Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 89.0%

Time: 15.3s

Alternatives: 10

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+207}:\\ \;\;\;\;w0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot 0.5}{\frac{d}{D}} \cdot \frac{\left(M \cdot 0.5\right) \cdot \frac{D}{d}}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
   (if (<= t_0 5e+207)
     (* w0 (sqrt t_0))
     (*
      w0
      (sqrt
       (- 1.0 (* h (* (/ (* M 0.5) (/ d D)) (/ (* (* M 0.5) (/ D d)) l)))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+207) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - (h * (((M * 0.5) / (d / D)) * (((M * 0.5) * (D / d)) / l)))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))
    if (t_0 <= 5d+207) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - (h * (((m * 0.5d0) / (d_1 / d)) * (((m * 0.5d0) * (d / d_1)) / l)))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+207) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (((M * 0.5) / (d / D)) * (((M * 0.5) * (D / d)) / l)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = 1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 5e+207:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - (h * (((M * 0.5) / (d / D)) * (((M * 0.5) * (D / d)) / l)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 5e+207)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(Float64(M * 0.5) / Float64(d / D)) * Float64(Float64(Float64(M * 0.5) * Float64(D / d)) / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = 1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 5e+207)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - (h * (((M * 0.5) / (d / D)) * (((M * 0.5) * (D / d)) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 5e+207], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(N[(M * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 4.9999999999999999e207

   1. Initial program 99.7%

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

   if 4.9999999999999999e207 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 42.6%

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

   3. Simplified 42.6%

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

      1. clear-num 42.5%

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

      2. un-div-inv 44.7%

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

      3. clear-num 44.7%

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

      4. un-div-inv 44.7%

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

      5. associate-/r/ 44.7%

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

   6. Applied egg-rr 44.7%

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

      1. associate-/r/ 62.6%

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

      2. associate-/r* 62.6%

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

      3. associate-/r/ 61.6%

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

      4. associate-*l/ 61.6%

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

   8. Simplified 61.6%

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

      1. unpow2 61.6%

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

      2. div-inv 61.6%

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

      3. metadata-eval 61.6%

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

      4. div-inv 61.6%

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

      5. metadata-eval 61.6%

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

   10. Applied egg-rr 61.6%

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

       1. associate-/l* 65.9%

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

       2. associate-*l* 65.9%

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

       3. associate-*l* 65.9%

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

   12. Applied egg-rr 65.9%

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

       1. *-commutative 65.9%

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

       2. clear-num 65.9%

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

       3. un-div-inv 65.9%

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

   14. Applied egg-rr 65.9%

       \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\frac{0.5 \cdot M}{\frac{d}{D}}} \cdot \frac{\frac{D}{d} \cdot \left(0.5 \cdot M\right)}{\ell}\right) \cdot h} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+207}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot 0.5}{\frac{d}{D}} \cdot \frac{\left(M \cdot 0.5\right) \cdot \frac{D}{d}}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right) \cdot 0.25}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-233}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\left(M \cdot D\right) \cdot \left(\frac{0.5}{d} \cdot \frac{D}{2 \cdot \frac{d}{M}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) (- INFINITY))
   (* w0 (sqrt (- 1.0 (* (* (* (/ D d) (/ D d)) (/ (* h (* M M)) l)) 0.25))))
   (if (<= (/ h l) -1e-233)
     (*
      w0
      (sqrt
       (- 1.0 (* (/ h l) (* (* M D) (* (/ 0.5 d) (/ D (* 2.0 (/ d M)))))))))
     w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - ((((D / d) * (D / d)) * ((h * (M * M)) / l)) * 0.25)));
	} else if ((h / l) <= -1e-233) {
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M * D) * ((0.5 / d) * (D / (2.0 * (d / M))))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - ((((D / d) * (D / d)) * ((h * (M * M)) / l)) * 0.25)));
	} else if ((h / l) <= -1e-233) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * ((M * D) * ((0.5 / d) * (D / (2.0 * (d / M))))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 - ((((D / d) * (D / d)) * ((h * (M * M)) / l)) * 0.25)))
	elif (h / l) <= -1e-233:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * ((M * D) * ((0.5 / d) * (D / (2.0 * (d / M))))))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(h * Float64(M * M)) / l)) * 0.25))));
	elseif (Float64(h / l) <= -1e-233)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(M * D) * Float64(Float64(0.5 / d) * Float64(D / Float64(2.0 * Float64(d / M)))))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 - ((((D / d) * (D / d)) * ((h * (M * M)) / l)) * 0.25)));
	elseif ((h / l) <= -1e-233)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M * D) * ((0.5 / d) * (D / (2.0 * (d / M))))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-233], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(N[(0.5 / d), $MachinePrecision] * N[(D / N[(2.0 * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -inf.0

   1. Initial program 30.5%

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

   3. Simplified 30.5%

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

      1. clear-num 30.5%

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

      2. un-div-inv 37.0%

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

      3. clear-num 37.0%

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

      4. un-div-inv 37.0%

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

      5. associate-/r/ 37.0%

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

   6. Applied egg-rr 37.0%

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

      1. associate-/r/ 74.6%

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

      2. associate-/r* 74.6%

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

      3. associate-/r/ 71.6%

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

      4. associate-*l/ 71.6%

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

   8. Simplified 71.6%

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

   9. Taylor expanded in D around 0 48.9%

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

       1. *-commutative 48.9%

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

       2. times-frac 42.5%

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

       3. unpow2 42.5%

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

       4. unpow2 42.5%

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

       5. times-frac 48.6%

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

       6. unpow2 48.6%

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

   11. Simplified 48.6%

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

       1. unpow2 48.6%

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

   13. Applied egg-rr 48.6%

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

       1. unpow2 48.6%

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

   15. Applied egg-rr 48.6%

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

   if -inf.0 < (/.f64 h l) < -9.99999999999999958e-234

   1. Initial program 83.4%

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

   3. Simplified 82.3%

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

      1. unpow2 82.3%

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

      2. associate-/r* 82.3%

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

      3. associate-*r/ 81.9%

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

      4. *-commutative 81.9%

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

      5. div-inv 81.9%

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

      6. associate-*l* 79.5%

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

      7. *-commutative 79.5%

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

      8. associate-/r* 79.5%

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

      9. metadata-eval 79.5%

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

      10. clear-num 79.5%

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

      11. un-div-inv 79.7%

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

      12. associate-/r/ 79.7%

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

   6. Applied egg-rr 79.7%

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

   if -9.99999999999999958e-234 < (/.f64 h l)

   1. Initial program 86.9%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in D around 0 92.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right) \cdot 0.25}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-233}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\left(M \cdot D\right) \cdot \left(\frac{0.5}{d} \cdot \frac{D}{2 \cdot \frac{d}{M}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 6.5 \cdot 10^{-127}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right) \cdot 0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 6.5e-127)
   w0
   (* w0 (sqrt (- 1.0 (* (* (* (/ D d) (/ D d)) (/ (* h (* M M)) l)) 0.25))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 6.5e-127) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((((D / d) * (D / d)) * ((h * (M * M)) / l)) * 0.25)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 6.5d-127) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((((d / d_1) * (d / d_1)) * ((h * (m * m)) / l)) * 0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 6.5e-127) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((((D / d) * (D / d)) * ((h * (M * M)) / l)) * 0.25)));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 6.5e-127:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((((D / d) * (D / d)) * ((h * (M * M)) / l)) * 0.25)))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 6.5e-127)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(h * Float64(M * M)) / l)) * 0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 6.5e-127)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((((D / d) * (D / d)) * ((h * (M * M)) / l)) * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 6.5e-127], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 6.49999999999999998e-127

   1. Initial program 81.2%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in D around 0 70.6%

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

   if 6.49999999999999998e-127 < M

   1. Initial program 73.8%

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

   3. Simplified 73.2%

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

      1. clear-num 73.2%

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

      2. un-div-inv 73.2%

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

      3. clear-num 73.2%

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

      4. un-div-inv 73.4%

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

      5. associate-/r/ 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. associate-/r/ 77.9%

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

      2. associate-/r* 77.9%

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

      3. associate-/r/ 77.6%

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

      4. associate-*l/ 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in D around 0 50.5%

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

       1. *-commutative 50.5%

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

       2. times-frac 51.6%

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

       3. unpow2 51.6%

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

       4. unpow2 51.6%

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

       5. times-frac 60.5%

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

       6. unpow2 60.5%

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

   11. Simplified 60.5%

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

       1. unpow2 60.5%

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

   13. Applied egg-rr 60.5%

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

       1. unpow2 60.5%

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

   15. Applied egg-rr 60.5%

       \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right) \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.5 \cdot 10^{-127}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right) \cdot 0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(M \cdot 0.5\right) \cdot \frac{D}{d}\\ w0 \cdot \sqrt{1 - h \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)} \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (* M 0.5) (/ D d))))
   (* w0 (sqrt (- 1.0 (* h (* t_0 (/ t_0 l))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * 0.5) * (D / d);
	return w0 * sqrt((1.0 - (h * (t_0 * (t_0 / 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
    real(8) :: t_0
    t_0 = (m * 0.5d0) * (d / d_1)
    code = w0 * sqrt((1.0d0 - (h * (t_0 * (t_0 / l)))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * 0.5) * (D / d);
	return w0 * Math.sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
}

Python

def code(w0, M, D, h, l, d):
	t_0 = (M * 0.5) * (D / d)
	return w0 * math.sqrt((1.0 - (h * (t_0 * (t_0 / l)))))

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * 0.5) * Float64(D / d))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(t_0 * Float64(t_0 / l))))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	t_0 = (M * 0.5) * (D / d);
	tmp = w0 * sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.5%

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

3. Simplified 77.6%

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

   1. clear-num 77.6%

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

   2. un-div-inv 78.3%

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

   3. clear-num 78.3%

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

   4. un-div-inv 78.4%

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

   5. associate-/r/ 78.4%

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

6. Applied egg-rr 78.4%

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

   1. associate-/r/ 85.1%

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

   2. associate-/r* 85.1%

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

   3. associate-/r/ 84.2%

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

   4. associate-*l/ 84.2%

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

8. Simplified 84.2%

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

   1. unpow2 84.2%

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

   2. div-inv 84.2%

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

   3. metadata-eval 84.2%

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

   4. div-inv 84.2%

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

   5. metadata-eval 84.2%

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

10. Applied egg-rr 84.2%

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

    1. associate-/l* 85.7%

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

    2. associate-*l* 85.7%

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

    3. associate-*l* 85.7%

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

12. Applied egg-rr 85.7%

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

13. Final simplification 85.7%

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

Alternative 5: 87.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ w0 \cdot \sqrt{1 - h \cdot \left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \frac{0.5 \cdot \left(D \cdot \frac{M}{d}\right)}{\ell}\right)} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (*
  w0
  (sqrt (- 1.0 (* h (* (* (* M 0.5) (/ D d)) (/ (* 0.5 (* D (/ M d))) l)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (h * (((M * 0.5) * (D / d)) * ((0.5 * (D * (M / d))) / 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 - (h * (((m * 0.5d0) * (d / d_1)) * ((0.5d0 * (d * (m / d_1))) / 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 - (h * (((M * 0.5) * (D / d)) * ((0.5 * (D * (M / d))) / l)))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (h * (((M * 0.5) * (D / d)) * ((0.5 * (D * (M / d))) / l)))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(Float64(M * 0.5) * Float64(D / d)) * Float64(Float64(0.5 * Float64(D * Float64(M / d))) / l))))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - (h * (((M * 0.5) * (D / d)) * ((0.5 * (D * (M / d))) / l)))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

3. Simplified 77.6%

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

   1. clear-num 77.6%

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

   2. un-div-inv 78.3%

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

   3. clear-num 78.3%

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

   4. un-div-inv 78.4%

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

   5. associate-/r/ 78.4%

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

6. Applied egg-rr 78.4%

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

   1. associate-/r/ 85.1%

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

   2. associate-/r* 85.1%

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

   3. associate-/r/ 84.2%

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

   4. associate-*l/ 84.2%

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

8. Simplified 84.2%

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

   1. unpow2 84.2%

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

   2. div-inv 84.2%

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

   3. metadata-eval 84.2%

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

   4. div-inv 84.2%

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

   5. metadata-eval 84.2%

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

10. Applied egg-rr 84.2%

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

    1. associate-/l* 85.7%

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

    2. associate-*l* 85.7%

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

    3. associate-*l* 85.7%

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

12. Applied egg-rr 85.7%

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

13. Taylor expanded in D around 0 85.5%

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

    1. associate-/l* 84.3%

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

15. Simplified 84.3%

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

16. Final simplification 84.3%

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

Alternative 6: 83.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ w0 \cdot \sqrt{1 - h \cdot \left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \left(D \cdot \frac{M \cdot \frac{0.5}{\ell}}{d}\right)\right)} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (*
  w0
  (sqrt (- 1.0 (* h (* (* (* M 0.5) (/ D d)) (* D (/ (* M (/ 0.5 l)) d))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (h * (((M * 0.5) * (D / d)) * (D * ((M * (0.5 / l)) / d))))));
}

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 - (h * (((m * 0.5d0) * (d / d_1)) * (d * ((m * (0.5d0 / l)) / d_1))))))
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 - (h * (((M * 0.5) * (D / d)) * (D * ((M * (0.5 / l)) / d))))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (h * (((M * 0.5) * (D / d)) * (D * ((M * (0.5 / l)) / d))))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(Float64(M * 0.5) * Float64(D / d)) * Float64(D * Float64(Float64(M * Float64(0.5 / l)) / d)))))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - (h * (((M * 0.5) * (D / d)) * (D * ((M * (0.5 / l)) / d))))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(D * N[(N[(M * N[(0.5 / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

3. Simplified 77.6%

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

   1. clear-num 77.6%

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

   2. un-div-inv 78.3%

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

   3. clear-num 78.3%

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

   4. un-div-inv 78.4%

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

   5. associate-/r/ 78.4%

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

6. Applied egg-rr 78.4%

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

   1. associate-/r/ 85.1%

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

   2. associate-/r* 85.1%

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

   3. associate-/r/ 84.2%

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

   4. associate-*l/ 84.2%

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

8. Simplified 84.2%

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

   1. unpow2 84.2%

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

   2. div-inv 84.2%

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

   3. metadata-eval 84.2%

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

   4. div-inv 84.2%

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

   5. metadata-eval 84.2%

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

10. Applied egg-rr 84.2%

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

    1. associate-/l* 85.7%

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

    2. associate-*l* 85.7%

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

    3. associate-*l* 85.7%

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

12. Applied egg-rr 85.7%

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

    1. *-un-lft-identity 85.7%

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

    2. associate-/l* 84.1%

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

14. Applied egg-rr 84.1%

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

    1. *-lft-identity 84.1%

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

    2. associate-*l/ 83.4%

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

    3. associate-/l* 83.0%

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

    4. *-commutative 83.0%

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

    5. associate-/l* 83.0%

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

16. Simplified 83.0%

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

17. Final simplification 83.0%

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

Alternative 7: 72.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 5.6 \cdot 10^{-63}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 5.6e-63)
   w0
   (+ w0 (* -0.125 (/ (* (pow (* M (/ D d)) 2.0) (* h w0)) l)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 5.6e-63) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((pow((M * (D / d)), 2.0) * (h * w0)) / l));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 5.6d-63) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((((m * (d / d_1)) ** 2.0d0) * (h * w0)) / l))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 5.6e-63) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((Math.pow((M * (D / d)), 2.0) * (h * w0)) / l));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 5.6e-63:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((math.pow((M * (D / d)), 2.0) * (h * w0)) / l))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 5.6e-63)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64((Float64(M * Float64(D / d)) ^ 2.0) * Float64(h * w0)) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 5.6e-63)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((((M * (D / d)) ^ 2.0) * (h * w0)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 5.6e-63], w0, N[(w0 + N[(-0.125 * N[(N[(N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * w0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.6000000000000005e-63

   1. Initial program 81.2%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in D around 0 72.0%

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

   if 5.6000000000000005e-63 < M

   1. Initial program 72.0%

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

   3. Simplified 70.8%

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

      1. associate-/r* 70.8%

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

      2. associate-*r/ 72.0%

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

      3. *-commutative 72.0%

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

      4. unpow2 72.0%

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

      5. clear-num 72.0%

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

      6. clear-num 72.0%

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

      7. frac-times 71.9%

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

      8. metadata-eval 71.9%

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

      9. times-frac 71.8%

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

      10. times-frac 71.9%

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

   6. Applied egg-rr 71.9%

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

   7. Taylor expanded in M around 0 38.8%

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

      1. +-commutative 38.8%

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

      2. fma-define 38.8%

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

      3. times-frac 37.6%

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

      4. associate-*r/ 38.9%

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

      5. unpow2 38.9%

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

      6. unpow2 38.9%

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

      7. times-frac 43.6%

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

      8. unpow2 43.6%

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

   9. Simplified 43.6%

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

       1. fma-undefine 43.6%

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

       2. pow2 43.6%

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

       3. associate-*r* 47.5%

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

       4. pow2 47.5%

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

       5. pow-prod-down 59.8%

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

   11. Applied egg-rr 59.8%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.6 \cdot 10^{-63}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(h \cdot w0\right)}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 4.5 \cdot 10^{-20}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 4.5e-20) w0 (* -0.125 (* (pow (* D (/ M d)) 2.0) (* h (/ w0 l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.5e-20) {
		tmp = w0;
	} else {
		tmp = -0.125 * (pow((D * (M / d)), 2.0) * (h * (w0 / l)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 4.5d-20) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * (m / d_1)) ** 2.0d0) * (h * (w0 / l)))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.5e-20) {
		tmp = w0;
	} else {
		tmp = -0.125 * (Math.pow((D * (M / d)), 2.0) * (h * (w0 / l)));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4.5e-20:
		tmp = w0
	else:
		tmp = -0.125 * (math.pow((D * (M / d)), 2.0) * (h * (w0 / l)))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4.5e-20)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64((Float64(D * Float64(M / d)) ^ 2.0) * Float64(h * Float64(w0 / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4.5e-20)
		tmp = w0;
	else
		tmp = -0.125 * (((D * (M / d)) ^ 2.0) * (h * (w0 / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 4.5e-20], w0, N[(-0.125 * N[(N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.5000000000000001e-20

   1. Initial program 82.4%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in D around 0 73.3%

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

   if 4.5000000000000001e-20 < M

   1. Initial program 66.8%

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

   3. Simplified 65.3%

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

      1. associate-/r* 65.3%

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

      2. associate-*r/ 66.8%

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

      3. *-commutative 66.8%

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

      4. unpow2 66.8%

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

      5. clear-num 66.7%

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

      6. clear-num 66.7%

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

      7. frac-times 66.7%

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

      8. metadata-eval 66.7%

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

      9. times-frac 66.6%

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

      10. times-frac 66.6%

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

   6. Applied egg-rr 66.6%

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

   7. Taylor expanded in M around 0 36.8%

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

      1. +-commutative 36.8%

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

      2. fma-define 36.8%

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

      3. times-frac 35.3%

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

      4. associate-*r/ 36.8%

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

      5. unpow2 36.8%

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

      6. unpow2 36.8%

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

      7. times-frac 37.6%

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

      8. unpow2 37.6%

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

   9. Simplified 37.6%

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

   10. Taylor expanded in D around inf 21.7%

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

       1. associate-*r* 21.8%

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

       2. times-frac 21.9%

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

       3. *-commutative 21.9%

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

       4. associate-/l* 21.8%

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

       5. unpow2 21.8%

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

       6. unpow2 21.8%

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

       7. unpow2 21.8%

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

       8. times-frac 22.4%

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

       9. swap-sqr 24.9%

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

       10. *-commutative 24.9%

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

       11. *-commutative 24.9%

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

       12. unpow2 24.9%

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

       13. associate-*l/ 25.1%

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

       14. associate-/l* 25.0%

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

       15. associate-/l* 24.9%

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

   12. Simplified 24.9%

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

Alternative 9: 60.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 2.8 \cdot 10^{-35}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{w0 \cdot w0}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 2.8e-35) w0 (sqrt (* w0 w0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.8e-35) {
		tmp = w0;
	} else {
		tmp = sqrt((w0 * w0));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 2.8d-35) then
        tmp = w0
    else
        tmp = sqrt((w0 * w0))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.8e-35) {
		tmp = w0;
	} else {
		tmp = Math.sqrt((w0 * w0));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.8e-35:
		tmp = w0
	else:
		tmp = math.sqrt((w0 * w0))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.8e-35)
		tmp = w0;
	else
		tmp = sqrt(Float64(w0 * w0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.8e-35)
		tmp = w0;
	else
		tmp = sqrt((w0 * w0));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 2.8e-35], w0, N[Sqrt[N[(w0 * w0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.8e-35

   1. Initial program 82.0%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in D around 0 73.2%

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

   if 2.8e-35 < M

   1. Initial program 68.7%

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

   3. Simplified 67.4%

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

      1. add-sqr-sqrt 28.5%

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

      2. sqrt-unprod 19.5%

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

      3. *-commutative 19.5%

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

      4. *-commutative 19.5%

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

      5. swap-sqr 19.4%

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

   6. Applied egg-rr 19.4%

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

      1. associate-*l/ 19.5%

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

      2. associate-/l* 19.6%

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

      3. associate-/r* 19.6%

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

      4. associate-/r/ 19.6%

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

      5. associate-*l/ 19.6%

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

   8. Simplified 19.6%

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

   9. Taylor expanded in h around 0 17.1%

      \[\leadsto \sqrt{\color{blue}{{w0}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 17.1%

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

   11. Applied egg-rr 17.1%

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

Alternative 10: 68.6% accurate, 216.0× speedup

Math

\[\begin{array}{l} \\ w0 \end{array} \]

FPCore

(FPCore (w0 M D h l d) :precision binary64 w0)

C

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

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
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Python

def code(w0, M, D, h, l, d):
	return w0

Julia

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

MATLAB

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

Wolfram

code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 78.5%

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

3. Simplified 77.6%

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

5. Taylor expanded in D around 0 65.4%

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

Reproduce

herbie shell --seed 2024118 
(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))))))