Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 87.6%

Time: 16.0s

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: 80.7% 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: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ d_m = \left|d\right| \\ [w0, M_m, D, h, l, d_m] = \mathsf{sort}([w0, M_m, D, h, l, d_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M_m \cdot D}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;w0 \cdot \left(\frac{D}{d_m} \cdot \sqrt{\frac{{M_m}^{2}}{\frac{\ell}{h}} \cdot -0.25}\right)\\ \mathbf{elif}\;t_0 \leq 0.5:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d_m)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M_m D) (* 2.0 d_m)) 2.0) (/ h l))))
   (if (<= t_0 (- INFINITY))
     (* w0 (* (/ D d_m) (sqrt (* (/ (pow M_m 2.0) (/ l h)) -0.25))))
     (if (<= t_0 0.5) (* w0 (sqrt (- 1.0 t_0))) w0))))

C

M_m = fabs(M);
d_m = fabs(d);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d_m);
double code(double w0, double M_m, double D, double h, double l, double d_m) {
	double t_0 = pow(((M_m * D) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = w0 * ((D / d_m) * sqrt(((pow(M_m, 2.0) / (l / h)) * -0.25)));
	} else if (t_0 <= 0.5) {
		tmp = w0 * sqrt((1.0 - t_0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

M_m = Math.abs(M);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D && D < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D, double h, double l, double d_m) {
	double t_0 = Math.pow(((M_m * D) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * ((D / d_m) * Math.sqrt(((Math.pow(M_m, 2.0) / (l / h)) * -0.25)));
	} else if (t_0 <= 0.5) {
		tmp = w0 * Math.sqrt((1.0 - t_0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
d_m = math.fabs(d)
[w0, M_m, D, h, l, d_m] = sort([w0, M_m, D, h, l, d_m])
def code(w0, M_m, D, h, l, d_m):
	t_0 = math.pow(((M_m * D) / (2.0 * d_m)), 2.0) * (h / l)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = w0 * ((D / d_m) * math.sqrt(((math.pow(M_m, 2.0) / (l / h)) * -0.25)))
	elif t_0 <= 0.5:
		tmp = w0 * math.sqrt((1.0 - t_0))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
d_m = abs(d)
w0, M_m, D, h, l, d_m = sort([w0, M_m, D, h, l, d_m])
function code(w0, M_m, D, h, l, d_m)
	t_0 = Float64((Float64(Float64(M_m * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(w0 * Float64(Float64(D / d_m) * sqrt(Float64(Float64((M_m ^ 2.0) / Float64(l / h)) * -0.25))));
	elseif (t_0 <= 0.5)
		tmp = Float64(w0 * sqrt(Float64(1.0 - t_0)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
d_m = abs(d);
w0, M_m, D, h, l, d_m = num2cell(sort([w0, M_m, D, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D, h, l, d_m)
	t_0 = (((M_m * D) / (2.0 * d_m)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = w0 * ((D / d_m) * sqrt((((M_m ^ 2.0) / (l / h)) * -0.25)));
	elseif (t_0 <= 0.5)
		tmp = w0 * sqrt((1.0 - t_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, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(w0 * N[(N[(D / d$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[Power[M$95$m, 2.0], $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(w0 * N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -inf.0

   1. Initial program 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in D around inf 51.8%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 51.8%

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot -0.25}} \]

      2. times-frac 51.7%

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \cdot -0.25} \]

   6. Simplified 51.7%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot -0.25}} \]
   7. Step-by-step derivation

      1. associate-*l* 51.7%

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{{D}^{2}}{{d}^{2}} \cdot \left(\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25\right)}} \]

      2. sqrt-prod 53.2%

         \[\leadsto w0 \cdot \color{blue}{\left(\sqrt{\frac{{D}^{2}}{{d}^{2}}} \cdot \sqrt{\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25}\right)} \]

      3. sqrt-div 53.2%

         \[\leadsto w0 \cdot \left(\color{blue}{\frac{\sqrt{{D}^{2}}}{\sqrt{{d}^{2}}}} \cdot \sqrt{\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25}\right) \]

      4. unpow2 53.2%

         \[\leadsto w0 \cdot \left(\frac{\sqrt{\color{blue}{D \cdot D}}}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25}\right) \]

      5. sqrt-prod 24.3%

         \[\leadsto w0 \cdot \left(\frac{\color{blue}{\sqrt{D} \cdot \sqrt{D}}}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25}\right) \]

      6. add-sqr-sqrt 24.3%

         \[\leadsto w0 \cdot \left(\frac{\color{blue}{D}}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25}\right) \]

      7. unpow2 24.3%

         \[\leadsto w0 \cdot \left(\frac{D}{\sqrt{\color{blue}{d \cdot d}}} \cdot \sqrt{\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25}\right) \]

      8. sqrt-prod 13.8%

         \[\leadsto w0 \cdot \left(\frac{D}{\color{blue}{\sqrt{d} \cdot \sqrt{d}}} \cdot \sqrt{\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25}\right) \]

      9. add-sqr-sqrt 30.5%

         \[\leadsto w0 \cdot \left(\frac{D}{\color{blue}{d}} \cdot \sqrt{\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25}\right) \]

      10. associate-/l* 30.5%

          \[\leadsto w0 \cdot \left(\frac{D}{d} \cdot \sqrt{\color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}} \cdot -0.25}\right) \]

   8. Applied egg-rr 30.5%

      \[\leadsto w0 \cdot \color{blue}{\left(\frac{D}{d} \cdot \sqrt{\frac{{M}^{2}}{\frac{\ell}{h}} \cdot -0.25}\right)} \]

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 0.5

   1. Initial program 99.9%

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

   if 0.5 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in M around 0 72.4%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \left(\frac{D}{d} \cdot \sqrt{\frac{{M}^{2}}{\frac{\ell}{h}} \cdot -0.25}\right)\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.5:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ d_m = \left|d\right| \\ [w0, M_m, D, h, l, d_m] = \mathsf{sort}([w0, M_m, D, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{+288}:\\ \;\;\;\;w0 \cdot \left(1 + \left(h \cdot \frac{{\left(M_m \cdot \frac{D}{d_m}\right)}^{2}}{\ell}\right) \cdot -0.125\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-281}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{M_m}{2 \cdot d_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d_m)
 :precision binary64
 (if (<= (/ h l) -2e+288)
   (* w0 (+ 1.0 (* (* h (/ (pow (* M_m (/ D d_m)) 2.0) l)) -0.125)))
   (if (<= (/ h l) -1e-281)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* D (/ M_m (* 2.0 d_m))) 2.0)))))
     w0)))

C

M_m = fabs(M);
d_m = fabs(d);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d_m);
double code(double w0, double M_m, double D, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -2e+288) {
		tmp = w0 * (1.0 + ((h * (pow((M_m * (D / d_m)), 2.0) / l)) * -0.125));
	} else if ((h / l) <= -1e-281) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D * (M_m / (2.0 * d_m))), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(M)
d_m = abs(d)
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if ((h / l) <= (-2d+288)) then
        tmp = w0 * (1.0d0 + ((h * (((m_m * (d / d_m)) ** 2.0d0) / l)) * (-0.125d0)))
    else if ((h / l) <= (-1d-281)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d * (m_m / (2.0d0 * d_m))) ** 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 && D < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -2e+288) {
		tmp = w0 * (1.0 + ((h * (Math.pow((M_m * (D / d_m)), 2.0) / l)) * -0.125));
	} else if ((h / l) <= -1e-281) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D * (M_m / (2.0 * d_m))), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
d_m = math.fabs(d)
[w0, M_m, D, h, l, d_m] = sort([w0, M_m, D, h, l, d_m])
def code(w0, M_m, D, h, l, d_m):
	tmp = 0
	if (h / l) <= -2e+288:
		tmp = w0 * (1.0 + ((h * (math.pow((M_m * (D / d_m)), 2.0) / l)) * -0.125))
	elif (h / l) <= -1e-281:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((D * (M_m / (2.0 * d_m))), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
d_m = abs(d)
w0, M_m, D, h, l, d_m = sort([w0, M_m, D, h, l, d_m])
function code(w0, M_m, D, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -2e+288)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(h * Float64((Float64(M_m * Float64(D / d_m)) ^ 2.0) / l)) * -0.125)));
	elseif (Float64(h / l) <= -1e-281)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D * Float64(M_m / Float64(2.0 * d_m))) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
d_m = abs(d);
w0, M_m, D, h, l, d_m = num2cell(sort([w0, M_m, D, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -2e+288)
		tmp = w0 * (1.0 + ((h * (((M_m * (D / d_m)) ^ 2.0) / l)) * -0.125));
	elseif ((h / l) <= -1e-281)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D * (M_m / (2.0 * d_m))) ^ 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, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -2e+288], N[(w0 * N[(1.0 + N[(N[(h * N[(N[Power[N[(M$95$m * N[(D / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-281], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M$95$m / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -2e288

   1. Initial program 42.0%

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

   3. Simplified 42.0%

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

   4. Taylor expanded in M around 0 41.9%

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

      1. *-commutative 41.9%

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

      2. times-frac 45.5%

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

   6. Simplified 45.5%

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

      1. add-sqr-sqrt 45.5%

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

      2. sqrt-div 45.5%

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

      3. unpow2 45.5%

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

      4. sqrt-prod 26.5%

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

      5. add-sqr-sqrt 34.2%

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

      6. unpow2 34.2%

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

      7. sqrt-prod 11.1%

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

      8. add-sqr-sqrt 33.7%

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

      9. sqrt-div 33.7%

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

      10. unpow2 33.7%

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

      11. sqrt-prod 30.2%

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

      12. add-sqr-sqrt 49.2%

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

      13. unpow2 49.2%

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

      14. sqrt-prod 30.5%

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

      15. add-sqr-sqrt 57.3%

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

   8. Applied egg-rr 57.3%

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

      1. expm1-log1p-u 15.5%

         \[\leadsto w0 \cdot \left(1 + \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{M}^{2} \cdot h}{\ell}\right)\right)}\right) \cdot -0.125\right) \]

      2. expm1-udef 15.5%

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

      3. associate-/l* 3.7%

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

   10. Applied egg-rr 3.7%

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

       1. expm1-def 3.7%

          \[\leadsto w0 \cdot \left(1 + \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{M}^{2}}{\frac{\ell}{h}}\right)\right)}\right) \cdot -0.125\right) \]

       2. expm1-log1p 41.9%

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

   12. Simplified 41.9%

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

       1. expm1-log1p-u 14.8%

          \[\leadsto w0 \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{{M}^{2}}{\frac{\ell}{h}}\right)\right)} \cdot -0.125\right) \]

       2. expm1-udef 14.8%

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

       3. associate-*r/ 14.8%

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

       4. pow2 14.8%

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

       5. pow-prod-down 15.0%

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

   14. Applied egg-rr 15.0%

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

       1. expm1-def 15.0%

          \[\leadsto w0 \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{D}{d} \cdot M\right)}^{2}}{\frac{\ell}{h}}\right)\right)} \cdot -0.125\right) \]

       2. expm1-log1p 42.2%

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

       3. associate-/r/ 64.3%

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

       4. *-commutative 64.3%

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

   16. Simplified 64.3%

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

   if -2e288 < (/.f64 h l) < -1e-281

   1. Initial program 86.1%

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

   3. Simplified 84.5%

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

   if -1e-281 < (/.f64 h l)

   1. Initial program 88.9%

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

   3. Simplified 87.9%

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

   4. Taylor expanded in M around 0 93.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{+288}:\\ \;\;\;\;w0 \cdot \left(1 + \left(h \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot -0.125\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-281}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3: 80.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Simplified 81.3%

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

4. Taylor expanded in M around 0 54.2%

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

   1. *-commutative 54.2%

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

   2. times-frac 55.9%

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

6. Simplified 55.9%

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

   1. add-sqr-sqrt 55.9%

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

   2. sqrt-div 55.9%

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

   3. unpow2 55.9%

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

   4. sqrt-prod 29.4%

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

   5. add-sqr-sqrt 49.1%

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

   6. unpow2 49.1%

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

   7. sqrt-prod 23.1%

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

   8. add-sqr-sqrt 49.7%

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

   9. sqrt-div 49.7%

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

   10. unpow2 49.7%

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

   11. sqrt-prod 29.3%

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

   12. add-sqr-sqrt 55.8%

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

   13. unpow2 55.8%

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

   14. sqrt-prod 31.6%

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

   15. add-sqr-sqrt 67.2%

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

8. Applied egg-rr 67.2%

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

   1. expm1-log1p-u 49.4%

      \[\leadsto w0 \cdot \left(1 + \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{M}^{2} \cdot h}{\ell}\right)\right)}\right) \cdot -0.125\right) \]

   2. expm1-udef 47.9%

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

   3. associate-/l* 46.4%

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

10. Applied egg-rr 46.4%

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

    1. expm1-def 47.9%

       \[\leadsto w0 \cdot \left(1 + \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{M}^{2}}{\frac{\ell}{h}}\right)\right)}\right) \cdot -0.125\right) \]

    2. expm1-log1p 66.1%

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

12. Simplified 66.1%

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

    1. expm1-log1p-u 52.5%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{{M}^{2}}{\frac{\ell}{h}}\right)\right)} \cdot -0.125\right) \]

    2. expm1-udef 52.5%

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

    3. associate-*r/ 52.9%

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

    4. pow2 52.9%

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

    5. pow-prod-down 61.0%

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

14. Applied egg-rr 61.0%

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

    1. expm1-def 61.0%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{D}{d} \cdot M\right)}^{2}}{\frac{\ell}{h}}\right)\right)} \cdot -0.125\right) \]

    2. expm1-log1p 75.4%

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

    3. associate-/r/ 79.2%

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

    4. *-commutative 79.2%

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

16. Simplified 79.2%

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

17. Final simplification 79.2%

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

Alternative 4: 68.2% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Simplified 81.3%

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

4. Taylor expanded in M around 0 67.7%

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

5. Final simplification 67.7%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023318 
(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))))))