Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 91.7%

Time: 14.2s

Alternatives: 10

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: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}\\ t_1 := \frac{d\_m}{D\_m \cdot M\_m}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;D\_m \cdot \frac{\sqrt{M\_m \cdot \left(M\_m \cdot \frac{\frac{h}{\ell \cdot -4}}{d\_m}\right)} \cdot w0}{\sqrt{d\_m}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{h}{\ell}}{4 \cdot \left(t\_1 \cdot t\_1\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(D\_m \cdot \left(h \cdot M\_m\right)\right) \cdot \left(\left(D\_m \cdot \frac{M\_m}{d\_m}\right) \cdot \frac{0.25}{d\_m}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)))
        (t_1 (/ d_m (* D_m M_m))))
   (if (<= t_0 (- INFINITY))
     (*
      D_m
      (/ (* (sqrt (* M_m (* M_m (/ (/ h (* l -4.0)) d_m)))) w0) (sqrt d_m)))
     (if (<= t_0 2e-6)
       (* w0 (sqrt (- 1.0 (/ (/ h l) (* 4.0 (* t_1 t_1))))))
       (*
        w0
        (sqrt
         (-
          1.0
          (/
           (* (* D_m (* h M_m)) (* (* D_m (/ M_m d_m)) (/ 0.25 d_m)))
           l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double t_1 = d_m / (D_m * M_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = D_m * ((sqrt((M_m * (M_m * ((h / (l * -4.0)) / d_m)))) * w0) / sqrt(d_m));
	} else if (t_0 <= 2e-6) {
		tmp = w0 * sqrt((1.0 - ((h / l) / (4.0 * (t_1 * t_1)))));
	} else {
		tmp = w0 * sqrt((1.0 - (((D_m * (h * M_m)) * ((D_m * (M_m / d_m)) * (0.25 / d_m))) / l)));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double t_1 = d_m / (D_m * M_m);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = D_m * ((Math.sqrt((M_m * (M_m * ((h / (l * -4.0)) / d_m)))) * w0) / Math.sqrt(d_m));
	} else if (t_0 <= 2e-6) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) / (4.0 * (t_1 * t_1)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((D_m * (h * M_m)) * ((D_m * (M_m / d_m)) * (0.25 / d_m))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)
	t_1 = d_m / (D_m * M_m)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = D_m * ((math.sqrt((M_m * (M_m * ((h / (l * -4.0)) / d_m)))) * w0) / math.sqrt(d_m))
	elif t_0 <= 2e-6:
		tmp = w0 * math.sqrt((1.0 - ((h / l) / (4.0 * (t_1 * t_1)))))
	else:
		tmp = w0 * math.sqrt((1.0 - (((D_m * (h * M_m)) * ((D_m * (M_m / d_m)) * (0.25 / d_m))) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l))
	t_1 = Float64(d_m / Float64(D_m * M_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(D_m * Float64(Float64(sqrt(Float64(M_m * Float64(M_m * Float64(Float64(h / Float64(l * -4.0)) / d_m)))) * w0) / sqrt(d_m)));
	elseif (t_0 <= 2e-6)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) / Float64(4.0 * Float64(t_1 * t_1))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D_m * Float64(h * M_m)) * Float64(Float64(D_m * Float64(M_m / d_m)) * Float64(0.25 / d_m))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = (((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l);
	t_1 = d_m / (D_m * M_m);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = D_m * ((sqrt((M_m * (M_m * ((h / (l * -4.0)) / d_m)))) * w0) / sqrt(d_m));
	elseif (t_0 <= 2e-6)
		tmp = w0 * sqrt((1.0 - ((h / l) / (4.0 * (t_1 * t_1)))));
	else
		tmp = w0 * sqrt((1.0 - (((D_m * (h * M_m)) * ((D_m * (M_m / d_m)) * (0.25 / d_m))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d$95$m / N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(D$95$m * N[(N[(N[Sqrt[N[(M$95$m * N[(M$95$m * N[(N[(h / N[(l * -4.0), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision] / N[Sqrt[d$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-6], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] / N[(4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D$95$m * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.25 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -inf.0

   1. Initial program 49.6%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in h around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]

   13. Applied egg-rr 0

       \[\leadsto expr\]

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 1.99999999999999991e-6

   1. Initial program 99.9%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   if 1.99999999999999991e-6 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.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

      \[\leadsto expr\]
   4. Add Preprocessing

   6. Applied egg-rr 0

      \[\leadsto expr\]

   8. Applied egg-rr 0

      \[\leadsto expr\]

   10. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 92.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}\\ t_1 := \frac{d\_m}{D\_m \cdot M\_m}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;D\_m \cdot \left(\sqrt{\frac{M\_m \cdot \frac{-0.25 \cdot \left(h \cdot \frac{M\_m}{d\_m}\right)}{\ell}}{d\_m}} \cdot w0\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{h}{\ell}}{4 \cdot \left(t\_1 \cdot t\_1\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(D\_m \cdot \left(h \cdot M\_m\right)\right) \cdot \left(\left(D\_m \cdot \frac{M\_m}{d\_m}\right) \cdot \frac{0.25}{d\_m}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)))
        (t_1 (/ d_m (* D_m M_m))))
   (if (<= t_0 (- INFINITY))
     (* D_m (* (sqrt (/ (* M_m (/ (* -0.25 (* h (/ M_m d_m))) l)) d_m)) w0))
     (if (<= t_0 2e-6)
       (* w0 (sqrt (- 1.0 (/ (/ h l) (* 4.0 (* t_1 t_1))))))
       (*
        w0
        (sqrt
         (-
          1.0
          (/
           (* (* D_m (* h M_m)) (* (* D_m (/ M_m d_m)) (/ 0.25 d_m)))
           l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double t_1 = d_m / (D_m * M_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = D_m * (sqrt(((M_m * ((-0.25 * (h * (M_m / d_m))) / l)) / d_m)) * w0);
	} else if (t_0 <= 2e-6) {
		tmp = w0 * sqrt((1.0 - ((h / l) / (4.0 * (t_1 * t_1)))));
	} else {
		tmp = w0 * sqrt((1.0 - (((D_m * (h * M_m)) * ((D_m * (M_m / d_m)) * (0.25 / d_m))) / l)));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double t_1 = d_m / (D_m * M_m);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = D_m * (Math.sqrt(((M_m * ((-0.25 * (h * (M_m / d_m))) / l)) / d_m)) * w0);
	} else if (t_0 <= 2e-6) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) / (4.0 * (t_1 * t_1)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((D_m * (h * M_m)) * ((D_m * (M_m / d_m)) * (0.25 / d_m))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)
	t_1 = d_m / (D_m * M_m)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = D_m * (math.sqrt(((M_m * ((-0.25 * (h * (M_m / d_m))) / l)) / d_m)) * w0)
	elif t_0 <= 2e-6:
		tmp = w0 * math.sqrt((1.0 - ((h / l) / (4.0 * (t_1 * t_1)))))
	else:
		tmp = w0 * math.sqrt((1.0 - (((D_m * (h * M_m)) * ((D_m * (M_m / d_m)) * (0.25 / d_m))) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l))
	t_1 = Float64(d_m / Float64(D_m * M_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(D_m * Float64(sqrt(Float64(Float64(M_m * Float64(Float64(-0.25 * Float64(h * Float64(M_m / d_m))) / l)) / d_m)) * w0));
	elseif (t_0 <= 2e-6)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) / Float64(4.0 * Float64(t_1 * t_1))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D_m * Float64(h * M_m)) * Float64(Float64(D_m * Float64(M_m / d_m)) * Float64(0.25 / d_m))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = (((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l);
	t_1 = d_m / (D_m * M_m);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = D_m * (sqrt(((M_m * ((-0.25 * (h * (M_m / d_m))) / l)) / d_m)) * w0);
	elseif (t_0 <= 2e-6)
		tmp = w0 * sqrt((1.0 - ((h / l) / (4.0 * (t_1 * t_1)))));
	else
		tmp = w0 * sqrt((1.0 - (((D_m * (h * M_m)) * ((D_m * (M_m / d_m)) * (0.25 / d_m))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d$95$m / N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(D$95$m * N[(N[Sqrt[N[(N[(M$95$m * N[(N[(-0.25 * N[(h * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-6], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] / N[(4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D$95$m * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.25 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -inf.0

   1. Initial program 49.6%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in h around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]

   12. Taylor expanded in M around 0 0

       \[\leadsto expr\]

   14. Simplified 0

       \[\leadsto expr\]

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 1.99999999999999991e-6

   1. Initial program 99.9%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   if 1.99999999999999991e-6 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.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

      \[\leadsto expr\]
   4. Add Preprocessing

   6. Applied egg-rr 0

      \[\leadsto expr\]

   8. Applied egg-rr 0

      \[\leadsto expr\]

   10. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-255}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{\frac{D\_m}{\frac{d\_m}{D\_m \cdot M\_m}} \cdot M\_m}{d\_m}}{-4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ h l) -1e-255)
   (*
    w0
    (sqrt
     (+ 1.0 (/ (* h (/ (/ (* (/ D_m (/ d_m (* D_m M_m))) M_m) d_m) -4.0)) l))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -1e-255) {
		tmp = w0 * sqrt((1.0 + ((h * ((((D_m / (d_m / (D_m * M_m))) * M_m) / d_m) / -4.0)) / l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((h / l) <= (-1d-255)) then
        tmp = w0 * sqrt((1.0d0 + ((h * ((((d_m / (d_m_1 / (d_m * m_m))) * m_m) / d_m_1) / (-4.0d0))) / l)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -1e-255) {
		tmp = w0 * Math.sqrt((1.0 + ((h * ((((D_m / (d_m / (D_m * M_m))) * M_m) / d_m) / -4.0)) / l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (h / l) <= -1e-255:
		tmp = w0 * math.sqrt((1.0 + ((h * ((((D_m / (d_m / (D_m * M_m))) * M_m) / d_m) / -4.0)) / l)))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -1e-255)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h * Float64(Float64(Float64(Float64(D_m / Float64(d_m / Float64(D_m * M_m))) * M_m) / d_m) / -4.0)) / l))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -1e-255)
		tmp = w0 * sqrt((1.0 + ((h * ((((D_m / (d_m / (D_m * M_m))) * M_m) / d_m) / -4.0)) / l)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e-255], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(h * N[(N[(N[(N[(D$95$m / N[(d$95$m / N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] / d$95$m), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1e-255

   1. Initial program 77.5%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   6. Applied egg-rr 0

      \[\leadsto expr\]

   if -1e-255 < (/.f64 h l)

   1. Initial program 89.1%

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

   3. Taylor expanded in M around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-255}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{M\_m}{d\_m} \cdot \frac{D\_m}{\frac{d\_m}{D\_m \cdot M\_m}}}{\ell \cdot -4} \cdot h}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ h l) -1e-255)
   (*
    w0
    (sqrt
     (+ 1.0 (* (/ (* (/ M_m d_m) (/ D_m (/ d_m (* D_m M_m)))) (* l -4.0)) h))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -1e-255) {
		tmp = w0 * sqrt((1.0 + ((((M_m / d_m) * (D_m / (d_m / (D_m * M_m)))) / (l * -4.0)) * h)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((h / l) <= (-1d-255)) then
        tmp = w0 * sqrt((1.0d0 + ((((m_m / d_m_1) * (d_m / (d_m_1 / (d_m * m_m)))) / (l * (-4.0d0))) * h)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -1e-255) {
		tmp = w0 * Math.sqrt((1.0 + ((((M_m / d_m) * (D_m / (d_m / (D_m * M_m)))) / (l * -4.0)) * h)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (h / l) <= -1e-255:
		tmp = w0 * math.sqrt((1.0 + ((((M_m / d_m) * (D_m / (d_m / (D_m * M_m)))) / (l * -4.0)) * h)))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -1e-255)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(M_m / d_m) * Float64(D_m / Float64(d_m / Float64(D_m * M_m)))) / Float64(l * -4.0)) * h))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -1e-255)
		tmp = w0 * sqrt((1.0 + ((((M_m / d_m) * (D_m / (d_m / (D_m * M_m)))) / (l * -4.0)) * h)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e-255], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(M$95$m / d$95$m), $MachinePrecision] * N[(D$95$m / N[(d$95$m / N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * -4.0), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1e-255

   1. Initial program 77.5%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   6. Applied egg-rr 0

      \[\leadsto expr\]

   if -1e-255 < (/.f64 h l)

   1. Initial program 89.1%

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

   3. Taylor expanded in M around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d\_m}\\ \mathbf{if}\;d\_m \leq 10^{+18}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{\frac{-4}{t\_0 \cdot t\_0}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(M\_m \cdot h\right) \cdot \left(\frac{\frac{D\_m}{d\_m}}{4} \cdot t\_0\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d_m))))
   (if (<= d_m 1e+18)
     (* w0 (sqrt (+ 1.0 (/ (/ h (/ -4.0 (* t_0 t_0))) l))))
     (* w0 (sqrt (- 1.0 (/ (* (* M_m h) (* (/ (/ D_m d_m) 4.0) t_0)) l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = D_m * (M_m / d_m);
	double tmp;
	if (d_m <= 1e+18) {
		tmp = w0 * sqrt((1.0 + ((h / (-4.0 / (t_0 * t_0))) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (((M_m * h) * (((D_m / d_m) / 4.0) * t_0)) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_m * (m_m / d_m_1)
    if (d_m_1 <= 1d+18) then
        tmp = w0 * sqrt((1.0d0 + ((h / ((-4.0d0) / (t_0 * t_0))) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (((m_m * h) * (((d_m / d_m_1) / 4.0d0) * t_0)) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = D_m * (M_m / d_m);
	double tmp;
	if (d_m <= 1e+18) {
		tmp = w0 * Math.sqrt((1.0 + ((h / (-4.0 / (t_0 * t_0))) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((M_m * h) * (((D_m / d_m) / 4.0) * t_0)) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = D_m * (M_m / d_m)
	tmp = 0
	if d_m <= 1e+18:
		tmp = w0 * math.sqrt((1.0 + ((h / (-4.0 / (t_0 * t_0))) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (((M_m * h) * (((D_m / d_m) / 4.0) * t_0)) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(D_m * Float64(M_m / d_m))
	tmp = 0.0
	if (d_m <= 1e+18)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h / Float64(-4.0 / Float64(t_0 * t_0))) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(M_m * h) * Float64(Float64(Float64(D_m / d_m) / 4.0) * t_0)) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = D_m * (M_m / d_m);
	tmp = 0.0;
	if (d_m <= 1e+18)
		tmp = w0 * sqrt((1.0 + ((h / (-4.0 / (t_0 * t_0))) / l)));
	else
		tmp = w0 * sqrt((1.0 - (((M_m * h) * (((D_m / d_m) / 4.0) * t_0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d$95$m, 1e+18], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(h / N[(-4.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(M$95$m * h), $MachinePrecision] * N[(N[(N[(D$95$m / d$95$m), $MachinePrecision] / 4.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 1e18

   1. Initial program 85.7%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   6. Applied egg-rr 0

      \[\leadsto expr\]

   8. Applied egg-rr 0

      \[\leadsto expr\]

   if 1e18 < d

   1. Initial program 75.7%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   6. Applied egg-rr 0

      \[\leadsto expr\]

   8. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d\_m}\\ w0 \cdot \sqrt{1 + \frac{\frac{h}{\frac{-4}{t\_0 \cdot t\_0}}}{\ell}} \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d_m))))
   (* w0 (sqrt (+ 1.0 (/ (/ h (/ -4.0 (* t_0 t_0))) l))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = D_m * (M_m / d_m);
	return w0 * sqrt((1.0 + ((h / (-4.0 / (t_0 * t_0))) / l)));
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    t_0 = d_m * (m_m / d_m_1)
    code = w0 * sqrt((1.0d0 + ((h / ((-4.0d0) / (t_0 * t_0))) / l)))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = D_m * (M_m / d_m);
	return w0 * Math.sqrt((1.0 + ((h / (-4.0 / (t_0 * t_0))) / l)));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = D_m * (M_m / d_m)
	return w0 * math.sqrt((1.0 + ((h / (-4.0 / (t_0 * t_0))) / l)))

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(D_m * Float64(M_m / d_m))
	return Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h / Float64(-4.0 / Float64(t_0 * t_0))) / l))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp = code(w0, M_m, D_m, h, l, d_m)
	t_0 = D_m * (M_m / d_m);
	tmp = w0 * sqrt((1.0 + ((h / (-4.0 / (t_0 * t_0))) / l)));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 + N[(N[(h / N[(-4.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 82.9%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

8. Applied egg-rr 0

   \[\leadsto expr\]
9. Add Preprocessing

Alternative 7: 77.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;d\_m \leq 1.92 \cdot 10^{-76}:\\ \;\;\;\;w0 \cdot \left(1 + \left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{\frac{M\_m}{d\_m}}{\frac{d\_m}{M\_m}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\frac{\frac{h}{d\_m \cdot d\_m}}{\frac{\ell}{M\_m \cdot M\_m}} \cdot \left(D\_m \cdot -0.125\right)\right) \cdot D\_m\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= d_m 1.92e-76)
   (*
    w0
    (+ 1.0 (* (* -0.125 (* D_m D_m)) (* (/ h l) (/ (/ M_m d_m) (/ d_m M_m))))))
   (*
    w0
    (+
     1.0
     (* (* (/ (/ h (* d_m d_m)) (/ l (* M_m M_m))) (* D_m -0.125)) D_m)))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 1.92e-76) {
		tmp = w0 * (1.0 + ((-0.125 * (D_m * D_m)) * ((h / l) * ((M_m / d_m) / (d_m / M_m)))));
	} else {
		tmp = w0 * (1.0 + ((((h / (d_m * d_m)) / (l / (M_m * M_m))) * (D_m * -0.125)) * D_m));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (d_m_1 <= 1.92d-76) then
        tmp = w0 * (1.0d0 + (((-0.125d0) * (d_m * d_m)) * ((h / l) * ((m_m / d_m_1) / (d_m_1 / m_m)))))
    else
        tmp = w0 * (1.0d0 + ((((h / (d_m_1 * d_m_1)) / (l / (m_m * m_m))) * (d_m * (-0.125d0))) * d_m))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 1.92e-76) {
		tmp = w0 * (1.0 + ((-0.125 * (D_m * D_m)) * ((h / l) * ((M_m / d_m) / (d_m / M_m)))));
	} else {
		tmp = w0 * (1.0 + ((((h / (d_m * d_m)) / (l / (M_m * M_m))) * (D_m * -0.125)) * D_m));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if d_m <= 1.92e-76:
		tmp = w0 * (1.0 + ((-0.125 * (D_m * D_m)) * ((h / l) * ((M_m / d_m) / (d_m / M_m)))))
	else:
		tmp = w0 * (1.0 + ((((h / (d_m * d_m)) / (l / (M_m * M_m))) * (D_m * -0.125)) * D_m))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (d_m <= 1.92e-76)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 * Float64(D_m * D_m)) * Float64(Float64(h / l) * Float64(Float64(M_m / d_m) / Float64(d_m / M_m))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(Float64(h / Float64(d_m * d_m)) / Float64(l / Float64(M_m * M_m))) * Float64(D_m * -0.125)) * D_m)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (d_m <= 1.92e-76)
		tmp = w0 * (1.0 + ((-0.125 * (D_m * D_m)) * ((h / l) * ((M_m / d_m) / (d_m / M_m)))));
	else
		tmp = w0 * (1.0 + ((((h / (d_m * d_m)) / (l / (M_m * M_m))) * (D_m * -0.125)) * D_m));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[d$95$m, 1.92e-76], N[(w0 * N[(1.0 + N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(N[(M$95$m / d$95$m), $MachinePrecision] / N[(d$95$m / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(N[(N[(h / N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(l / N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * -0.125), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.91999999999999992e-76

   1. Initial program 84.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in h around 0 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]

   if 1.91999999999999992e-76 < d

   1. Initial program 78.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in h around 0 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;d\_m \leq 3.8 \cdot 10^{-88}:\\ \;\;\;\;w0 \cdot \left(1 + \left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{\frac{M\_m}{d\_m}}{\frac{d\_m}{M\_m}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= d_m 3.8e-88)
   (*
    w0
    (+ 1.0 (* (* -0.125 (* D_m D_m)) (* (/ h l) (/ (/ M_m d_m) (/ d_m M_m))))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 3.8e-88) {
		tmp = w0 * (1.0 + ((-0.125 * (D_m * D_m)) * ((h / l) * ((M_m / d_m) / (d_m / M_m)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (d_m_1 <= 3.8d-88) then
        tmp = w0 * (1.0d0 + (((-0.125d0) * (d_m * d_m)) * ((h / l) * ((m_m / d_m_1) / (d_m_1 / m_m)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 3.8e-88) {
		tmp = w0 * (1.0 + ((-0.125 * (D_m * D_m)) * ((h / l) * ((M_m / d_m) / (d_m / M_m)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if d_m <= 3.8e-88:
		tmp = w0 * (1.0 + ((-0.125 * (D_m * D_m)) * ((h / l) * ((M_m / d_m) / (d_m / M_m)))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (d_m <= 3.8e-88)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 * Float64(D_m * D_m)) * Float64(Float64(h / l) * Float64(Float64(M_m / d_m) / Float64(d_m / M_m))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (d_m <= 3.8e-88)
		tmp = w0 * (1.0 + ((-0.125 * (D_m * D_m)) * ((h / l) * ((M_m / d_m) / (d_m / M_m)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[d$95$m, 3.8e-88], N[(w0 * N[(1.0 + N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(N[(M$95$m / d$95$m), $MachinePrecision] / N[(d$95$m / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 3.80000000000000011e-88

   1. Initial program 84.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in h around 0 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]

   if 3.80000000000000011e-88 < d

   1. Initial program 79.2%

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

   3. Taylor expanded in M around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 70.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.05 \cdot 10^{+154}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left(\frac{h \cdot \frac{M\_m \cdot M\_m}{d\_m}}{\frac{\ell}{-0.125}} \cdot D\_m\right) \cdot \frac{D\_m}{d\_m}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 1.05e+154)
   w0
   (* w0 (* (* (/ (* h (/ (* M_m M_m) d_m)) (/ l -0.125)) D_m) (/ D_m d_m)))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 1.05e+154) {
		tmp = w0;
	} else {
		tmp = w0 * ((((h * ((M_m * M_m) / d_m)) / (l / -0.125)) * D_m) * (D_m / d_m));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 1.05d+154) then
        tmp = w0
    else
        tmp = w0 * ((((h * ((m_m * m_m) / d_m_1)) / (l / (-0.125d0))) * d_m) * (d_m / d_m_1))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 1.05e+154) {
		tmp = w0;
	} else {
		tmp = w0 * ((((h * ((M_m * M_m) / d_m)) / (l / -0.125)) * D_m) * (D_m / d_m));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 1.05e+154:
		tmp = w0
	else:
		tmp = w0 * ((((h * ((M_m * M_m) / d_m)) / (l / -0.125)) * D_m) * (D_m / d_m))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 1.05e+154)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(Float64(Float64(Float64(h * Float64(Float64(M_m * M_m) / d_m)) / Float64(l / -0.125)) * D_m) * Float64(D_m / d_m)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 1.05e+154)
		tmp = w0;
	else
		tmp = w0 * ((((h * ((M_m * M_m) / d_m)) / (l / -0.125)) * D_m) * (D_m / d_m));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 1.05e+154], w0, N[(w0 * N[(N[(N[(N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] / N[(l / -0.125), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.04999999999999997e154

   1. Initial program 84.3%

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

   3. Taylor expanded in M around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 1.04999999999999997e154 < M

   1. Initial program 70.4%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in h around 0 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]

   10. Taylor expanded in D around inf 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]

   14. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.7% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in M around 0 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(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))))))