Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 86.6%

Time: 18.3s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.3% 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: 86.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+26}:\\ \;\;\;\;w0 \cdot {\left(e^{0.25 \cdot \left(\log \left(-0.25 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{d}\right)}^{2}\right)\right) + -2 \cdot \log \left(\frac{1}{D\_m}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e+26)
   (*
    w0
    (pow
     (exp
      (*
       0.25
       (+
        (log (* -0.25 (* (/ h l) (pow (/ M_m d) 2.0))))
        (* -2.0 (log (/ 1.0 D_m))))))
     2.0))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+26) {
		tmp = w0 * pow(exp((0.25 * (log((-0.25 * ((h / l) * pow((M_m / d), 2.0)))) + (-2.0 * log((1.0 / D_m)))))), 2.0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    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
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-2d+26)) then
        tmp = w0 * (exp((0.25d0 * (log(((-0.25d0) * ((h / l) * ((m_m / d) ** 2.0d0)))) + ((-2.0d0) * log((1.0d0 / d_m)))))) ** 2.0d0)
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+26) {
		tmp = w0 * Math.pow(Math.exp((0.25 * (Math.log((-0.25 * ((h / l) * Math.pow((M_m / d), 2.0)))) + (-2.0 * Math.log((1.0 / D_m)))))), 2.0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+26:
		tmp = w0 * math.pow(math.exp((0.25 * (math.log((-0.25 * ((h / l) * math.pow((M_m / d), 2.0)))) + (-2.0 * math.log((1.0 / D_m)))))), 2.0)
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+26)
		tmp = Float64(w0 * (exp(Float64(0.25 * Float64(log(Float64(-0.25 * Float64(Float64(h / l) * (Float64(M_m / d) ^ 2.0)))) + Float64(-2.0 * log(Float64(1.0 / D_m)))))) ^ 2.0));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+26)
		tmp = w0 * (exp((0.25 * (log((-0.25 * ((h / l) * ((M_m / d) ^ 2.0)))) + (-2.0 * log((1.0 / D_m)))))) ^ 2.0);
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+26], N[(w0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(-0.25 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M$95$m / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[N[(1.0 / D$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e26

   1. Initial program 62.8%

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

   3. Simplified 66.0%

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

   6. Applied egg-rr 66.0%

      \[\leadsto w0 \cdot \color{blue}{{\left({\left(1 - \frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in D around inf 24.2%

      \[\leadsto w0 \cdot {\color{blue}{\left(e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}}^{2} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 24.2%

         \[\leadsto w0 \cdot {\left(e^{0.25 \cdot \left(\log \color{blue}{\left(1 \cdot \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)} + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      2. log-prod 24.2%

         \[\leadsto w0 \cdot {\left(e^{0.25 \cdot \left(\color{blue}{\left(\log 1 + \log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)} + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      3. metadata-eval 24.2%

         \[\leadsto w0 \cdot {\left(e^{0.25 \cdot \left(\left(\color{blue}{0} + \log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      4. *-commutative 24.2%

         \[\leadsto w0 \cdot {\left(e^{0.25 \cdot \left(\left(0 + \log \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot -0.25\right)}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      5. times-frac 26.6%

         \[\leadsto w0 \cdot {\left(e^{0.25 \cdot \left(\left(0 + \log \left(\color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)} \cdot -0.25\right)\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      6. unpow2 26.6%

         \[\leadsto w0 \cdot {\left(e^{0.25 \cdot \left(\left(0 + \log \left(\left(\frac{\color{blue}{M \cdot M}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25\right)\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      7. unpow2 26.6%

         \[\leadsto w0 \cdot {\left(e^{0.25 \cdot \left(\left(0 + \log \left(\left(\frac{M \cdot M}{\color{blue}{d \cdot d}} \cdot \frac{h}{\ell}\right) \cdot -0.25\right)\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      8. frac-times 36.0%

         \[\leadsto w0 \cdot {\left(e^{0.25 \cdot \left(\left(0 + \log \left(\left(\color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)} \cdot \frac{h}{\ell}\right) \cdot -0.25\right)\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      9. pow2 36.0%

         \[\leadsto w0 \cdot {\left(e^{0.25 \cdot \left(\left(0 + \log \left(\left(\color{blue}{{\left(\frac{M}{d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25\right)\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

   9. Applied egg-rr 36.0%

      \[\leadsto w0 \cdot {\left(e^{0.25 \cdot \left(\color{blue}{\left(0 + \log \left(\left({\left(\frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.25\right)\right)} + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]
   10. Step-by-step derivation

       1. +-lft-identity 36.0%

          \[\leadsto w0 \cdot {\left(e^{0.25 \cdot \left(\color{blue}{\log \left(\left({\left(\frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.25\right)} + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

       2. *-commutative 36.0%

          \[\leadsto w0 \cdot {\left(e^{0.25 \cdot \left(\log \color{blue}{\left(-0.25 \cdot \left({\left(\frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

       3. *-commutative 36.0%

          \[\leadsto w0 \cdot {\left(e^{0.25 \cdot \left(\log \left(-0.25 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(\frac{M}{d}\right)}^{2}\right)}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

   11. Simplified 36.0%

       \[\leadsto w0 \cdot {\left(e^{0.25 \cdot \left(\color{blue}{\log \left(-0.25 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{d}\right)}^{2}\right)\right)} + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

   if -2.0000000000000001e26 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.5%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in D around 0 97.1%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+26}:\\ \;\;\;\;w0 \cdot {\left(e^{0.25 \cdot \left(\log \left(-0.25 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{d}\right)}^{2}\right)\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.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 86.5%

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

   3. Simplified 87.7%

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

   if +inf.0 < (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.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 18.8%

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

   5. Taylor expanded in D around 0 74.5%

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

4. Final simplification 86.9%

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

Alternative 3: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-290}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ h l) -5e-290)
   (* w0 (fma -0.125 (* (/ h l) (pow (* D_m (/ M_m d)) 2.0)) 1.0))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-290) {
		tmp = w0 * fma(-0.125, ((h / l) * pow((D_m * (M_m / d)), 2.0)), 1.0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -5e-290)
		tmp = Float64(w0 * fma(-0.125, Float64(Float64(h / l) * (Float64(D_m * Float64(M_m / d)) ^ 2.0)), 1.0));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-290], N[(w0 * N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -5.0000000000000001e-290

   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 79.3%

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

   5. Taylor expanded in D around 0 46.9%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 46.9%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot 0.25}} \]

      2. associate-/l* 46.3%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)} \cdot 0.25} \]

      3. times-frac 48.8%

         \[\leadsto w0 \cdot \sqrt{1 - \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right) \cdot 0.25} \]

   7. Simplified 48.8%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({D}^{2} \cdot \left(\frac{{M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)\right) \cdot 0.25}} \]
   8. Step-by-step derivation

      1. associate-*r/ 49.5%

         \[\leadsto w0 \cdot \sqrt{1 - \left({D}^{2} \cdot \color{blue}{\frac{\frac{{M}^{2}}{{d}^{2}} \cdot h}{\ell}}\right) \cdot 0.25} \]

      2. add-sqr-sqrt 49.5%

         \[\leadsto w0 \cdot \sqrt{1 - \left({D}^{2} \cdot \frac{\color{blue}{\left(\sqrt{\frac{{M}^{2}}{{d}^{2}}} \cdot \sqrt{\frac{{M}^{2}}{{d}^{2}}}\right)} \cdot h}{\ell}\right) \cdot 0.25} \]

      3. pow2 49.5%

         \[\leadsto w0 \cdot \sqrt{1 - \left({D}^{2} \cdot \frac{\color{blue}{{\left(\sqrt{\frac{{M}^{2}}{{d}^{2}}}\right)}^{2}} \cdot h}{\ell}\right) \cdot 0.25} \]

      4. sqrt-div 49.5%

         \[\leadsto w0 \cdot \sqrt{1 - \left({D}^{2} \cdot \frac{{\color{blue}{\left(\frac{\sqrt{{M}^{2}}}{\sqrt{{d}^{2}}}\right)}}^{2} \cdot h}{\ell}\right) \cdot 0.25} \]

      5. sqrt-pow1 59.9%

         \[\leadsto w0 \cdot \sqrt{1 - \left({D}^{2} \cdot \frac{{\left(\frac{\color{blue}{{M}^{\left(\frac{2}{2}\right)}}}{\sqrt{{d}^{2}}}\right)}^{2} \cdot h}{\ell}\right) \cdot 0.25} \]

      6. metadata-eval 59.9%

         \[\leadsto w0 \cdot \sqrt{1 - \left({D}^{2} \cdot \frac{{\left(\frac{{M}^{\color{blue}{1}}}{\sqrt{{d}^{2}}}\right)}^{2} \cdot h}{\ell}\right) \cdot 0.25} \]

      7. pow1 59.9%

         \[\leadsto w0 \cdot \sqrt{1 - \left({D}^{2} \cdot \frac{{\left(\frac{\color{blue}{M}}{\sqrt{{d}^{2}}}\right)}^{2} \cdot h}{\ell}\right) \cdot 0.25} \]

      8. sqrt-pow1 65.9%

         \[\leadsto w0 \cdot \sqrt{1 - \left({D}^{2} \cdot \frac{{\left(\frac{M}{\color{blue}{{d}^{\left(\frac{2}{2}\right)}}}\right)}^{2} \cdot h}{\ell}\right) \cdot 0.25} \]

      9. metadata-eval 65.9%

         \[\leadsto w0 \cdot \sqrt{1 - \left({D}^{2} \cdot \frac{{\left(\frac{M}{{d}^{\color{blue}{1}}}\right)}^{2} \cdot h}{\ell}\right) \cdot 0.25} \]

      10. pow1 65.9%

          \[\leadsto w0 \cdot \sqrt{1 - \left({D}^{2} \cdot \frac{{\left(\frac{M}{\color{blue}{d}}\right)}^{2} \cdot h}{\ell}\right) \cdot 0.25} \]

   9. Applied egg-rr 65.9%

      \[\leadsto w0 \cdot \sqrt{1 - \left({D}^{2} \cdot \color{blue}{\frac{{\left(\frac{M}{d}\right)}^{2} \cdot h}{\ell}}\right) \cdot 0.25} \]

   10. Taylor expanded in D around 0 45.2%

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

       1. +-commutative 45.2%

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

       2. fma-define 45.2%

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

       3. associate-*r* 47.0%

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

       4. times-frac 46.0%

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

       5. associate-/l* 46.6%

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

       6. unpow2 46.6%

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

       7. unpow2 46.6%

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

       8. unpow2 46.6%

          \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{\color{blue}{d \cdot d}}\right) \cdot \frac{h}{\ell}, 1\right) \]

       9. times-frac 60.7%

          \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \left(\left(D \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)}\right) \cdot \frac{h}{\ell}, 1\right) \]

       10. swap-sqr 69.6%

           \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \frac{h}{\ell}, 1\right) \]

       11. unpow2 69.6%

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

   12. Simplified 69.6%

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

   if -5.0000000000000001e-290 < (/.f64 h l)

   1. Initial program 87.3%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in D around 0 95.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-290}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M\_m \cdot \frac{D\_m \cdot 0.5}{d}\right)}^{2}}{\ell}} \end{array} \]

FPCore

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

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0 * sqrt((1.0 - (h * (pow((M_m * ((D_m * 0.5) / d)), 2.0) / l))));
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    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
    code = w0 * sqrt((1.0d0 - (h * (((m_m * ((d_m * 0.5d0) / d)) ** 2.0d0) / l))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (h * (Math.pow((M_m * ((D_m * 0.5) / d)), 2.0) / l))));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	return w0 * math.sqrt((1.0 - (h * (math.pow((M_m * ((D_m * 0.5) / d)), 2.0) / l))))

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(M_m * Float64(Float64(D_m * 0.5) / d)) ^ 2.0) / l)))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp = code(w0, M_m, D_m, h, l, d)
	tmp = w0 * sqrt((1.0 - (h * (((M_m * ((D_m * 0.5) / d)) ^ 2.0) / l))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.1%

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

3. Simplified 83.4%

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

6. Applied egg-rr 83.4%

   \[\leadsto w0 \cdot \color{blue}{\left(1 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}\right)} \]
7. Step-by-step derivation

   1. *-lft-identity 83.4%

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

   2. associate-*l/ 84.7%

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

   3. associate-/l* 85.4%

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

   4. associate-*r/ 85.4%

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

   5. associate-/l* 83.2%

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

   6. *-commutative 83.2%

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

   7. associate-*l* 83.2%

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

   8. *-commutative 83.2%

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

   9. associate-/l* 84.0%

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

8. Simplified 84.0%

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

9. Final simplification 84.0%

   \[\leadsto w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\ell}} \]
10. Add Preprocessing

Alternative 5: 71.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.65 \cdot 10^{+115}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(\frac{M\_m \cdot D\_m}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= d 1.65e+115)
   (+ w0 (* -0.125 (* (pow (/ (* M_m D_m) d) 2.0) (/ (* h w0) l))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 1.65e+115) {
		tmp = w0 + (-0.125 * (pow(((M_m * D_m) / d), 2.0) * ((h * w0) / l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    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
    real(8) :: tmp
    if (d <= 1.65d+115) then
        tmp = w0 + ((-0.125d0) * ((((m_m * d_m) / d) ** 2.0d0) * ((h * w0) / l)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 1.65e+115) {
		tmp = w0 + (-0.125 * (Math.pow(((M_m * D_m) / d), 2.0) * ((h * w0) / l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if d <= 1.65e+115:
		tmp = w0 + (-0.125 * (math.pow(((M_m * D_m) / d), 2.0) * ((h * w0) / l)))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (d <= 1.65e+115)
		tmp = Float64(w0 + Float64(-0.125 * Float64((Float64(Float64(M_m * D_m) / d) ^ 2.0) * Float64(Float64(h * w0) / l))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (d <= 1.65e+115)
		tmp = w0 + (-0.125 * ((((M_m * D_m) / d) ^ 2.0) * ((h * w0) / l)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[d, 1.65e+115], N[(w0 + N[(-0.125 * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * w0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 1.65000000000000003e115

   1. Initial program 80.1%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in D around 0 48.2%

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

      1. associate-/l* 47.7%

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

      2. times-frac 47.7%

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

   7. Simplified 47.7%

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

      1. add-sqr-sqrt 47.7%

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

      2. sqrt-div 47.7%

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

      3. sqrt-pow1 38.0%

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

      4. metadata-eval 38.0%

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

      5. pow1 38.0%

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

      6. sqrt-pow1 38.7%

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

      7. metadata-eval 38.7%

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

      8. pow1 38.7%

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

      9. sqrt-div 38.7%

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

      10. sqrt-pow1 44.3%

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

      11. metadata-eval 44.3%

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

      12. pow1 44.3%

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

      13. sqrt-pow1 61.8%

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

      14. metadata-eval 61.8%

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

      15. pow1 61.8%

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

   9. Applied egg-rr 61.8%

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

       1. frac-times 47.7%

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

       2. unpow2 47.7%

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

       3. unpow2 47.7%

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

       4. rem-cube-cbrt 47.7%

          \[\leadsto w0 + -0.125 \cdot \left({D}^{2} \cdot \left(\frac{{M}^{2}}{{d}^{2}} \cdot \color{blue}{{\left(\sqrt[3]{\frac{h \cdot w0}{\ell}}\right)}^{3}}\right)\right) \]

       5. pow1 47.7%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{{\left({D}^{2} \cdot \left(\frac{{M}^{2}}{{d}^{2}} \cdot {\left(\sqrt[3]{\frac{h \cdot w0}{\ell}}\right)}^{3}\right)\right)}^{1}} \]

       6. rem-cube-cbrt 47.7%

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

       7. unpow2 47.7%

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

       8. unpow2 47.7%

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

       9. frac-times 61.8%

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

       10. associate-*r* 61.8%

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

       11. pow2 61.8%

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

       12. pow-prod-down 71.7%

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

       13. associate-/l* 69.2%

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

   11. Applied egg-rr 69.2%

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

       1. unpow1 69.2%

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

       2. associate-*r/ 67.7%

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

   13. Simplified 67.7%

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

   14. Taylor expanded in h around 0 70.3%

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

   if 1.65000000000000003e115 < d

   1. Initial program 85.1%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in D around 0 87.4%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.65 \cdot 10^{+115}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d}\\ \mathbf{if}\;d \leq 3.6 \cdot 10^{+100}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d))))
   (if (<= d 3.6e+100) (+ w0 (* -0.125 (* (* t_0 t_0) (* h (/ w0 l))))) w0)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M_m / d);
	double tmp;
	if (d <= 3.6e+100) {
		tmp = w0 + (-0.125 * ((t_0 * t_0) * (h * (w0 / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_m * (m_m / d)
    if (d <= 3.6d+100) then
        tmp = w0 + ((-0.125d0) * ((t_0 * t_0) * (h * (w0 / l))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M_m / d);
	double tmp;
	if (d <= 3.6e+100) {
		tmp = w0 + (-0.125 * ((t_0 * t_0) * (h * (w0 / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = D_m * (M_m / d)
	tmp = 0
	if d <= 3.6e+100:
		tmp = w0 + (-0.125 * ((t_0 * t_0) * (h * (w0 / l))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m * Float64(M_m / d))
	tmp = 0.0
	if (d <= 3.6e+100)
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(t_0 * t_0) * Float64(h * Float64(w0 / l)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = D_m * (M_m / d);
	tmp = 0.0;
	if (d <= 3.6e+100)
		tmp = w0 + (-0.125 * ((t_0 * t_0) * (h * (w0 / l))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 3.6e+100], N[(w0 + N[(-0.125 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if d < 3.6e100

   1. Initial program 80.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in D around 0 47.9%

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

      1. associate-/l* 47.4%

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

      2. times-frac 47.4%

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

   7. Simplified 47.4%

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

      1. add-sqr-sqrt 47.4%

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

      2. sqrt-div 47.4%

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

      3. sqrt-pow1 37.5%

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

      4. metadata-eval 37.5%

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

      5. pow1 37.5%

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

      6. sqrt-pow1 38.1%

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

      7. metadata-eval 38.1%

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

      8. pow1 38.1%

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

      9. sqrt-div 38.1%

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

      10. sqrt-pow1 43.9%

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

      11. metadata-eval 43.9%

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

      12. pow1 43.9%

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

      13. sqrt-pow1 61.8%

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

      14. metadata-eval 61.8%

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

      15. pow1 61.8%

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

   9. Applied egg-rr 61.8%

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

       1. frac-times 47.4%

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

       2. unpow2 47.4%

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

       3. unpow2 47.4%

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

       4. rem-cube-cbrt 47.4%

          \[\leadsto w0 + -0.125 \cdot \left({D}^{2} \cdot \left(\frac{{M}^{2}}{{d}^{2}} \cdot \color{blue}{{\left(\sqrt[3]{\frac{h \cdot w0}{\ell}}\right)}^{3}}\right)\right) \]

       5. pow1 47.4%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{{\left({D}^{2} \cdot \left(\frac{{M}^{2}}{{d}^{2}} \cdot {\left(\sqrt[3]{\frac{h \cdot w0}{\ell}}\right)}^{3}\right)\right)}^{1}} \]

       6. rem-cube-cbrt 47.4%

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

       7. unpow2 47.4%

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

       8. unpow2 47.4%

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

       9. frac-times 61.8%

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

       10. associate-*r* 61.8%

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

       11. pow2 61.8%

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

       12. pow-prod-down 72.0%

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

       13. associate-/l* 69.8%

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

   11. Applied egg-rr 69.8%

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

       1. unpow1 69.8%

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

       2. associate-*r/ 68.3%

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

   13. Simplified 68.3%

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

       1. unpow2 68.3%

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

       2. associate-/l* 68.3%

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

       3. associate-/l* 69.8%

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

   15. Applied egg-rr 69.8%

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

   if 3.6e100 < d

   1. Initial program 84.8%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in D around 0 85.2%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 3.6 \cdot 10^{+100}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.8% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

3. Simplified 83.4%

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

5. Taylor expanded in D around 0 66.1%

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

6. Final simplification 66.1%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(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))))))