Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 86.0%

Time: 12.5s

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: 81.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.0% accurate, 1.0× speedup

Math

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

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 (* (/ (pow (* (/ M_m d) (/ D 2.0)) 2.0) l) h)))))

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 - ((pow(((M_m / d) * (D / 2.0)), 2.0) / l) * h)));
}

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 - (((((m_m / d_1) * (d / 2.0d0)) ** 2.0d0) / l) * h)))
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 - ((Math.pow(((M_m / d) * (D / 2.0)), 2.0) / l) * h)));
}

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 - ((math.pow(((M_m / d) * (D / 2.0)), 2.0) / l) * h)))

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(Float64((Float64(Float64(M_m / d) * Float64(D / 2.0)) ^ 2.0) / l) * h))))
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 - (((((M_m / d) * (D / 2.0)) ^ 2.0) / l) * h)));
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[(N[(N[Power[N[(N[(M$95$m / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.7%

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

3. Simplified 84.9%

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

   1. frac-times 83.7%

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

   2. associate-*l/ 83.4%

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

   3. clear-num 83.3%

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

   4. un-div-inv 84.4%

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

   5. associate-*l/ 84.8%

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

   6. *-commutative 84.8%

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

   7. times-frac 84.4%

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

5. Applied egg-rr 84.4%

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

   1. associate-/r/ 87.7%

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

7. Simplified 87.7%

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

8. Final simplification 87.7%

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

Alternative 2: 69.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 3.2 \cdot 10^{-259} \lor \neg \left(d \leq 3.2 \cdot 10^{-227}\right) \land d \leq 5.4 \cdot 10^{-10}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{\frac{{\left(M_m \cdot D\right)}^{2}}{\ell}}{d} \cdot \frac{w0 \cdot h}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (or (<= d 3.2e-259) (and (not (<= d 3.2e-227)) (<= d 5.4e-10)))
   (+ w0 (* -0.125 (* (/ (/ (pow (* M_m D) 2.0) l) d) (/ (* w0 h) d))))
   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) {
	double tmp;
	if ((d <= 3.2e-259) || (!(d <= 3.2e-227) && (d <= 5.4e-10))) {
		tmp = w0 + (-0.125 * (((pow((M_m * D), 2.0) / l) / d) * ((w0 * h) / d)));
	} else {
		tmp = w0;
	}
	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_1 <= 3.2d-259) .or. (.not. (d_1 <= 3.2d-227)) .and. (d_1 <= 5.4d-10)) then
        tmp = w0 + ((-0.125d0) * (((((m_m * d) ** 2.0d0) / l) / d_1) * ((w0 * h) / d_1)))
    else
        tmp = w0
    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 <= 3.2e-259) || (!(d <= 3.2e-227) && (d <= 5.4e-10))) {
		tmp = w0 + (-0.125 * (((Math.pow((M_m * D), 2.0) / l) / d) * ((w0 * h) / d)));
	} else {
		tmp = w0;
	}
	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 <= 3.2e-259) or (not (d <= 3.2e-227) and (d <= 5.4e-10)):
		tmp = w0 + (-0.125 * (((math.pow((M_m * D), 2.0) / l) / d) * ((w0 * h) / d)))
	else:
		tmp = w0
	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 <= 3.2e-259) || (!(d <= 3.2e-227) && (d <= 5.4e-10)))
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64((Float64(M_m * D) ^ 2.0) / l) / d) * Float64(Float64(w0 * h) / d))));
	else
		tmp = w0;
	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 <= 3.2e-259) || (~((d <= 3.2e-227)) && (d <= 5.4e-10)))
		tmp = w0 + (-0.125 * (((((M_m * D) ^ 2.0) / l) / d) * ((w0 * h) / d)));
	else
		tmp = w0;
	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[Or[LessEqual[d, 3.2e-259], And[N[Not[LessEqual[d, 3.2e-227]], $MachinePrecision], LessEqual[d, 5.4e-10]]], N[(w0 + N[(-0.125 * N[(N[(N[(N[Power[N[(M$95$m * D), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision] * N[(N[(w0 * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 3.19999999999999988e-259 or 3.2000000000000001e-227 < d < 5.4e-10

   1. Initial program 82.2%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in M around 0 47.9%

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

      1. add-exp-log 36.7%

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

      2. associate-*r* 37.3%

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

      3. pow-prod-down 43.9%

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

   6. Applied egg-rr 43.9%

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

      1. rem-exp-log 60.3%

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

      2. *-commutative 60.3%

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

      3. frac-times 59.1%

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

      4. associate-*r/ 60.3%

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

      5. unpow2 60.3%

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

      6. times-frac 66.9%

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

   8. Applied egg-rr 66.9%

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

   if 3.19999999999999988e-259 < d < 3.2000000000000001e-227 or 5.4e-10 < d

   1. Initial program 87.0%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in M around 0 83.5%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 3.2 \cdot 10^{-259} \lor \neg \left(d \leq 3.2 \cdot 10^{-227}\right) \land d \leq 5.4 \cdot 10^{-10}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{\frac{{\left(M \cdot D\right)}^{2}}{\ell}}{d} \cdot \frac{w0 \cdot h}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3: 70.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 7.6 \cdot 10^{-10}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{\frac{{\left(M_m \cdot D\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 7.6e-10)
   (+ w0 (* -0.125 (/ (/ (* (pow (* M_m D) 2.0) (/ h (/ l w0))) d) d)))
   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) {
	double tmp;
	if (d <= 7.6e-10) {
		tmp = w0 + (-0.125 * (((pow((M_m * D), 2.0) * (h / (l / w0))) / d) / d));
	} else {
		tmp = w0;
	}
	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_1 <= 7.6d-10) then
        tmp = w0 + ((-0.125d0) * (((((m_m * d) ** 2.0d0) * (h / (l / w0))) / d_1) / d_1))
    else
        tmp = w0
    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 <= 7.6e-10) {
		tmp = w0 + (-0.125 * (((Math.pow((M_m * D), 2.0) * (h / (l / w0))) / d) / d));
	} else {
		tmp = w0;
	}
	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 <= 7.6e-10:
		tmp = w0 + (-0.125 * (((math.pow((M_m * D), 2.0) * (h / (l / w0))) / d) / d))
	else:
		tmp = w0
	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 <= 7.6e-10)
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64((Float64(M_m * D) ^ 2.0) * Float64(h / Float64(l / w0))) / d) / d)));
	else
		tmp = w0;
	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 <= 7.6e-10)
		tmp = w0 + (-0.125 * (((((M_m * D) ^ 2.0) * (h / (l / w0))) / d) / d));
	else
		tmp = w0;
	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, 7.6e-10], N[(w0 + N[(-0.125 * N[(N[(N[(N[Power[N[(M$95$m * D), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / N[(l / w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 7.5999999999999996e-10

   1. Initial program 81.4%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in M around 0 45.1%

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

      1. add-exp-log 34.2%

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

      2. associate-*r* 34.7%

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

      3. pow-prod-down 41.4%

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

   6. Applied egg-rr 41.4%

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

      1. rem-exp-log 56.6%

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

      2. *-commutative 56.6%

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

      3. frac-times 55.5%

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

      4. associate-*r/ 56.6%

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

      5. unpow2 56.6%

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

      6. times-frac 65.0%

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

   8. Applied egg-rr 65.0%

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

      1. associate-*l/ 65.0%

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

      2. associate-*r/ 66.8%

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

      3. associate-*l/ 66.9%

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

      4. associate-*r/ 63.1%

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

      5. associate-/l* 64.2%

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

   10. Simplified 64.2%

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

   if 7.5999999999999996e-10 < d

   1. Initial program 90.0%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in M around 0 87.5%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 7.6 \cdot 10^{-10}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{\frac{{\left(M \cdot D\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4: 67.2% 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 83.7%

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

3. Simplified 84.9%

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

4. Taylor expanded in M around 0 67.4%

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

5. Final simplification 67.4%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023331 
(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))))))