Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 88.7%

Time: 17.1s

Alternatives: 3

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: 81.1% 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: 88.7% accurate, 1.8× speedup

Math

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

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 (* (* (/ (* 0.5 M_m) d) D) (/ (* D (* 0.5 (/ M_m d))) 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 * ((((0.5 * M_m) / d) * D) * ((D * (0.5 * (M_m / d))) / 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 * ((((0.5d0 * m_m) / d_1) * d) * ((d * (0.5d0 * (m_m / d_1))) / 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 * ((((0.5 * M_m) / d) * D) * ((D * (0.5 * (M_m / d))) / 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 * ((((0.5 * M_m) / d) * D) * ((D * (0.5 * (M_m / d))) / 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(Float64(Float64(0.5 * M_m) / d) * D) * Float64(Float64(D * Float64(0.5 * Float64(M_m / d))) / 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 * ((((0.5 * M_m) / d) * D) * ((D * (0.5 * (M_m / d))) / 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[(N[(N[(0.5 * M$95$m), $MachinePrecision] / d), $MachinePrecision] * D), $MachinePrecision] * N[(N[(D * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.0%

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

3. Simplified 82.0%

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

6. Applied egg-rr 82.0%

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

   1. unpow1 82.0%

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

   2. associate-*l/ 88.4%

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

   3. associate-/l* 87.9%

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

   4. associate-*r/ 88.9%

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

   5. times-frac 87.9%

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

   6. associate-*r/ 88.9%

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

8. Simplified 88.9%

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

   1. associate-*r/ 87.9%

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

   2. unpow2 87.9%

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

   3. *-un-lft-identity 87.9%

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

   4. times-frac 90.6%

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

   5. div-inv 90.6%

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

   6. metadata-eval 90.6%

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

   7. associate-*l* 90.6%

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

   8. div-inv 90.6%

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

   9. metadata-eval 90.6%

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

   10. associate-*l* 90.6%

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

10. Applied egg-rr 90.6%

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

    1. /-rgt-identity 90.6%

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

    2. *-commutative 90.6%

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

    3. associate-*r/ 90.6%

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

12. Applied egg-rr 90.6%

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

Alternative 2: 68.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(w0 \cdot \left(1 + w0 \cdot \left(w0 \cdot 0.3333333333333333 - 0.5\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= M_m 7.2e+15)
   w0
   (expm1 (* w0 (+ 1.0 (* w0 (- (* w0 0.3333333333333333) 0.5)))))))

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 (M_m <= 7.2e+15) {
		tmp = w0;
	} else {
		tmp = expm1((w0 * (1.0 + (w0 * ((w0 * 0.3333333333333333) - 0.5)))));
	}
	return tmp;
}

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 (M_m <= 7.2e+15) {
		tmp = w0;
	} else {
		tmp = Math.expm1((w0 * (1.0 + (w0 * ((w0 * 0.3333333333333333) - 0.5)))));
	}
	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 M_m <= 7.2e+15:
		tmp = w0
	else:
		tmp = math.expm1((w0 * (1.0 + (w0 * ((w0 * 0.3333333333333333) - 0.5)))))
	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 (M_m <= 7.2e+15)
		tmp = w0;
	else
		tmp = expm1(Float64(w0 * Float64(1.0 + Float64(w0 * Float64(Float64(w0 * 0.3333333333333333) - 0.5)))));
	end
	return 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[M$95$m, 7.2e+15], w0, N[(Exp[N[(w0 * N[(1.0 + N[(w0 * N[(N[(w0 * 0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.2e15

   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 82.3%

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

   5. Taylor expanded in D around 0 73.5%

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

   if 7.2e15 < M

   1. Initial program 84.5%

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

   3. Simplified 80.8%

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

      1. associate-*r/ 83.1%

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

      2. clear-num 83.1%

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

      3. *-commutative 83.1%

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

      4. associate-/r* 83.1%

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

   6. Applied egg-rr 83.1%

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

      1. associate-/r/ 83.0%

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

      2. associate-*r/ 86.7%

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

      3. times-frac 83.0%

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

      4. associate-*r/ 86.7%

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

   8. Simplified 86.7%

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

      1. expm1-log1p-u 57.5%

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

      2. associate-*l/ 57.5%

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

      3. *-un-lft-identity 57.5%

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

      4. associate-/l* 54.2%

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

      5. div-inv 54.2%

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

      6. metadata-eval 54.2%

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

   10. Applied egg-rr 54.2%

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

   11. Taylor expanded in h around 0 22.2%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + w0\right)}\right) \]
   12. Step-by-step derivation

       1. log1p-define 31.9%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(w0\right)}\right) \]

   13. Simplified 31.9%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(w0\right)}\right) \]

   14. Taylor expanded in w0 around 0 29.6%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{w0 \cdot \left(1 + w0 \cdot \left(0.3333333333333333 \cdot w0 - 0.5\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(w0 \cdot \left(1 + w0 \cdot \left(w0 \cdot 0.3333333333333333 - 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.3% 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 83.0%

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

3. Simplified 82.0%

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

5. Taylor expanded in D around 0 67.9%

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

Reproduce

herbie shell --seed 2024158 
(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))))))