Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 86.8%

Time: 15.8s

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: 81.4% 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: 86.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

3. Simplified 81.7%

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

   1. associate-*r/ 87.6%

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

   2. add-sqr-sqrt 87.6%

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

   3. pow2 87.6%

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

   4. unpow2 87.6%

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

   5. sqrt-prod 51.9%

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

   6. add-sqr-sqrt 87.6%

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

   7. div-inv 87.6%

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

   8. metadata-eval 87.6%

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

6. Applied egg-rr 87.6%

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

7. Final simplification 87.6%

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

Alternative 2: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 2.85 \cdot 10^{+224}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{{\left(D\_m \cdot M\right)}^{2}}{{d}^{2}} \cdot \frac{w0 \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= D_m 2.85e+224)
   w0
   (* -0.125 (* (/ (pow (* D_m M) 2.0) (pow d 2.0)) (/ (* w0 h) l)))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 2.85e+224) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((pow((D_m * M), 2.0) / pow(d, 2.0)) * ((w0 * h) / l));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d_m <= 2.85d+224) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((((d_m * m) ** 2.0d0) / (d ** 2.0d0)) * ((w0 * h) / l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 2.85e+224) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((Math.pow((D_m * M), 2.0) / Math.pow(d, 2.0)) * ((w0 * h) / l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	tmp = 0
	if D_m <= 2.85e+224:
		tmp = w0
	else:
		tmp = -0.125 * ((math.pow((D_m * M), 2.0) / math.pow(d, 2.0)) * ((w0 * h) / l))
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (D_m <= 2.85e+224)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64((Float64(D_m * M) ^ 2.0) / (d ^ 2.0)) * Float64(Float64(w0 * h) / l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	tmp = 0.0;
	if (D_m <= 2.85e+224)
		tmp = w0;
	else
		tmp = -0.125 * ((((D_m * M) ^ 2.0) / (d ^ 2.0)) * ((w0 * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[D$95$m, 2.85e+224], w0, N[(-0.125 * N[(N[(N[Power[N[(D$95$m * M), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.84999999999999999e224

   1. Initial program 82.8%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in D around 0 70.9%

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

   if 2.84999999999999999e224 < D

   1. Initial program 51.5%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in D around 0 25.6%

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

      1. *-commutative 25.6%

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

      2. associate-*r* 26.6%

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

   7. Simplified 26.6%

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

      1. *-un-lft-identity 26.6%

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

      2. add-sqr-sqrt 0.6%

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

      3. times-frac 0.6%

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

      4. sqrt-prod 0.6%

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

      5. unpow2 0.6%

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

      6. sqrt-prod 0.6%

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

      7. add-sqr-sqrt 0.6%

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

      8. *-commutative 0.6%

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

      9. pow-prod-down 0.6%

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

      10. sqrt-prod 0.6%

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

      11. unpow2 0.6%

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

      12. sqrt-prod 0.6%

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

      13. add-sqr-sqrt 9.3%

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

   9. Applied egg-rr 9.3%

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

       1. times-frac 9.3%

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

   11. Simplified 9.3%

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

   12. Taylor expanded in d around 0 25.8%

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

       1. associate-*r* 26.6%

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

       2. unpow2 26.6%

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

       3. unpow2 26.6%

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

       4. swap-sqr 27.0%

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

       5. unpow2 27.0%

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

       6. times-frac 26.9%

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

   14. Simplified 26.9%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.85 \cdot 10^{+224}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}} \cdot \frac{w0 \cdot h}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 * ((((d_m * 0.5d0) / (d / m)) ** 2.0d0) / l))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

3. Simplified 81.7%

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

   1. associate-*r/ 87.6%

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

   2. add-sqr-sqrt 87.6%

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

   3. pow2 87.6%

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

   4. unpow2 87.6%

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

   5. sqrt-prod 51.9%

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

   6. add-sqr-sqrt 87.6%

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

   7. div-inv 87.6%

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

   8. metadata-eval 87.6%

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

6. Applied egg-rr 87.6%

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

   1. log1p-expm1-u 82.6%

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

   2. log1p-expm1-u 87.6%

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

   3. associate-*r/ 81.7%

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

   4. expm1-log1p-u 61.3%

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

   5. expm1-udef 61.3%

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

   6. log1p-udef 61.3%

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

   7. add-exp-log 81.7%

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

   8. expm1-log1p-u 81.5%

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

   9. expm1-udef 81.4%

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

8. Applied egg-rr 81.1%

   \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\frac{\left(D \cdot 0.5\right) \cdot M}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)} - 1\right)} \]
9. Step-by-step derivation

   1. expm1-def 81.1%

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

   2. expm1-log1p 81.3%

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

   3. associate-*r/ 87.2%

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

   4. associate-*l/ 86.1%

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

   5. *-commutative 86.1%

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

   6. associate-/l* 86.9%

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

10. Simplified 86.9%

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

11. Final simplification 86.9%

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

Alternative 4: 68.3% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

3. Simplified 81.7%

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

5. Taylor expanded in D around 0 68.6%

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

6. Final simplification 68.6%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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))))))