Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 87.8%

Time: 19.7s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{h}\\ \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D}{\frac{d}{M} \cdot 2}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \frac{t\_0}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (* D (* 0.5 (/ M d))) (sqrt h))))
   (if (<= h -4e-310)
     (* w0 (sqrt (- 1.0 (/ (* h (pow (/ D (* (/ d M) 2.0)) 2.0)) l))))
     (* w0 (sqrt (- 1.0 (* t_0 (/ t_0 l))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (D * (0.5 * (M / d))) * sqrt(h);
	double tmp;
	if (h <= -4e-310) {
		tmp = w0 * sqrt((1.0 - ((h * pow((D / ((d / M) * 2.0)), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_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 = (d * (0.5d0 * (m / d_1))) * sqrt(h)
    if (h <= (-4d-310)) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((d / ((d_1 / m) * 2.0d0)) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (t_0 * (t_0 / 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 = (D * (0.5 * (M / d))) * Math.sqrt(h);
	double tmp;
	if (h <= -4e-310) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((D / ((d / M) * 2.0)), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_0 / l))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = (D * (0.5 * (M / d))) * math.sqrt(h)
	tmp = 0
	if h <= -4e-310:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((D / ((d / M) * 2.0)), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_0 / l))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(D * Float64(0.5 * Float64(M / d))) * sqrt(h))
	tmp = 0.0
	if (h <= -4e-310)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(D / Float64(Float64(d / M) * 2.0)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (D * (0.5 * (M / d))) * sqrt(h);
	tmp = 0.0;
	if (h <= -4e-310)
		tmp = w0 * sqrt((1.0 - ((h * ((D / ((d / M) * 2.0)) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -4e-310], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(D / N[(N[(d / M), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -3.999999999999988e-310

   1. Initial program 74.8%

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

   3. Simplified 77.3%

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

      1. unpow2 77.3%

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

      2. unpow2 77.3%

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

      3. associate-/r* 77.3%

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

      4. associate-*r/ 74.8%

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

      5. *-commutative 74.8%

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

      6. associate-*r/ 79.8%

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

      7. *-commutative 79.8%

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

      8. associate-*l/ 81.4%

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

      9. associate-/r* 81.4%

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

      10. clear-num 81.4%

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

      11. associate-*l/ 81.4%

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

      12. *-un-lft-identity 81.4%

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

      13. associate-/r/ 81.4%

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

   6. Applied egg-rr 81.4%

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

   if -3.999999999999988e-310 < h

   1. Initial program 74.7%

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

   3. Simplified 74.7%

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

      1. unpow2 74.7%

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

      2. unpow2 74.7%

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

      3. associate-/r* 74.7%

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

      4. associate-*r/ 74.7%

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

      5. *-commutative 74.7%

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

      6. associate-*r/ 81.7%

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

      7. *-commutative 81.7%

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

      8. associate-*l/ 81.7%

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

      9. associate-/r* 81.7%

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

      10. clear-num 81.7%

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

      11. associate-*l/ 81.7%

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

      12. *-un-lft-identity 81.7%

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

      13. associate-/r/ 81.7%

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

   6. Applied egg-rr 81.7%

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

      1. add-sqr-sqrt 81.7%

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

      2. *-un-lft-identity 81.7%

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

      3. times-frac 81.6%

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

   8. Applied egg-rr 86.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D}{\frac{d}{M} \cdot 2}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{h}\right) \cdot \frac{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D M) (* d 2.0)) 2.0) (/ h l)) -2e-13)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* D (/ (/ M 2.0) d)) 2.0)))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((D * M) / (d * 2.0)), 2.0) * (h / l)) <= -2e-13) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D * ((M / 2.0) / d)), 2.0))));
	} 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) :: tmp
    if (((((d * m) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= (-2d-13)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d * ((m / 2.0d0) / d_1)) ** 2.0d0))))
    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 tmp;
	if ((Math.pow(((D * M) / (d * 2.0)), 2.0) * (h / l)) <= -2e-13) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D * ((M / 2.0) / d)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((D * M) / (d * 2.0)), 2.0) * (h / l)) <= -2e-13:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((D * ((M / 2.0) / d)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -2e-13)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D * Float64(Float64(M / 2.0) / d)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((D * M) / (d * 2.0)) ^ 2.0) * (h / l)) <= -2e-13)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D * ((M / 2.0) / d)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-13], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -2.0000000000000001e-13

   1. Initial program 57.1%

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

   3. Simplified 58.1%

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

   if -2.0000000000000001e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 84.3%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in D around 0 94.3%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D}{\frac{d}{M} \cdot 2}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{h}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= h -4e-310)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (/ D (* (/ d M) 2.0)) 2.0)) l))))
   (* w0 (sqrt (- 1.0 (/ (pow (* (* D (* 0.5 (/ M d))) (sqrt h)) 2.0) l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (h <= -4e-310) {
		tmp = w0 * sqrt((1.0 - ((h * pow((D / ((d / M) * 2.0)), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (pow(((D * (0.5 * (M / d))) * sqrt(h)), 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 (h <= (-4d-310)) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((d / ((d_1 / m) * 2.0d0)) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - ((((d * (0.5d0 * (m / d_1))) * sqrt(h)) ** 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 (h <= -4e-310) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((D / ((d / M) * 2.0)), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow(((D * (0.5 * (M / d))) * Math.sqrt(h)), 2.0) / l)));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if h <= -4e-310:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((D / ((d / M) * 2.0)), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (math.pow(((D * (0.5 * (M / d))) * math.sqrt(h)), 2.0) / l)))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (h <= -4e-310)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(D / Float64(Float64(d / M) * 2.0)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(D * Float64(0.5 * Float64(M / d))) * sqrt(h)) ^ 2.0) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (h <= -4e-310)
		tmp = w0 * sqrt((1.0 - ((h * ((D / ((d / M) * 2.0)) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - ((((D * (0.5 * (M / d))) * sqrt(h)) ^ 2.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[h, -4e-310], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(D / N[(N[(d / M), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -3.999999999999988e-310

   1. Initial program 74.8%

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

   3. Simplified 77.3%

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

      1. unpow2 77.3%

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

      2. unpow2 77.3%

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

      3. associate-/r* 77.3%

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

      4. associate-*r/ 74.8%

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

      5. *-commutative 74.8%

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

      6. associate-*r/ 79.8%

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

      7. *-commutative 79.8%

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

      8. associate-*l/ 81.4%

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

      9. associate-/r* 81.4%

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

      10. clear-num 81.4%

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

      11. associate-*l/ 81.4%

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

      12. *-un-lft-identity 81.4%

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

      13. associate-/r/ 81.4%

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

   6. Applied egg-rr 81.4%

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

   if -3.999999999999988e-310 < h

   1. Initial program 74.7%

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

   3. Simplified 74.7%

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

      1. unpow2 74.7%

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

      2. unpow2 74.7%

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

      3. associate-/r* 74.7%

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

      4. associate-*r/ 74.7%

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

      5. *-commutative 74.7%

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

      6. associate-*r/ 81.7%

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

      7. *-commutative 81.7%

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

      8. associate-*l/ 81.7%

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

      9. associate-/r* 81.7%

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

      10. clear-num 81.7%

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

      11. associate-*l/ 81.7%

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

      12. *-un-lft-identity 81.7%

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

      13. associate-/r/ 81.7%

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

   6. Applied egg-rr 81.7%

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

      1. add-sqr-sqrt 81.7%

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

      2. pow2 81.7%

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

      3. *-commutative 81.7%

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

      4. div-inv 81.7%

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

      5. associate-*l/ 81.7%

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

      6. *-commutative 81.7%

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

      7. clear-num 81.7%

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

      8. sqrt-prod 81.7%

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

      9. unpow2 81.7%

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

      10. sqrt-prod 54.3%

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

      11. add-sqr-sqrt 85.2%

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

      12. *-un-lft-identity 85.2%

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

      13. times-frac 85.2%

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

      14. metadata-eval 85.2%

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

   8. Applied egg-rr 85.2%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D}{\frac{d}{M} \cdot 2}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{h}\right)}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (/ (* h (pow (/ D (* (/ d M) 2.0)) 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 / ((d / M) * 2.0)), 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 / ((d_1 / m) * 2.0d0)) ** 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 / ((d / M) * 2.0)), 2.0)) / l)));
}

Python

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

Julia

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

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((h * ((D / ((d / M) * 2.0)) ^ 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[(D / N[(N[(d / M), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.7%

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

3. Simplified 75.9%

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

   1. unpow2 75.9%

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

   2. unpow2 75.9%

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

   3. associate-/r* 75.9%

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

   4. associate-*r/ 74.7%

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

   5. *-commutative 74.7%

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

   6. associate-*r/ 80.8%

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

   7. *-commutative 80.8%

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

   8. associate-*l/ 81.5%

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

   9. associate-/r* 81.5%

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

   10. clear-num 81.5%

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

   11. associate-*l/ 81.5%

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

   12. *-un-lft-identity 81.5%

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

   13. associate-/r/ 81.5%

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

6. Applied egg-rr 81.5%

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

7. Final simplification 81.5%

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

Alternative 5: 64.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 3000000000000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(w0 \cdot \left(1 + w0 \cdot \left(w0 \cdot 0.3333333333333333 - 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 3000000000000.0)
   w0
   (expm1 (* w0 (+ 1.0 (* w0 (- (* w0 0.3333333333333333) 0.5)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3000000000000.0) {
		tmp = w0;
	} else {
		tmp = expm1((w0 * (1.0 + (w0 * ((w0 * 0.3333333333333333) - 0.5)))));
	}
	return tmp;
}

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3000000000000.0) {
		tmp = w0;
	} else {
		tmp = Math.expm1((w0 * (1.0 + (w0 * ((w0 * 0.3333333333333333) - 0.5)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 3000000000000.0:
		tmp = w0
	else:
		tmp = math.expm1((w0 * (1.0 + (w0 * ((w0 * 0.3333333333333333) - 0.5)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 3000000000000.0)
		tmp = w0;
	else
		tmp = expm1(Float64(w0 * Float64(1.0 + Float64(w0 * Float64(Float64(w0 * 0.3333333333333333) - 0.5)))));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 3000000000000.0], w0, N[(Exp[N[(w0 * N[(1.0 + N[(w0 * N[(N[(w0 * 0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3e12

   1. Initial program 76.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in D around 0 68.5%

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

   if 3e12 < M

   1. Initial program 68.4%

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

   3. Simplified 70.2%

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

   6. Applied egg-rr 21.1%

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

      1. expm1-define 43.3%

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

      2. *-commutative 43.3%

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

      3. associate-*l/ 43.7%

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

      4. associate-/l* 47.2%

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

      5. associate-*l/ 47.2%

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

      6. *-commutative 47.2%

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

      7. associate-*l/ 47.2%

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

      8. associate-/l/ 47.2%

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

      9. associate-/r/ 47.2%

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

      10. associate-*l/ 47.2%

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

      11. *-commutative 47.2%

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

      12. associate-*r/ 45.4%

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

      13. *-commutative 45.4%

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

      14. associate-/l/ 45.4%

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

      15. associate-/l* 47.2%

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

   8. Simplified 47.2%

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

   9. Taylor expanded in h around 0 8.0%

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

       1. log1p-define 30.6%

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

   11. Simplified 30.6%

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

   12. Taylor expanded in w0 around 0 37.6%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3000000000000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(w0 \cdot \left(1 + w0 \cdot \left(w0 \cdot 0.3333333333333333 - 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.7% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 3.2 \cdot 10^{+95}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(w0 + -1\right) \cdot \left(1 - w0\right)}{2 - w0}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 3.2e+95) w0 (/ (+ 1.0 (* (+ w0 -1.0) (- 1.0 w0))) (- 2.0 w0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.2e+95) {
		tmp = w0;
	} else {
		tmp = (1.0 + ((w0 + -1.0) * (1.0 - w0))) / (2.0 - 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 <= 3.2d+95) then
        tmp = w0
    else
        tmp = (1.0d0 + ((w0 + (-1.0d0)) * (1.0d0 - w0))) / (2.0d0 - 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 <= 3.2e+95) {
		tmp = w0;
	} else {
		tmp = (1.0 + ((w0 + -1.0) * (1.0 - w0))) / (2.0 - w0);
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 3.2e+95:
		tmp = w0
	else:
		tmp = (1.0 + ((w0 + -1.0) * (1.0 - w0))) / (2.0 - w0)
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 3.2e+95)
		tmp = w0;
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(w0 + -1.0) * Float64(1.0 - w0))) / Float64(2.0 - w0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 3.2e+95)
		tmp = w0;
	else
		tmp = (1.0 + ((w0 + -1.0) * (1.0 - w0))) / (2.0 - w0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 3.2e+95], w0, N[(N[(1.0 + N[(N[(w0 + -1.0), $MachinePrecision] * N[(1.0 - w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 - w0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.2000000000000001e95

   1. Initial program 74.9%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in D around 0 66.9%

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

   if 3.2000000000000001e95 < M

   1. Initial program 73.5%

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

   3. Simplified 76.5%

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

   6. Applied egg-rr 24.8%

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

      1. expm1-define 44.9%

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

      2. *-commutative 44.9%

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

      3. associate-*l/ 45.3%

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

      4. associate-/l* 51.1%

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

      5. associate-*l/ 51.1%

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

      6. *-commutative 51.1%

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

      7. associate-*l/ 51.1%

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

      8. associate-/l/ 51.1%

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

      9. associate-/r/ 51.1%

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

      10. associate-*l/ 51.1%

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

      11. *-commutative 51.1%

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

      12. associate-*r/ 48.0%

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

      13. *-commutative 48.0%

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

      14. associate-/l/ 48.0%

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

      15. associate-/l* 51.1%

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

   8. Simplified 51.1%

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

   9. Taylor expanded in h around 0 8.2%

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

       1. log1p-define 28.6%

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

   11. Simplified 28.6%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(w0\right)}\right) \]
   12. Step-by-step derivation

       1. expm1-undefine 8.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(w0\right)} - 1} \]

       2. log1p-undefine 8.2%

          \[\leadsto e^{\color{blue}{\log \left(1 + w0\right)}} - 1 \]

       3. rem-exp-log 19.1%

          \[\leadsto \color{blue}{\left(1 + w0\right)} - 1 \]

   13. Applied egg-rr 19.1%

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

       1. associate--l+ 19.1%

          \[\leadsto \color{blue}{1 + \left(w0 - 1\right)} \]

       2. flip-+ 19.2%

          \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(w0 - 1\right) \cdot \left(w0 - 1\right)}{1 - \left(w0 - 1\right)}} \]

       3. metadata-eval 19.2%

          \[\leadsto \frac{\color{blue}{1} - \left(w0 - 1\right) \cdot \left(w0 - 1\right)}{1 - \left(w0 - 1\right)} \]

       4. sub-neg 19.2%

          \[\leadsto \frac{1 - \color{blue}{\left(w0 + \left(-1\right)\right)} \cdot \left(w0 - 1\right)}{1 - \left(w0 - 1\right)} \]

       5. metadata-eval 19.2%

          \[\leadsto \frac{1 - \left(w0 + \color{blue}{-1}\right) \cdot \left(w0 - 1\right)}{1 - \left(w0 - 1\right)} \]

       6. sub-neg 19.2%

          \[\leadsto \frac{1 - \left(w0 + -1\right) \cdot \color{blue}{\left(w0 + \left(-1\right)\right)}}{1 - \left(w0 - 1\right)} \]

       7. metadata-eval 19.2%

          \[\leadsto \frac{1 - \left(w0 + -1\right) \cdot \left(w0 + \color{blue}{-1}\right)}{1 - \left(w0 - 1\right)} \]

       8. sub-neg 19.2%

          \[\leadsto \frac{1 - \left(w0 + -1\right) \cdot \left(w0 + -1\right)}{1 - \color{blue}{\left(w0 + \left(-1\right)\right)}} \]

       9. metadata-eval 19.2%

          \[\leadsto \frac{1 - \left(w0 + -1\right) \cdot \left(w0 + -1\right)}{1 - \left(w0 + \color{blue}{-1}\right)} \]

   15. Applied egg-rr 19.2%

       \[\leadsto \color{blue}{\frac{1 - \left(w0 + -1\right) \cdot \left(w0 + -1\right)}{1 - \left(w0 + -1\right)}} \]
   16. Step-by-step derivation

       1. sub-neg 19.2%

          \[\leadsto \frac{\color{blue}{1 + \left(-\left(w0 + -1\right) \cdot \left(w0 + -1\right)\right)}}{1 - \left(w0 + -1\right)} \]

       2. distribute-rgt-neg-in 19.2%

          \[\leadsto \frac{1 + \color{blue}{\left(w0 + -1\right) \cdot \left(-\left(w0 + -1\right)\right)}}{1 - \left(w0 + -1\right)} \]

       3. +-commutative 19.2%

          \[\leadsto \frac{1 + \color{blue}{\left(-1 + w0\right)} \cdot \left(-\left(w0 + -1\right)\right)}{1 - \left(w0 + -1\right)} \]

       4. +-commutative 19.2%

          \[\leadsto \frac{1 + \left(-1 + w0\right) \cdot \left(-\color{blue}{\left(-1 + w0\right)}\right)}{1 - \left(w0 + -1\right)} \]

       5. distribute-neg-in 19.2%

          \[\leadsto \frac{1 + \left(-1 + w0\right) \cdot \color{blue}{\left(\left(--1\right) + \left(-w0\right)\right)}}{1 - \left(w0 + -1\right)} \]

       6. metadata-eval 19.2%

          \[\leadsto \frac{1 + \left(-1 + w0\right) \cdot \left(\color{blue}{1} + \left(-w0\right)\right)}{1 - \left(w0 + -1\right)} \]

       7. sub-neg 19.2%

          \[\leadsto \frac{1 + \left(-1 + w0\right) \cdot \color{blue}{\left(1 - w0\right)}}{1 - \left(w0 + -1\right)} \]

       8. +-commutative 19.2%

          \[\leadsto \frac{1 + \left(-1 + w0\right) \cdot \left(1 - w0\right)}{1 - \color{blue}{\left(-1 + w0\right)}} \]

       9. associate--r+ 19.2%

          \[\leadsto \frac{1 + \left(-1 + w0\right) \cdot \left(1 - w0\right)}{\color{blue}{\left(1 - -1\right) - w0}} \]

       10. metadata-eval 19.2%

           \[\leadsto \frac{1 + \left(-1 + w0\right) \cdot \left(1 - w0\right)}{\color{blue}{2} - w0} \]

   17. Simplified 19.2%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.2 \cdot 10^{+95}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(w0 + -1\right) \cdot \left(1 - w0\right)}{2 - w0}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% 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 74.7%

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

3. Simplified 75.9%

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

5. Taylor expanded in D around 0 63.4%

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

6. Final simplification 63.4%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024054 
(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))))))