_m * 0.5d0) / (d / d_m)) ** 2.0d0) * h) / 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) {
	return w0 * Math.sqrt((1.0 - ((Math.pow(((M_m * 0.5) / (d / D_m)), 2.0) * h) / 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):
	return w0 * math.sqrt((1.0 - ((math.pow(((M_m * 0.5) / (d / D_m)), 2.0) * h) / 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)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(Float64(M_m * 0.5) / Float64(d / D_m)) ^ 2.0) * h) / 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)
	tmp = w0 * sqrt((1.0 - (((((M_m * 0.5) / (d / D_m)) ^ 2.0) * h) / 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_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(N[(M$95$m * 0.5), $MachinePrecision] / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

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

3. Simplified 79.2%

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

   1. associate-*r/ 87.5%

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

   2. *-commutative 87.5%

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

   3. associate-*l/ 86.6%

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

   4. associate-*r/ 88.1%

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

   5. div-inv 88.1%

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

   6. metadata-eval 88.1%

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

6. Applied egg-rr 88.1%

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

   1. clear-num 88.1%

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

   2. un-div-inv 88.1%

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

8. Applied egg-rr 88.1%

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

9. Final simplification 88.1%

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

Alternative 4: 86.2% accurate, 1.0× speedup

Math

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

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 (sqrt (- 1.0 (/ (* h (pow (* (* M_m 0.5) (/ D_m d)) 2.0)) 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) {
	return w0 * sqrt((1.0 - ((h * pow(((M_m * 0.5) * (D_m / d)), 2.0)) / 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
    code = w0 * sqrt((1.0d0 - ((h * (((m_m * 0.5d0) * (d_m / d)) ** 2.0d0)) / 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) {
	return w0 * Math.sqrt((1.0 - ((h * Math.pow(((M_m * 0.5) * (D_m / d)), 2.0)) / 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):
	return w0 * math.sqrt((1.0 - ((h * math.pow(((M_m * 0.5) * (D_m / d)), 2.0)) / 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)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(M_m * 0.5) * Float64(D_m / d)) ^ 2.0)) / 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)
	tmp = w0 * sqrt((1.0 - ((h * (((M_m * 0.5) * (D_m / d)) ^ 2.0)) / 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_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

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

3. Simplified 79.2%

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

   1. associate-*r/ 87.5%

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

   2. *-commutative 87.5%

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

   3. associate-*l/ 86.6%

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

   4. associate-*r/ 88.1%

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

   5. div-inv 88.1%

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

   6. metadata-eval 88.1%

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

6. Applied egg-rr 88.1%

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

7. Final simplification 88.1%

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

Alternative 5: 79.8% accurate, 1.0× speedup

Math

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

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
 (fma -0.125 (/ (* h (* w0 (pow (* D_m (/ M_m d)) 2.0))) l) 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 fma(-0.125, ((h * (w0 * pow((D_m * (M_m / d)), 2.0))) / l), 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 fma(-0.125, Float64(Float64(h * Float64(w0 * (Float64(D_m * Float64(M_m / d)) ^ 2.0))) / l), 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_] := N[(-0.125 * N[(N[(h * N[(w0 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + w0), $MachinePrecision]

Derivation

1. Initial program 78.6%

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

3. Simplified 79.2%

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

5. Taylor expanded in D around 0 51.0%

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

   1. +-commutative 51.0%

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

   2. fma-define 51.0%

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

   3. associate-*r* 51.8%

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

   4. unpow2 51.8%

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

   5. unpow2 51.8%

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

   6. swap-sqr 63.0%

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

   7. unpow2 63.0%

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

7. Simplified 63.0%

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

   1. *-commutative 63.0%

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

   2. times-frac 60.6%

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

   3. *-commutative 60.6%

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

9. Applied egg-rr 60.6%

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

    1. frac-times 63.0%

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

    2. unpow2 63.0%

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

    3. *-commutative 63.0%

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

    4. associate-*l* 60.2%

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

    5. *-commutative 60.2%

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

    6. associate-/r* 61.5%

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

    7. *-commutative 61.5%

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

    8. associate-*l* 64.2%

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

    9. *-commutative 64.2%

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

    10. unpow2 64.2%

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

11. Applied egg-rr 64.2%

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

12. Taylor expanded in h around 0 52.7%

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

    1. associate-*r* 53.4%

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

    2. *-commutative 53.4%

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

    3. unpow2 53.4%

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

    4. unpow2 53.4%

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

    5. swap-sqr 64.2%

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

    6. unpow2 64.2%

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

    7. associate-*l/ 63.9%

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

    8. *-commutative 63.9%

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

    9. associate-*l* 71.3%

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

    10. unpow2 71.3%

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

    11. unpow2 71.3%

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

    12. times-frac 80.2%

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

    13. *-commutative 80.2%

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

    14. associate-*r/ 80.2%

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

    15. *-commutative 80.2%

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

    16. associate-*r/ 81.0%

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

    17. unpow2 81.0%

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

14. Simplified 81.0%

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

15. Final simplification 81.0%

    \[\leadsto \mathsf{fma}\left(-0.125, \frac{h \cdot \left(w0 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}{\ell}, w0\right) \]
16. Add Preprocessing

Alternative 6: 68.5% 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 78.6%

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

3. Simplified 79.2%

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

5. Taylor expanded in D around 0 68.7%

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

6. Final simplification 68.7%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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))))))