Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 87.4%

Time: 17.4s

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: 81.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+281}:\\ \;\;\;\;w0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
   (if (<= t_0 5e+281)
     (* w0 (sqrt t_0))
     (* 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) {
	double t_0 = 1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+281) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow(((D / d) * (M / 2.0)), 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 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))
    if (t_0 <= 5d+281) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - ((h * (((d / d_1) * (m / 2.0d0)) ** 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(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+281) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow(((D / d) * (M / 2.0)), 2.0)) / l)));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = 1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 5e+281:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow(((D / d) * (M / 2.0)), 2.0)) / l)))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 5e+281)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0)) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = 1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 5e+281)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - ((h * (((D / d) * (M / 2.0)) ^ 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[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+281], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $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))) < 5.00000000000000016e281

   1. Initial program 99.9%

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

   if 5.00000000000000016e281 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 39.8%

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

   3. Simplified 42.4%

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

      1. associate-*r/ 62.7%

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

      2. *-commutative 62.7%

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

      3. frac-times 60.2%

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

      4. *-commutative 60.2%

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

      5. times-frac 62.7%

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

   6. Applied egg-rr 62.7%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+281}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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))) < +inf.0

   1. Initial program 89.8%

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

   if +inf.0 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 0.0%

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

   3. Simplified 4.5%

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

   5. Taylor expanded in D around 0 53.5%

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

4. Final simplification 86.7%

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

Alternative 3: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-212}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -2e-212)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D 2.0) (/ M d)) 2.0)))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -2e-212) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D / 2.0) * (M / 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 ((h / l) <= (-2d-212)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d / 2.0d0) * (m / 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 ((h / l) <= -2e-212) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D / 2.0) * (M / d)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -2e-212:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D / 2.0) * (M / d)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -2e-212)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D / 2.0) * Float64(M / 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 ((h / l) <= -2e-212)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D / 2.0) * (M / d)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -2e-212], 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], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.99999999999999991e-212

   1. Initial program 75.8%

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

   3. Simplified 77.0%

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

   if -1.99999999999999991e-212 < (/.f64 h l)

   1. Initial program 88.4%

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

   3. Simplified 87.6%

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

   5. Taylor expanded in D around 0 92.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-212}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-212}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D \cdot 0.5}{\frac{d}{M}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -2e-212)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ (* D 0.5) (/ d M)) 2.0)))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -2e-212) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D * 0.5) / (d / M)), 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 ((h / l) <= (-2d-212)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d * 0.5d0) / (d_1 / m)) ** 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 ((h / l) <= -2e-212) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D * 0.5) / (d / M)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -2e-212:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D * 0.5) / (d / M)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -2e-212)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D * 0.5) / Float64(d / M)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -2e-212)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D * 0.5) / (d / M)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -2e-212], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D * 0.5), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.99999999999999991e-212

   1. Initial program 75.8%

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

   3. Simplified 77.0%

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

      1. clear-num 76.9%

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

      2. un-div-inv 76.9%

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

      3. div-inv 76.9%

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

      4. metadata-eval 76.9%

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

   6. Applied egg-rr 76.9%

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

   if -1.99999999999999991e-212 < (/.f64 h l)

   1. Initial program 88.4%

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

   3. Simplified 87.6%

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

   5. Taylor expanded in D around 0 92.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-212}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D \cdot 0.5}{\frac{d}{M}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{+27}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{w0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d) :precision binary64 (if (<= M 5e+27) w0 (log (exp w0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 5e+27) {
		tmp = w0;
	} else {
		tmp = log(exp(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 <= 5d+27) then
        tmp = w0
    else
        tmp = log(exp(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 <= 5e+27) {
		tmp = w0;
	} else {
		tmp = Math.log(Math.exp(w0));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 5e+27:
		tmp = w0
	else:
		tmp = math.log(math.exp(w0))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 5e+27)
		tmp = w0;
	else
		tmp = log(exp(w0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 5e+27)
		tmp = w0;
	else
		tmp = log(exp(w0));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 5e+27], w0, N[Log[N[Exp[w0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.99999999999999979e27

   1. Initial program 84.0%

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

   3. Simplified 83.8%

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

   5. Taylor expanded in D around 0 74.7%

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

   if 4.99999999999999979e27 < M

   1. Initial program 72.4%

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

   3. Simplified 74.7%

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

   6. Applied egg-rr 22.6%

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

   8. Simplified 49.7%

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

   9. Taylor expanded in h around 0 11.8%

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

       1. log1p-define 28.4%

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

   11. Simplified 28.4%

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

       1. add-log-exp 16.7%

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

       2. expm1-log1p-u 31.6%

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

   13. Applied egg-rr 31.6%

       \[\leadsto \color{blue}{\log \left(e^{w0}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{+27}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{w0}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{+69}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(w0 + -1\right)}^{2}}{2 - w0}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 4e+69) w0 (/ (- 1.0 (pow (+ w0 -1.0) 2.0)) (- 2.0 w0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4e+69) {
		tmp = w0;
	} else {
		tmp = (1.0 - pow((w0 + -1.0), 2.0)) / (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 <= 4d+69) then
        tmp = w0
    else
        tmp = (1.0d0 - ((w0 + (-1.0d0)) ** 2.0d0)) / (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 <= 4e+69) {
		tmp = w0;
	} else {
		tmp = (1.0 - Math.pow((w0 + -1.0), 2.0)) / (2.0 - w0);
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4e+69:
		tmp = w0
	else:
		tmp = (1.0 - math.pow((w0 + -1.0), 2.0)) / (2.0 - w0)
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4e+69)
		tmp = w0;
	else
		tmp = Float64(Float64(1.0 - (Float64(w0 + -1.0) ^ 2.0)) / Float64(2.0 - w0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4e+69)
		tmp = w0;
	else
		tmp = (1.0 - ((w0 + -1.0) ^ 2.0)) / (2.0 - w0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 4e+69], w0, N[(N[(1.0 - N[Power[N[(w0 + -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 - w0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.0000000000000003e69

   1. Initial program 83.7%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in D around 0 73.0%

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

   if 4.0000000000000003e69 < M

   1. Initial program 71.6%

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

   3. Simplified 74.4%

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

   6. Applied egg-rr 27.2%

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

   8. Simplified 55.3%

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

   9. Taylor expanded in h around 0 13.9%

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

       1. log1p-define 31.3%

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

   11. Simplified 31.3%

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

       1. expm1-undefine 13.9%

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

       2. log1p-undefine 13.9%

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

       3. rem-exp-log 18.9%

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

   13. Applied egg-rr 18.9%

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

       1. associate--l+ 18.9%

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

   15. Simplified 18.9%

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

       1. flip-+ 32.5%

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

       2. metadata-eval 32.5%

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

       3. div-sub 32.5%

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

       4. sub-neg 32.5%

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

       5. metadata-eval 32.5%

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

       6. pow2 32.5%

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

       7. sub-neg 32.5%

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

       8. metadata-eval 32.5%

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

       9. sub-neg 32.5%

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

       10. metadata-eval 32.5%

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

   17. Applied egg-rr 32.5%

       \[\leadsto \color{blue}{\frac{1}{1 - \left(w0 + -1\right)} - \frac{{\left(w0 + -1\right)}^{2}}{1 - \left(w0 + -1\right)}} \]
   18. Step-by-step derivation

       1. div-sub 32.5%

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

       2. +-commutative 32.5%

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

       3. +-commutative 32.5%

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

       4. associate--r+ 32.5%

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

       5. metadata-eval 32.5%

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

   19. Simplified 32.5%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{+69}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(w0 + -1\right)}^{2}}{2 - w0}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.5% 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 82.1%

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

3. Simplified 82.3%

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

5. Taylor expanded in D around 0 68.2%

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

6. Final simplification 68.2%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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))))))