Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.5% → 90.1%

Time: 13.5s

Alternatives: 10

Speedup: 0.8×

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: 80.5% 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: 90.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\left(\left|M \cdot D\right| \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \cdot \left(\frac{1}{d\_m} \cdot w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d_m)) 2.0) (/ h l)) -2e+41)
   (* (* (fabs (* M D)) (sqrt (* (/ h l) -0.25))) (* (/ 1.0 d_m) w0))
   w0))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+41) {
		tmp = (fabs((M * D)) * sqrt(((h / l) * -0.25))) * ((1.0 / d_m) * w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_m)) ** 2.0d0) * (h / l)) <= (-2d+41)) then
        tmp = (abs((m * d)) * sqrt(((h / l) * (-0.25d0)))) * ((1.0d0 / d_m) * w0)
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+41) {
		tmp = (Math.abs((M * D)) * Math.sqrt(((h / l) * -0.25))) * ((1.0 / d_m) * w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+41:
		tmp = (math.fabs((M * D)) * math.sqrt(((h / l) * -0.25))) * ((1.0 / d_m) * w0)
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+41)
		tmp = Float64(Float64(abs(Float64(M * D)) * sqrt(Float64(Float64(h / l) * -0.25))) * Float64(Float64(1.0 / d_m) * w0));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -2e+41)
		tmp = (abs((M * D)) * sqrt(((h / l) * -0.25))) * ((1.0 / d_m) * w0);
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+41], N[(N[(N[Abs[N[(M * D), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(h / l), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / d$95$m), $MachinePrecision] * w0), $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)) < -2.00000000000000001e41

   1. Initial program 68.1%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 34.8

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

   5. Simplified 34.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   7. Applied egg-rr 47.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 47.1

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 49.5

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

   9. Applied egg-rr 49.5%

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

   11. Applied egg-rr 49.7%

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

   if -2.00000000000000001e41 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.9%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 96.6%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 96.6

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

   7. Applied egg-rr 96.6%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\left(\left|M \cdot D\right| \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \cdot \left(\frac{1}{d} \cdot w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{d\_m} \cdot \left(\left(\left|M \cdot D\right| \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \cdot w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d_m)) 2.0) (/ h l)) -2e+41)
   (* (/ 1.0 d_m) (* (* (fabs (* M D)) (sqrt (* (/ h l) -0.25))) w0))
   w0))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+41) {
		tmp = (1.0 / d_m) * ((fabs((M * D)) * sqrt(((h / l) * -0.25))) * w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_m)) ** 2.0d0) * (h / l)) <= (-2d+41)) then
        tmp = (1.0d0 / d_m) * ((abs((m * d)) * sqrt(((h / l) * (-0.25d0)))) * w0)
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+41) {
		tmp = (1.0 / d_m) * ((Math.abs((M * D)) * Math.sqrt(((h / l) * -0.25))) * w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+41:
		tmp = (1.0 / d_m) * ((math.fabs((M * D)) * math.sqrt(((h / l) * -0.25))) * w0)
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+41)
		tmp = Float64(Float64(1.0 / d_m) * Float64(Float64(abs(Float64(M * D)) * sqrt(Float64(Float64(h / l) * -0.25))) * w0));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -2e+41)
		tmp = (1.0 / d_m) * ((abs((M * D)) * sqrt(((h / l) * -0.25))) * w0);
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+41], N[(N[(1.0 / d$95$m), $MachinePrecision] * N[(N[(N[Abs[N[(M * D), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(h / l), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * w0), $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)) < -2.00000000000000001e41

   1. Initial program 68.1%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 34.8

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

   5. Simplified 34.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   7. Applied egg-rr 47.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 47.1

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 49.5

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

   9. Applied egg-rr 49.5%

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

   11. Applied egg-rr 48.6%

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

   if -2.00000000000000001e41 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.9%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 96.6%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 96.6

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

   7. Applied egg-rr 96.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\left(\left|M \cdot D\right| \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \cdot w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\frac{\left(\left|M \cdot D\right| \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \cdot w0}{d\_m}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d_m)) 2.0) (/ h l)) -2e+41)
   (/ (* (* (fabs (* M D)) (sqrt (* (/ h l) -0.25))) w0) d_m)
   w0))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+41) {
		tmp = ((fabs((M * D)) * sqrt(((h / l) * -0.25))) * w0) / d_m;
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_m)) ** 2.0d0) * (h / l)) <= (-2d+41)) then
        tmp = ((abs((m * d)) * sqrt(((h / l) * (-0.25d0)))) * w0) / d_m
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+41) {
		tmp = ((Math.abs((M * D)) * Math.sqrt(((h / l) * -0.25))) * w0) / d_m;
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+41:
		tmp = ((math.fabs((M * D)) * math.sqrt(((h / l) * -0.25))) * w0) / d_m
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+41)
		tmp = Float64(Float64(Float64(abs(Float64(M * D)) * sqrt(Float64(Float64(h / l) * -0.25))) * w0) / d_m);
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -2e+41)
		tmp = ((abs((M * D)) * sqrt(((h / l) * -0.25))) * w0) / d_m;
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+41], N[(N[(N[(N[Abs[N[(M * D), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(h / l), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * w0), $MachinePrecision] / d$95$m), $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)) < -2.00000000000000001e41

   1. Initial program 68.1%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 34.8

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

   5. Simplified 34.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   7. Applied egg-rr 47.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 47.1

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 49.5

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

   9. Applied egg-rr 49.5%

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

   11. Applied egg-rr 48.6%

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

   if -2.00000000000000001e41 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.9%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 96.6%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 96.6

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

   7. Applied egg-rr 96.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\frac{\left(\left|M \cdot D\right| \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \cdot w0}{d}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+41}:\\ \;\;\;\;w0 \cdot \frac{\left|M \cdot D\right| \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}}{d\_m}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d_m)) 2.0) (/ h l)) -2e+41)
   (* w0 (/ (* (fabs (* M D)) (sqrt (* (/ h l) -0.25))) d_m))
   w0))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+41) {
		tmp = w0 * ((fabs((M * D)) * sqrt(((h / l) * -0.25))) / d_m);
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_m)) ** 2.0d0) * (h / l)) <= (-2d+41)) then
        tmp = w0 * ((abs((m * d)) * sqrt(((h / l) * (-0.25d0)))) / d_m)
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+41) {
		tmp = w0 * ((Math.abs((M * D)) * Math.sqrt(((h / l) * -0.25))) / d_m);
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+41:
		tmp = w0 * ((math.fabs((M * D)) * math.sqrt(((h / l) * -0.25))) / d_m)
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+41)
		tmp = Float64(w0 * Float64(Float64(abs(Float64(M * D)) * sqrt(Float64(Float64(h / l) * -0.25))) / d_m));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -2e+41)
		tmp = w0 * ((abs((M * D)) * sqrt(((h / l) * -0.25))) / d_m);
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+41], N[(w0 * N[(N[(N[Abs[N[(M * D), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(h / l), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / d$95$m), $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)) < -2.00000000000000001e41

   1. Initial program 68.1%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 34.8

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

   5. Simplified 34.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   7. Applied egg-rr 47.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 47.1

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 49.5

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

   9. Applied egg-rr 49.5%

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

   11. Applied egg-rr 48.6%

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

   if -2.00000000000000001e41 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.9%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 96.6%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 96.6

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

   7. Applied egg-rr 96.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+41}:\\ \;\;\;\;w0 \cdot \frac{\left|M \cdot D\right| \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}}{d}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+86}:\\ \;\;\;\;w0 \cdot \left(\left|M \cdot D\right| \cdot \sqrt{\frac{h \cdot -0.25}{d\_m \cdot \left(d\_m \cdot \ell\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d_m)) 2.0) (/ h l)) -4e+86)
   (* w0 (* (fabs (* M D)) (sqrt (/ (* h -0.25) (* d_m (* d_m l))))))
   w0))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -4e+86) {
		tmp = w0 * (fabs((M * D)) * sqrt(((h * -0.25) / (d_m * (d_m * l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_m)) ** 2.0d0) * (h / l)) <= (-4d+86)) then
        tmp = w0 * (abs((m * d)) * sqrt(((h * (-0.25d0)) / (d_m * (d_m * l)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -4e+86) {
		tmp = w0 * (Math.abs((M * D)) * Math.sqrt(((h * -0.25) / (d_m * (d_m * l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -4e+86:
		tmp = w0 * (math.fabs((M * D)) * math.sqrt(((h * -0.25) / (d_m * (d_m * l)))))
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -4e+86)
		tmp = Float64(w0 * Float64(abs(Float64(M * D)) * sqrt(Float64(Float64(h * -0.25) / Float64(d_m * Float64(d_m * l))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -4e+86)
		tmp = w0 * (abs((M * D)) * sqrt(((h * -0.25) / (d_m * (d_m * l)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e+86], N[(w0 * N[(N[Abs[N[(M * D), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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)) < -4.0000000000000001e86

   1. Initial program 67.2%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 35.7

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

   5. Simplified 35.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   7. Applied egg-rr 47.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 47.0

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 49.5

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

   9. Applied egg-rr 49.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-sqrt.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 49.5

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

   11. Applied egg-rr 60.8%

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

   if -4.0000000000000001e86 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 95.6%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 95.6

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

   7. Applied egg-rr 95.6%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+86}:\\ \;\;\;\;w0 \cdot \left(\left|M \cdot D\right| \cdot \sqrt{\frac{h \cdot -0.25}{d \cdot \left(d \cdot \ell\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \left(h \cdot -0.125\right) \cdot \frac{M \cdot \left(M \cdot w0\right)}{d\_m \cdot \left(d\_m \cdot \ell\right)}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d_m)) 2.0) (/ h l)) -4e-6)
   (fma (* D D) (* (* h -0.125) (/ (* M (* M w0)) (* d_m (* d_m l)))) w0)
   w0))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -4e-6) {
		tmp = fma((D * D), ((h * -0.125) * ((M * (M * w0)) / (d_m * (d_m * l)))), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -4e-6)
		tmp = fma(Float64(D * D), Float64(Float64(h * -0.125) * Float64(Float64(M * Float64(M * w0)) / Float64(d_m * Float64(d_m * l)))), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e-6], N[(N[(D * D), $MachinePrecision] * N[(N[(h * -0.125), $MachinePrecision] * N[(N[(M * N[(M * w0), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $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)) < -3.99999999999999982e-6

   1. Initial program 68.5%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} + w0} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}} + w0 \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} \cdot \frac{-1}{8} + w0 \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)} + w0 \]

      5. *-commutative N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} + w0 \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right)} \]

   5. Simplified 37.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot \color{blue}{\left(M \cdot w0\right)}\right)\right)}{\left(d \cdot d\right) \cdot \ell}, w0\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot \left(M \cdot w0\right)\right)}\right)}{\left(d \cdot d\right) \cdot \ell}, w0\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \color{blue}{\left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)}}{\left(d \cdot d\right) \cdot \ell}, w0\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}, w0\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \color{blue}{\left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)}}{\left(d \cdot d\right) \cdot \ell}, w0\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\color{blue}{\left(\frac{-1}{8} \cdot h\right) \cdot \left(M \cdot \left(M \cdot w0\right)\right)}}{\left(d \cdot d\right) \cdot \ell}, w0\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\left(\frac{-1}{8} \cdot h\right) \cdot \left(M \cdot \left(M \cdot w0\right)\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}, w0\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\left(\frac{-1}{8} \cdot h\right) \cdot \frac{M \cdot \left(M \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell}}, w0\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\left(\frac{-1}{8} \cdot h\right) \cdot \frac{M \cdot \left(M \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell}}, w0\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\left(h \cdot \frac{-1}{8}\right)} \cdot \frac{M \cdot \left(M \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell}, w0\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\left(h \cdot \frac{-1}{8}\right)} \cdot \frac{M \cdot \left(M \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell}, w0\right) \]

      12. lower-/.f64 37.9

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

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \left(h \cdot \frac{-1}{8}\right) \cdot \frac{M \cdot \left(M \cdot w0\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}, w0\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \left(h \cdot \frac{-1}{8}\right) \cdot \frac{M \cdot \left(M \cdot w0\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}, w0\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \left(h \cdot \frac{-1}{8}\right) \cdot \frac{M \cdot \left(M \cdot w0\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}, w0\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \left(h \cdot \frac{-1}{8}\right) \cdot \frac{M \cdot \left(M \cdot w0\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}, w0\right) \]

      17. lower-*.f64 38.5

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

   7. Applied egg-rr 38.5%

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

   if -3.99999999999999982e-6 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.9%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 96.8%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 96.8

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

   7. Applied egg-rr 96.8%

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

Alternative 7: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+98}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{-0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell \cdot \left(d\_m \cdot d\_m\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d_m)) 2.0) (/ h l)) -2e+98)
   (* w0 (fma (* D D) (/ (* -0.125 (* h (* M M))) (* l (* d_m d_m))) 1.0))
   w0))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+98) {
		tmp = w0 * fma((D * D), ((-0.125 * (h * (M * M))) / (l * (d_m * d_m))), 1.0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+98)
		tmp = Float64(w0 * fma(Float64(D * D), Float64(Float64(-0.125 * Float64(h * Float64(M * M))) / Float64(l * Float64(d_m * d_m))), 1.0));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+98], N[(w0 * N[(N[(D * D), $MachinePrecision] * N[(N[(-0.125 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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)) < -2e98

   1. Initial program 66.4%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

         \[\leadsto w0 \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto w0 \cdot \left(\color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}} + 1\right) \]

      3. associate-/l* N/A

         \[\leadsto w0 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)} \cdot \frac{-1}{8} + 1\right) \]

      4. associate-*r* N/A

         \[\leadsto w0 \cdot \left(\color{blue}{{D}^{2} \cdot \left(\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)} + 1\right) \]

      5. *-commutative N/A

         \[\leadsto w0 \cdot \left({D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right)} \]

      7. unpow2 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\color{blue}{D \cdot D}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\color{blue}{D \cdot D}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right) \]

      9. associate-*r/ N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}, 1\right) \]

      10. lower-/.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\color{blue}{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}}{{d}^{2} \cdot \ell}, 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2} \cdot \ell}, 1\right) \]

      13. unpow2 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)}{{d}^{2} \cdot \ell}, 1\right) \]

      14. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)}{{d}^{2} \cdot \ell}, 1\right) \]

      15. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}, 1\right) \]

      16. unpow2 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}, 1\right) \]

      17. lower-*.f64 35.5

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

   5. Simplified 35.5%

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

   if -2e98 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.2%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 94.7%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 94.7

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

   7. Applied egg-rr 94.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+98}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{-0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+98}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\ell \cdot \left(d\_m \cdot d\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d_m)) 2.0) (/ h l)) -2e+98)
   (* (* D D) (/ (* -0.125 (* w0 (* h (* M M)))) (* l (* d_m d_m))))
   w0))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+98) {
		tmp = (D * D) * ((-0.125 * (w0 * (h * (M * M)))) / (l * (d_m * d_m)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_m)) ** 2.0d0) * (h / l)) <= (-2d+98)) then
        tmp = (d * d) * (((-0.125d0) * (w0 * (h * (m * m)))) / (l * (d_m * d_m)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+98) {
		tmp = (D * D) * ((-0.125 * (w0 * (h * (M * M)))) / (l * (d_m * d_m)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+98:
		tmp = (D * D) * ((-0.125 * (w0 * (h * (M * M)))) / (l * (d_m * d_m)))
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+98)
		tmp = Float64(Float64(D * D) * Float64(Float64(-0.125 * Float64(w0 * Float64(h * Float64(M * M)))) / Float64(l * Float64(d_m * d_m))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -2e+98)
		tmp = (D * D) * ((-0.125 * (w0 * (h * (M * M)))) / (l * (d_m * d_m)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+98], N[(N[(D * D), $MachinePrecision] * N[(N[(-0.125 * N[(w0 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $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)) < -2e98

   1. Initial program 66.4%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} + w0} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}} + w0 \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} \cdot \frac{-1}{8} + w0 \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)} + w0 \]

      5. *-commutative N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} + w0 \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right)} \]

   5. Simplified 39.6%

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

   6. Taylor expanded in D around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)} \]

      4. *-commutative N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} \]

      6. unpow2 N/A

         \[\leadsto \color{blue}{\left(D \cdot D\right)} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(D \cdot D\right)} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right) \]

      8. associate-*r/ N/A

         \[\leadsto \left(D \cdot D\right) \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(D \cdot D\right) \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(D \cdot D\right) \cdot \frac{\color{blue}{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}}{{d}^{2} \cdot \ell} \]

      11. associate-*r* N/A

          \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \left(\color{blue}{\left({M}^{2} \cdot h\right)} \cdot w0\right)}{{d}^{2} \cdot \ell} \]

      14. unpow2 N/A

          \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot w0\right)}{{d}^{2} \cdot \ell} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot w0\right)}{{d}^{2} \cdot \ell} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

      17. unpow2 N/A

          \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} \]

      18. lower-*.f64 35.4

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

   8. Simplified 35.4%

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

   if -2e98 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.2%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 94.7%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 94.7

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

   7. Applied egg-rr 94.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+98}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 4 \cdot 10^{-102}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(D \cdot -0.125\right) \cdot \frac{M \cdot \left(h \cdot \left(M \cdot w0\right)\right)}{d\_m \cdot \left(d\_m \cdot \ell\right)}, D, w0\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= (* M D) 4e-102)
   w0
   (fma (* (* D -0.125) (/ (* M (* h (* M w0))) (* d_m (* d_m l)))) D w0)))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((M * D) <= 4e-102) {
		tmp = w0;
	} else {
		tmp = fma(((D * -0.125) * ((M * (h * (M * w0))) / (d_m * (d_m * l)))), D, w0);
	}
	return tmp;
}

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (Float64(M * D) <= 4e-102)
		tmp = w0;
	else
		tmp = fma(Float64(Float64(D * -0.125) * Float64(Float64(M * Float64(h * Float64(M * w0))) / Float64(d_m * Float64(d_m * l)))), D, w0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[N[(M * D), $MachinePrecision], 4e-102], w0, N[(N[(N[(D * -0.125), $MachinePrecision] * N[(N[(M * N[(h * N[(M * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * D + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 3.99999999999999973e-102

   1. Initial program 85.0%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 76.4%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 76.4

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

   7. Applied egg-rr 76.4%

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

   if 3.99999999999999973e-102 < (*.f64 M D)

   1. Initial program 76.3%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} + w0} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}} + w0 \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} \cdot \frac{-1}{8} + w0 \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)} + w0 \]

      5. *-commutative N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} + w0 \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right)} \]

   5. Simplified 36.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(D \cdot D\right)} \cdot \frac{\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell} + w0 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot \color{blue}{\left(M \cdot w0\right)}\right)\right)}{\left(d \cdot d\right) \cdot \ell} + w0 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot \left(M \cdot w0\right)\right)}\right)}{\left(d \cdot d\right) \cdot \ell} + w0 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \color{blue}{\left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)}}{\left(d \cdot d\right) \cdot \ell} + w0 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(D \cdot D\right) \cdot \frac{\color{blue}{\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)}}{\left(d \cdot d\right) \cdot \ell} + w0 \]

      6. lift-*.f64 N/A

         \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} + w0 \]

      7. lift-*.f64 N/A

         \[\leadsto \left(D \cdot D\right) \cdot \frac{\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}} + w0 \]

      8. lift-/.f64 N/A

         \[\leadsto \left(D \cdot D\right) \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}} + w0 \]

      9. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(D \cdot D\right)} \cdot \frac{\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell} + w0 \]

      10. associate-*l* N/A

          \[\leadsto \color{blue}{D \cdot \left(D \cdot \frac{\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right)} + w0 \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(D \cdot \frac{\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right) \cdot D} + w0 \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(D \cdot \frac{\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}, D, w0\right)} \]

   7. Applied egg-rr 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(D \cdot -0.125\right) \cdot \frac{M \cdot \left(\left(M \cdot w0\right) \cdot h\right)}{d \cdot \left(d \cdot \ell\right)}, D, w0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 4 \cdot 10^{-102}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(D \cdot -0.125\right) \cdot \frac{M \cdot \left(h \cdot \left(M \cdot w0\right)\right)}{d \cdot \left(d \cdot \ell\right)}, D, w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.1% accurate, 157.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ w0 \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m) :precision binary64 w0)

C

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

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

d_m = abs(d);
function tmp = code(w0, M, D, h, l, d_m)
	tmp = w0;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := w0

Derivation

1. Initial program 82.8%

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

3. Taylor expanded in M around 0

   \[\leadsto w0 \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 69.8%

   \[\leadsto w0 \cdot \color{blue}{1} \]
6. Step-by-step derivation

   1. *-rgt-identity 69.8

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

7. Applied egg-rr 69.8%

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

Reproduce

herbie shell --seed 2024211 
(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))))))