Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.2% → 86.4%

Time: 12.1s

Alternatives: 7

Speedup: 1.7×

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: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 10^{+287}:\\ \;\;\;\;w0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* D M) (* d 2.0)) 2.0) (/ h l)))))
   (if (<= t_0 1e+287)
     (* w0 (sqrt t_0))
     (* w0 (sqrt (- 1.0 (* h (/ (pow (* M (/ 0.5 (/ d D))) 2.0) l))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (pow(((D * M) / (d * 2.0)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1e+287) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - (h * (pow((M * (0.5 / (d / D))), 2.0) / 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 - ((((d * m) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l))
    if (t_0 <= 1d+287) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - (h * (((m * (0.5d0 / (d_1 / d))) ** 2.0d0) / 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(((D * M) / (d * 2.0)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1e+287) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow((M * (0.5 / (d / D))), 2.0) / l))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = 1.0 - (math.pow(((D * M) / (d * 2.0)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 1e+287:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow((M * (0.5 / (d / D))), 2.0) / l))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 1e+287)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(M * Float64(0.5 / Float64(d / D))) ^ 2.0) / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = 1.0 - ((((D * M) / (d * 2.0)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 1e+287)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - (h * (((M * (0.5 / (d / D))) ^ 2.0) / 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[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+287], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(M * N[(0.5 / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 1.0000000000000001e287

   1. Initial program 99.9%

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

   if 1.0000000000000001e287 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 41.0%

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

   3. Simplified 44.5%

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

      1. frac-times 41.0%

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

      2. *-commutative 41.0%

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

      3. clear-num 41.0%

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

      4. un-div-inv 42.2%

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

      5. *-commutative 42.2%

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

      6. frac-times 45.7%

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

      7. frac-times 42.2%

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

      8. *-commutative 42.2%

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

      9. div-inv 42.2%

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

      10. *-commutative 42.2%

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

      11. associate-/r* 42.2%

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

      12. metadata-eval 42.2%

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

   5. Applied egg-rr 42.2%

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

      1. associate-/r/ 60.2%

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

      2. *-commutative 60.2%

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

      3. *-commutative 60.2%

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

      4. associate-*r* 63.7%

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

      5. *-commutative 63.7%

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

      6. associate-/r/ 63.7%

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

   7. Simplified 63.7%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+287}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 2: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.22:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 1.22)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D 2.0) (/ M d)) 2.0)))))
   (* w0 (sqrt (- 1.0 (* h (/ (pow (* M (/ 0.5 (/ d D))) 2.0) l)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.22) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D / 2.0) * (M / d)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 - (h * (pow((M * (0.5 / (d / D))), 2.0) / 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 (d_1 <= 1.22d0) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d / 2.0d0) * (m / d_1)) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 - (h * (((m * (0.5d0 / (d_1 / d))) ** 2.0d0) / 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 (d <= 1.22) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D / 2.0) * (M / d)), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow((M * (0.5 / (d / D))), 2.0) / l))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 1.22:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D / 2.0) * (M / d)), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow((M * (0.5 / (d / D))), 2.0) / l))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 1.22)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D / 2.0) * Float64(M / d)) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(M * Float64(0.5 / Float64(d / D))) ^ 2.0) / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 1.22)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D / 2.0) * (M / d)) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 - (h * (((M * (0.5 / (d / D))) ^ 2.0) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 1.22], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / 2.0), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(M * N[(0.5 / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.21999999999999997

   1. Initial program 78.7%

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

   3. Simplified 76.6%

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

   if 1.21999999999999997 < d

   1. Initial program 85.3%

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

   3. Simplified 86.8%

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

      1. frac-times 85.3%

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

      2. *-commutative 85.3%

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

      3. clear-num 85.2%

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

      4. un-div-inv 85.2%

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

      5. *-commutative 85.2%

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

      6. frac-times 86.8%

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

      7. frac-times 85.2%

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

      8. *-commutative 85.2%

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

      9. div-inv 85.2%

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

      10. *-commutative 85.2%

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

      11. associate-/r* 85.2%

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

      12. metadata-eval 85.2%

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

   5. Applied egg-rr 85.2%

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

      1. associate-/r/ 89.5%

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

      2. *-commutative 89.5%

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

      3. *-commutative 89.5%

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

      4. associate-*r* 92.1%

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

      5. *-commutative 92.1%

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

      6. associate-/r/ 92.1%

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

   7. Simplified 92.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.22:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 3: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\ell}} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* h (/ (pow (* M (/ 0.5 (/ d D))) 2.0) l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (h * (pow((M * (0.5 / (d / D))), 2.0) / 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
    code = w0 * sqrt((1.0d0 - (h * (((m * (0.5d0 / (d_1 / d))) ** 2.0d0) / 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 * (Math.pow((M * (0.5 / (d / D))), 2.0) / l))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (h * (math.pow((M * (0.5 / (d / D))), 2.0) / l))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(M * Float64(0.5 / Float64(d / D))) ^ 2.0) / l)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - (h * (((M * (0.5 / (d / D))) ^ 2.0) / l))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(M * N[(0.5 / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.6%

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

3. Simplified 79.5%

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

   1. frac-times 80.6%

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

   2. *-commutative 80.6%

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

   3. clear-num 80.6%

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

   4. un-div-inv 81.0%

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

   5. *-commutative 81.0%

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

   6. frac-times 79.9%

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

   7. frac-times 81.0%

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

   8. *-commutative 81.0%

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

   9. div-inv 81.0%

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

   10. *-commutative 81.0%

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

   11. associate-/r* 81.0%

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

   12. metadata-eval 81.0%

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

5. Applied egg-rr 81.0%

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

   1. associate-/r/ 85.4%

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

   2. *-commutative 85.4%

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

   3. *-commutative 85.4%

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

   4. associate-*r* 85.8%

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

   5. *-commutative 85.8%

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

   6. associate-/r/ 85.7%

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

7. Simplified 85.7%

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

8. Final simplification 85.7%

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

Alternative 4: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (/ (* h (pow (* (* D M) (/ 0.5 d)) 2.0)) l)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((h * pow(((D * M) * (0.5 / d)), 2.0)) / 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
    code = w0 * sqrt((1.0d0 - ((h * (((d * m) * (0.5d0 / d_1)) ** 2.0d0)) / 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 * Math.pow(((D * M) * (0.5 / d)), 2.0)) / l)));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - ((h * math.pow(((D * M) * (0.5 / d)), 2.0)) / l)))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(D * M) * Float64(0.5 / d)) ^ 2.0)) / l))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((h * (((D * M) * (0.5 / d)) ^ 2.0)) / l)));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(D * M), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.6%

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

3. Simplified 79.5%

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

   1. *-commutative 79.5%

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

   2. frac-times 80.6%

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

   3. *-commutative 80.6%

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

   4. associate-*l/ 86.5%

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

   5. *-commutative 86.5%

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

   6. frac-times 85.4%

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

   7. frac-times 86.5%

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

   8. *-commutative 86.5%

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

   9. div-inv 86.5%

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

   10. *-commutative 86.5%

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

   11. associate-/r* 86.5%

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

   12. metadata-eval 86.5%

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

5. Applied egg-rr 86.5%

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

6. Final simplification 86.5%

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

Alternative 5: 83.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{d \cdot \frac{2}{D}}\\ \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-223}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(t_0 \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ M (* d (/ 2.0 D)))))
   (if (<= (/ h l) -2e-223) (* w0 (sqrt (- 1.0 (* (/ h l) (* t_0 t_0))))) w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M / (d * (2.0 / D));
	double tmp;
	if ((h / l) <= -2e-223) {
		tmp = w0 * sqrt((1.0 - ((h / l) * (t_0 * t_0))));
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = m / (d_1 * (2.0d0 / d))
    if ((h / l) <= (-2d-223)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (t_0 * t_0))))
    else
        tmp = 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 t_0 = M / (d * (2.0 / D));
	double tmp;
	if ((h / l) <= -2e-223) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * (t_0 * t_0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = M / (d * (2.0 / D))
	tmp = 0
	if (h / l) <= -2e-223:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * (t_0 * t_0))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(M / Float64(d * Float64(2.0 / D)))
	tmp = 0.0
	if (Float64(h / l) <= -2e-223)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(t_0 * t_0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = M / (d * (2.0 / D));
	tmp = 0.0;
	if ((h / l) <= -2e-223)
		tmp = w0 * sqrt((1.0 - ((h / l) * (t_0 * t_0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(M / N[(d * N[(2.0 / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], -2e-223], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.9999999999999999e-223

   1. Initial program 76.8%

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

   3. Simplified 76.1%

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

      1. clear-num 76.0%

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

      2. frac-times 77.5%

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

      3. *-un-lft-identity 77.5%

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

   5. Applied egg-rr 77.5%

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

      1. unpow2 77.5%

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

   7. Applied egg-rr 77.5%

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

   if -1.9999999999999999e-223 < (/.f64 h l)

   1. Initial program 84.7%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in D around 0 93.6%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-223}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\frac{M}{d \cdot \frac{2}{D}} \cdot \frac{M}{d \cdot \frac{2}{D}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 6: 66.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 4.7 \cdot 10^{-138}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{\left(w0 \cdot h\right) \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{\ell \cdot {d}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 4.7e-138)
   w0
   (+ w0 (* -0.125 (/ (* (* w0 h) (* (* D M) (* D M))) (* l (pow d 2.0)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.7e-138) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((w0 * h) * ((D * M) * (D * M))) / (l * pow(d, 2.0))));
	}
	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 <= 4.7d-138) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((w0 * h) * ((d * m) * (d * m))) / (l * (d_1 ** 2.0d0))))
    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 <= 4.7e-138) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((w0 * h) * ((D * M) * (D * M))) / (l * Math.pow(d, 2.0))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4.7e-138:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (((w0 * h) * ((D * M) * (D * M))) / (l * math.pow(d, 2.0))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4.7e-138)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(w0 * h) * Float64(Float64(D * M) * Float64(D * M))) / Float64(l * (d ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4.7e-138)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((w0 * h) * ((D * M) * (D * M))) / (l * (d ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 4.7e-138], w0, N[(w0 + N[(-0.125 * N[(N[(N[(w0 * h), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.7000000000000001e-138

   1. Initial program 82.9%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in D around 0 77.5%

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

   if 4.7000000000000001e-138 < M

   1. Initial program 76.8%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in D around 0 50.6%

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

      1. expm1-log1p-u 33.9%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)\right)}}{{d}^{2} \cdot \ell} \]

      2. expm1-udef 33.9%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)} - 1}}{{d}^{2} \cdot \ell} \]

      3. associate-*r* 38.1%

         \[\leadsto w0 + -0.125 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}\right)} - 1}{{d}^{2} \cdot \ell} \]

      4. pow-prod-down 43.3%

         \[\leadsto w0 + -0.125 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)\right)} - 1}{{d}^{2} \cdot \ell} \]

   6. Applied egg-rr 43.3%

      \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)\right)} - 1}}{{d}^{2} \cdot \ell} \]
   7. Step-by-step derivation

      1. expm1-def 43.4%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)\right)\right)}}{{d}^{2} \cdot \ell} \]

      2. expm1-log1p 62.2%

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

      3. *-commutative 62.2%

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

   8. Simplified 62.2%

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

      1. unpow2 62.2%

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

   10. Applied egg-rr 62.2%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.7 \cdot 10^{-138}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{\left(w0 \cdot h\right) \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{\ell \cdot {d}^{2}}\\ \end{array} \]

Alternative 7: 66.9% 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 80.6%

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

3. Simplified 79.5%

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

4. Taylor expanded in D around 0 69.1%

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

5. Final simplification 69.1%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023311 
(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))))))