Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 89.3%

Time: 17.6s

Alternatives: 5

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: 81.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - (((((0.5d0 * d) / (d_1 / m_m)) * h) / l) * (0.5d0 * (d / (d_1 / m_m))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

3. Simplified 81.4%

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

   1. associate-*r/ 85.9%

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

   2. add-sqr-sqrt 85.9%

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

   3. pow2 85.9%

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

   4. unpow2 85.9%

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

   5. sqrt-prod 56.2%

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

   6. add-sqr-sqrt 85.9%

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

   7. div-inv 85.9%

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

   8. frac-times 85.9%

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

   9. *-commutative 85.9%

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

   10. *-un-lft-identity 85.9%

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

6. Applied egg-rr 85.9%

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

   1. pow2 85.9%

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

   2. associate-*r/ 81.4%

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

   3. associate-*l* 82.6%

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

   4. *-commutative 82.6%

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

   5. clear-num 82.2%

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

   6. div-inv 82.2%

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

   7. *-un-lft-identity 82.2%

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

   8. associate-/l* 82.2%

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

   9. times-frac 82.2%

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

   10. metadata-eval 82.2%

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

   11. clear-num 82.2%

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

   12. div-inv 82.8%

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

   13. *-un-lft-identity 82.8%

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

   14. associate-/l* 82.8%

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

   15. times-frac 82.8%

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

   16. metadata-eval 82.8%

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

8. Applied egg-rr 82.8%

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

   1. associate-*r/ 88.3%

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

   2. associate-*r/ 88.3%

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

10. Applied egg-rr 88.3%

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

Alternative 2: 84.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := D \cdot \frac{M\_m}{d \cdot 2}\\ \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(t\_0 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (* D (/ M_m (* d 2.0)))))
   (if (<= (/ h l) -5e-324) (* w0 (sqrt (- 1.0 (* (/ h l) (* t_0 t_0))))) w0)))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = D * (M_m / (d * 2.0));
	double tmp;
	if ((h / l) <= -5e-324) {
		tmp = w0 * sqrt((1.0 - ((h / l) * (t_0 * t_0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * (m_m / (d_1 * 2.0d0))
    if ((h / l) <= (-5d-324)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (t_0 * t_0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = D * (M_m / (d * 2.0));
	double tmp;
	if ((h / l) <= -5e-324) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * (t_0 * t_0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	t_0 = D * (M_m / (d * 2.0))
	tmp = 0
	if (h / l) <= -5e-324:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * (t_0 * t_0))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(D * Float64(M_m / Float64(d * 2.0)))
	tmp = 0.0
	if (Float64(h / l) <= -5e-324)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(t_0 * t_0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	t_0 = D * (M_m / (d * 2.0));
	tmp = 0.0;
	if ((h / l) <= -5e-324)
		tmp = w0 * sqrt((1.0 - ((h / l) * (t_0 * t_0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(D * N[(M$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], -5e-324], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -4.94066e-324

   1. Initial program 77.6%

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

   3. Simplified 78.2%

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

      1. unpow-prod-down 65.5%

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

      2. add-sqr-sqrt 65.5%

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

      3. unpow-prod-down 65.5%

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

      4. unpow2 65.5%

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

      5. sqrt-prod 41.0%

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

      6. add-sqr-sqrt 54.3%

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

      7. div-inv 54.4%

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

      8. frac-times 54.3%

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

      9. *-commutative 54.3%

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

      10. *-un-lft-identity 54.3%

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

      11. unpow-prod-down 64.4%

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

      12. unpow2 64.4%

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

      13. sqrt-prod 51.1%

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

      14. add-sqr-sqrt 78.2%

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

      15. div-inv 78.3%

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

      16. frac-times 78.2%

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

      17. *-commutative 78.2%

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

      18. *-un-lft-identity 78.2%

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

   6. Applied egg-rr 78.2%

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

   if -4.94066e-324 < (/.f64 h l)

   1. Initial program 85.8%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in D around 0 95.6%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\left(D \cdot \frac{M}{d \cdot 2}\right) \cdot \left(D \cdot \frac{M}{d \cdot 2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{D}{\frac{d}{M\_m}}\\ \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(t\_0 \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ D (/ d M_m)))))
   (if (<= (/ h l) -5e-324) (* w0 (sqrt (- 1.0 (* t_0 (* t_0 (/ h l)))))) w0)))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = 0.5 * (D / (d / M_m));
	double tmp;
	if ((h / l) <= -5e-324) {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (d / (d_1 / m_m))
    if ((h / l) <= (-5d-324)) then
        tmp = w0 * sqrt((1.0d0 - (t_0 * (t_0 * (h / l)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = 0.5 * (D / (d / M_m));
	double tmp;
	if ((h / l) <= -5e-324) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	t_0 = 0.5 * (D / (d / M_m))
	tmp = 0
	if (h / l) <= -5e-324:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(0.5 * Float64(D / Float64(d / M_m)))
	tmp = 0.0
	if (Float64(h / l) <= -5e-324)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(h / l))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	t_0 = 0.5 * (D / (d / M_m));
	tmp = 0.0;
	if ((h / l) <= -5e-324)
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(0.5 * N[(D / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], -5e-324], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -4.94066e-324

   1. Initial program 77.6%

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

   3. Simplified 78.2%

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

      1. associate-*r/ 80.4%

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

      2. add-sqr-sqrt 80.4%

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

      3. pow2 80.4%

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

      4. unpow2 80.4%

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

      5. sqrt-prod 53.9%

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

      6. add-sqr-sqrt 80.4%

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

      7. div-inv 80.4%

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

      8. frac-times 80.4%

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

      9. *-commutative 80.4%

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

      10. *-un-lft-identity 80.4%

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

   6. Applied egg-rr 80.4%

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

      1. pow2 80.4%

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

      2. associate-*r/ 78.2%

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

      3. associate-*l* 78.2%

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

      4. *-commutative 78.2%

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

      5. clear-num 77.6%

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

      6. div-inv 77.6%

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

      7. *-un-lft-identity 77.6%

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

      8. associate-/l* 77.6%

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

      9. times-frac 77.6%

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

      10. metadata-eval 77.6%

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

      11. clear-num 77.6%

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

      12. div-inv 77.9%

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

      13. *-un-lft-identity 77.9%

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

      14. associate-/l* 77.9%

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

      15. times-frac 77.9%

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

      16. metadata-eval 77.9%

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

   8. Applied egg-rr 77.9%

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

   if -4.94066e-324 < (/.f64 h l)

   1. Initial program 85.8%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in D around 0 95.6%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 270000000000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(w0 \cdot \left(1 + w0 \cdot \left(w0 \cdot 0.3333333333333333 - 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 270000000000.0)
   w0
   (expm1 (* w0 (+ 1.0 (* w0 (- (* w0 0.3333333333333333) 0.5)))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 270000000000.0) {
		tmp = w0;
	} else {
		tmp = expm1((w0 * (1.0 + (w0 * ((w0 * 0.3333333333333333) - 0.5)))));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 270000000000.0) {
		tmp = w0;
	} else {
		tmp = Math.expm1((w0 * (1.0 + (w0 * ((w0 * 0.3333333333333333) - 0.5)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 270000000000.0:
		tmp = w0
	else:
		tmp = math.expm1((w0 * (1.0 + (w0 * ((w0 * 0.3333333333333333) - 0.5)))))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 270000000000.0)
		tmp = w0;
	else
		tmp = expm1(Float64(w0 * Float64(1.0 + Float64(w0 * Float64(Float64(w0 * 0.3333333333333333) - 0.5)))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 270000000000.0], w0, N[(Exp[N[(w0 * N[(1.0 + N[(w0 * N[(N[(w0 * 0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.7e11

   1. Initial program 81.6%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in D around 0 72.2%

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

   if 2.7e11 < M

   1. Initial program 78.4%

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

   3. Simplified 78.5%

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

      1. unpow-prod-down 69.9%

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

      2. add-sqr-sqrt 69.9%

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

      3. unpow-prod-down 69.9%

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

      4. unpow2 69.9%

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

      5. sqrt-prod 32.8%

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

      6. add-sqr-sqrt 61.3%

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

      7. div-inv 61.3%

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

      8. frac-times 61.3%

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

      9. *-commutative 61.3%

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

      10. *-un-lft-identity 61.3%

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

      11. unpow-prod-down 69.9%

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

      12. unpow2 69.9%

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

      13. sqrt-prod 37.2%

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

      14. add-sqr-sqrt 78.5%

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

      15. div-inv 78.5%

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

      16. frac-times 78.5%

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

      17. *-commutative 78.5%

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

      18. *-un-lft-identity 78.5%

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

   6. Applied egg-rr 78.5%

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

      1. expm1-log1p-u 55.0%

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

      2. *-commutative 55.0%

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

   8. Applied egg-rr 58.1%

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

   9. Taylor expanded in h around 0 12.0%

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

       1. log1p-define 45.6%

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

   11. Simplified 45.6%

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

   12. Taylor expanded in w0 around 0 45.0%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 270000000000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(w0 \cdot \left(1 + w0 \cdot \left(w0 \cdot 0.3333333333333333 - 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

3. Simplified 81.4%

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

5. Taylor expanded in D around 0 68.5%

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

Reproduce

herbie shell --seed 2024178 
(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))))))