Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.5% → 86.2%

Time: 12.1s

Alternatives: 7

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.2% accurate, 0.5× speedup

Math

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

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<=
      (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
      INFINITY)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D 2.0) (/ M d)) 2.0)))))
   w0))

C

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

Java

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

Python

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

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= Inf)
		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

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))))) <= Inf)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D / 2.0) * (M / d)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[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], Infinity], 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 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < +inf.0

   1. Initial program 86.1%

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

   3. Simplified 86.5%

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

   if +inf.0 < (*.f64 w0 (sqrt.f64 (-.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 13.0%

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

   4. Taylor expanded in D around 0 76.6%

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

4. Final simplification 85.6%

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

Alternative 2: 71.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4.6 \cdot 10^{-119}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 4.6e-119)
   w0
   (* w0 (sqrt (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (/ (* h (* M M)) l))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.6e-119) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((h * (M * M)) / l)))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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 <= 4.6d-119) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * ((h * (m * m)) / l)))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.6e-119) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((h * (M * M)) / l)))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4.6e-119:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((h * (M * M)) / l)))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4.6e-119)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(h * Float64(M * M)) / l))))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4.6e-119)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((h * (M * M)) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 4.6e-119], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.59999999999999987e-119

   1. Initial program 82.0%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in D around 0 70.3%

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

   if 4.59999999999999987e-119 < M

   1. Initial program 70.8%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in D around 0 52.0%

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

      1. times-frac 50.9%

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

      2. unpow2 50.9%

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

      3. unpow2 50.9%

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

      4. unpow2 50.9%

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

   6. Simplified 50.9%

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

   7. Taylor expanded in D around 0 50.9%

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

      1. unpow2 50.9%

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

      2. unpow2 50.9%

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

      3. times-frac 62.4%

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

   9. Simplified 62.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.6 \cdot 10^{-119}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)}\\ \end{array} \]

Alternative 3: 70.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.25 \cdot 10^{-118}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left(\frac{M \cdot D}{d \cdot d} \cdot \frac{M \cdot D}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.25e-118)
   w0
   (* w0 (+ 1.0 (* -0.125 (* h (* (/ (* M D) (* d d)) (/ (* M D) l))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.25e-118) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (h * (((M * D) / (d * d)) * ((M * D) / l)))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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 <= 1.25d-118) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (h * (((m * d) / (d_1 * d_1)) * ((m * d) / l)))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.25e-118) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (h * (((M * D) / (d * d)) * ((M * D) / l)))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.25e-118:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (h * (((M * D) / (d * d)) * ((M * D) / l)))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.25e-118)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(h * Float64(Float64(Float64(M * D) / Float64(d * d)) * Float64(Float64(M * D) / l))))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.25e-118)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (h * (((M * D) / (d * d)) * ((M * D) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.25e-118], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(h * N[(N[(N[(M * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.25000000000000004e-118

   1. Initial program 82.0%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in D around 0 70.3%

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

   if 1.25000000000000004e-118 < M

   1. Initial program 70.8%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in D around 0 49.2%

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

      1. *-commutative 49.2%

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

      2. times-frac 49.3%

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

      3. unpow2 49.3%

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

      4. unpow2 49.3%

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

      5. unpow2 49.3%

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

   6. Simplified 49.3%

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

   7. Taylor expanded in D around 0 49.2%

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

      1. associate-*r* 50.4%

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

      2. unpow2 50.4%

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

      3. unpow2 50.4%

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

      4. unpow2 50.4%

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

      5. associate-*r* 55.6%

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

      6. associate-*l/ 55.6%

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

      7. *-commutative 55.6%

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

      8. unswap-sqr 62.0%

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

      9. associate-*r* 56.9%

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

   9. Simplified 56.9%

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

       1. times-frac 58.5%

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

   11. Applied egg-rr 58.5%

       \[\leadsto w0 \cdot \left(1 + \left(h \cdot \color{blue}{\left(\frac{D \cdot M}{d \cdot d} \cdot \frac{D \cdot M}{\ell}\right)}\right) \cdot -0.125\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.25 \cdot 10^{-118}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left(\frac{M \cdot D}{d \cdot d} \cdot \frac{M \cdot D}{\ell}\right)\right)\right)\\ \end{array} \]

Alternative 4: 70.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-119}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell} \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 5e-119)
   w0
   (* w0 (+ 1.0 (* (/ (* (* (/ D d) (/ D d)) (* h (* M M))) l) -0.125)))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 5e-119) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (((((D / d) * (D / d)) * (h * (M * M))) / l) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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-119) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + (((((d / d_1) * (d / d_1)) * (h * (m * m))) / l) * (-0.125d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 5e-119) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (((((D / d) * (D / d)) * (h * (M * M))) / l) * -0.125));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 5e-119:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (((((D / d) * (D / d)) * (h * (M * M))) / l) * -0.125))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 5e-119)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(h * Float64(M * M))) / l) * -0.125)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 5e-119)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (((((D / d) * (D / d)) * (h * (M * M))) / l) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 5e-119], w0, N[(w0 * N[(1.0 + N[(N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.99999999999999993e-119

   1. Initial program 82.0%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in D around 0 70.3%

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

   if 4.99999999999999993e-119 < M

   1. Initial program 70.8%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in D around 0 49.2%

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

      1. *-commutative 49.2%

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

      2. times-frac 49.3%

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

      3. unpow2 49.3%

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

      4. unpow2 49.3%

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

      5. unpow2 49.3%

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

   6. Simplified 49.3%

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

      1. associate-*r/ 49.3%

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

      2. times-frac 56.2%

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

      3. *-commutative 56.2%

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

   8. Applied egg-rr 56.2%

      \[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell}} \cdot -0.125\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-119}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell} \cdot -0.125\right)\\ \end{array} \]

Alternative 5: 68.4% accurate, 10.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.1 \cdot 10^{-38}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(w0 \cdot M\right) \cdot \left(\left(M \cdot h\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d \cdot \ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 3.1e-38)
   w0
   (* -0.125 (* (* w0 M) (* (* M h) (* (/ D d) (/ D (* d l))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.1e-38) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((w0 * M) * ((M * h) * ((D / d) * (D / (d * l)))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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 <= 3.1d-38) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((w0 * m) * ((m * h) * ((d / d_1) * (d / (d_1 * l)))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.1e-38) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((w0 * M) * ((M * h) * ((D / d) * (D / (d * l)))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 3.1e-38:
		tmp = w0
	else:
		tmp = -0.125 * ((w0 * M) * ((M * h) * ((D / d) * (D / (d * l)))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 3.1e-38)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(w0 * M) * Float64(Float64(M * h) * Float64(Float64(D / d) * Float64(D / Float64(d * l))))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 3.1e-38)
		tmp = w0;
	else
		tmp = -0.125 * ((w0 * M) * ((M * h) * ((D / d) * (D / (d * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 3.1e-38], w0, N[(-0.125 * N[(N[(w0 * M), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(D / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.09999999999999983e-38

   1. Initial program 82.5%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in D around 0 70.8%

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

   if 3.09999999999999983e-38 < M

   1. Initial program 66.5%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in D around 0 44.4%

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

      1. +-commutative 44.4%

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

      2. *-commutative 44.4%

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

      3. fma-def 44.4%

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

      4. associate-*r* 45.9%

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

      5. unpow2 45.9%

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

      6. unpow2 45.9%

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

      7. swap-sqr 53.8%

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

      8. associate-/l* 53.6%

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

      9. swap-sqr 45.7%

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

      10. unpow2 45.7%

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

      11. associate-*l* 49.0%

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

   6. Simplified 49.0%

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

   7. Taylor expanded in D around inf 30.7%

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

      1. unpow2 30.7%

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

      2. associate-*r* 32.5%

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

      3. associate-/l* 32.6%

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

      4. unpow2 32.6%

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

      5. associate-*r* 30.8%

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

      6. associate-*r* 30.8%

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

      7. *-commutative 30.8%

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

      8. unpow2 30.8%

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

      9. associate-*r* 31.2%

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

   9. Simplified 31.2%

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

       1. pow1 31.2%

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

       2. associate-*l* 33.1%

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

   11. Applied egg-rr 33.1%

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

       1. pow1 33.1%

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

       2. associate-/r/ 32.9%

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

       3. associate-*l* 33.1%

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

       4. *-commutative 33.1%

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

       5. associate-*r* 33.2%

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

       6. times-frac 34.0%

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

   13. Applied egg-rr 34.0%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.1 \cdot 10^{-38}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(w0 \cdot M\right) \cdot \left(\left(M \cdot h\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d \cdot \ell}\right)\right)\right)\\ \end{array} \]

Alternative 6: 68.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.12 \cdot 10^{-38}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\frac{D}{d} \cdot \frac{D}{d}}{\ell} \cdot \left(\left(M \cdot h\right) \cdot \left(w0 \cdot M\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 2.12e-38)
   w0
   (* -0.125 (* (/ (* (/ D d) (/ D d)) l) (* (* M h) (* w0 M))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.12e-38) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((((D / d) * (D / d)) / l) * ((M * h) * (w0 * M)));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 2.12d-38) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((((d / d_1) * (d / d_1)) / l) * ((m * h) * (w0 * m)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.12e-38) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((((D / d) * (D / d)) / l) * ((M * h) * (w0 * M)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.12e-38:
		tmp = w0
	else:
		tmp = -0.125 * ((((D / d) * (D / d)) / l) * ((M * h) * (w0 * M)))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.12e-38)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) / l) * Float64(Float64(M * h) * Float64(w0 * M))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.12e-38)
		tmp = w0;
	else
		tmp = -0.125 * ((((D / d) * (D / d)) / l) * ((M * h) * (w0 * M)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 2.12e-38], w0, N[(-0.125 * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] * N[(w0 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.12000000000000001e-38

   1. Initial program 82.5%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in D around 0 70.8%

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

   if 2.12000000000000001e-38 < M

   1. Initial program 66.5%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in D around 0 44.4%

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

      1. +-commutative 44.4%

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

      2. *-commutative 44.4%

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

      3. fma-def 44.4%

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

      4. associate-*r* 45.9%

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

      5. unpow2 45.9%

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

      6. unpow2 45.9%

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

      7. swap-sqr 53.8%

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

      8. associate-/l* 53.6%

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

      9. swap-sqr 45.7%

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

      10. unpow2 45.7%

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

      11. associate-*l* 49.0%

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

   6. Simplified 49.0%

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

   7. Taylor expanded in D around inf 30.7%

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

      1. unpow2 30.7%

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

      2. associate-*r* 32.5%

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

      3. associate-/l* 32.6%

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

      4. unpow2 32.6%

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

      5. associate-*r* 30.8%

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

      6. associate-*r* 30.8%

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

      7. *-commutative 30.8%

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

      8. unpow2 30.8%

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

      9. associate-*r* 31.2%

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

   9. Simplified 31.2%

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

       1. associate-/r/ 31.1%

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

       2. associate-*l* 32.9%

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

       3. associate-*l* 33.1%

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

       4. *-commutative 33.1%

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

   11. Applied egg-rr 33.1%

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

   12. Taylor expanded in D around 0 31.2%

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

       1. unpow2 31.2%

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

       2. associate-/r* 29.9%

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

       3. unpow2 29.9%

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

       4. times-frac 30.8%

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

   14. Simplified 30.8%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.12 \cdot 10^{-38}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\frac{D}{d} \cdot \frac{D}{d}}{\ell} \cdot \left(\left(M \cdot h\right) \cdot \left(w0 \cdot M\right)\right)\right)\\ \end{array} \]

Alternative 7: 68.0% accurate, 216.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d) :precision binary64 w0)

C

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

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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

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

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return w0
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 78.4%

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

3. Simplified 79.9%

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

4. Taylor expanded in D around 0 61.0%

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

5. Final simplification 61.0%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023290 
(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))))))