Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.5% → 85.6%

Time: 16.0s

Alternatives: 5

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 85.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

3. Simplified 85.1%

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

   1. associate-*r/ 89.1%

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

   2. associate-/r* 89.1%

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

6. Applied egg-rr 89.1%

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

   1. unpow2 89.1%

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

8. Applied egg-rr 89.1%

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

9. Taylor expanded in M around 0 89.1%

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

    1. associate-*r/ 89.1%

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

    2. *-commutative 89.1%

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

    3. associate-*r/ 89.1%

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

11. Simplified 89.1%

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

Alternative 2: 73.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 6.5 \cdot 10^{-78}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\frac{D \cdot M}{2 \cdot d} \cdot \frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 6.5e-78)
   w0
   (*
    w0
    (sqrt
     (- 1.0 (* (/ h l) (* (/ (* D M) (* 2.0 d)) (/ (* D (* M 0.5)) d))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 6.5e-78) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D * M) / (2.0 * d)) * ((D * (M * 0.5)) / d)))));
	}
	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 <= 6.5d-78) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d * m) / (2.0d0 * d_1)) * ((d * (m * 0.5d0)) / d_1)))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 6.5e-78) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * (((D * M) / (2.0 * d)) * ((D * (M * 0.5)) / d)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 6.5e-78:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * (((D * M) / (2.0 * d)) * ((D * (M * 0.5)) / d)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 6.5e-78)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(D * M) / Float64(2.0 * d)) * Float64(Float64(D * Float64(M * 0.5)) / d))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 6.5e-78)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D * M) / (2.0 * d)) * ((D * (M * 0.5)) / d)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 6.5e-78], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 6.5000000000000003e-78

   1. Initial program 84.5%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in D around 0 74.2%

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

   if 6.5000000000000003e-78 < D

   1. Initial program 82.2%

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

   3. Simplified 83.7%

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

      1. associate-*r/ 87.7%

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

      2. associate-/r* 87.7%

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

   6. Applied egg-rr 87.7%

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

      1. unpow2 87.7%

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

   8. Applied egg-rr 87.7%

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

   9. Taylor expanded in M around 0 87.7%

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

       1. associate-*r/ 87.7%

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

       2. *-commutative 87.7%

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

       3. associate-*r/ 87.8%

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

   11. Simplified 87.8%

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

       1. *-un-lft-identity 87.8%

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

       2. associate-/l* 83.7%

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

       3. associate-*l* 83.7%

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

       4. *-commutative 83.7%

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

   13. Applied egg-rr 83.7%

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

       1. *-lft-identity 83.7%

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

       2. *-commutative 83.7%

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

       3. associate-*r* 83.7%

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

       4. associate-*r/ 82.2%

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

       5. associate-*r/ 82.3%

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

       6. associate-*r/ 82.2%

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

   15. Simplified 82.2%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 6.5 \cdot 10^{-78}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\frac{D \cdot M}{2 \cdot d} \cdot \frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 4.9 \cdot 10^{+164}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \left(w0 \cdot h\right)}{\ell \cdot \left(d \cdot d\right)}, w0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 4.9e+164)
   w0
   (fma -0.125 (/ (* (* (* D M) (* D M)) (* w0 h)) (* l (* d d))) w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 4.9e+164) {
		tmp = w0;
	} else {
		tmp = fma(-0.125, ((((D * M) * (D * M)) * (w0 * h)) / (l * (d * d))), w0);
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 4.9e+164)
		tmp = w0;
	else
		tmp = fma(-0.125, Float64(Float64(Float64(Float64(D * M) * Float64(D * M)) * Float64(w0 * h)) / Float64(l * Float64(d * d))), w0);
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 4.9e+164], w0, N[(-0.125 * N[(N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * N[(w0 * h), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 4.8999999999999999e164

   1. Initial program 85.7%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in D around 0 74.5%

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

   if 4.8999999999999999e164 < D

   1. Initial program 66.1%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in D around 0 30.7%

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

      1. +-commutative 30.7%

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

      2. fma-define 30.7%

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

      3. associate-*r* 35.2%

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

      4. unpow2 35.2%

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

      5. unpow2 35.2%

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

      6. swap-sqr 59.5%

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

      7. unpow2 59.5%

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

      8. *-commutative 59.5%

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

   7. Simplified 59.5%

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

      1. unpow2 59.5%

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

   9. Applied egg-rr 59.5%

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

       1. unpow2 59.5%

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

   11. Applied egg-rr 59.5%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 4.9 \cdot 10^{+164}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \left(w0 \cdot h\right)}{\ell \cdot \left(d \cdot d\right)}, w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 7 \cdot 10^{+24}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{w0}{d \cdot d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 7e+24)
   w0
   (* -0.125 (* (* (* D M) (* D M)) (* (/ h l) (/ w0 (* d d)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 7e+24) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * M) * (D * M)) * ((h / l) * (w0 / (d * d))));
	}
	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 <= 7d+24) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * m) * (d * m)) * ((h / l) * (w0 / (d_1 * d_1))))
    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 <= 7e+24) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * M) * (D * M)) * ((h / l) * (w0 / (d * d))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 7e+24:
		tmp = w0
	else:
		tmp = -0.125 * (((D * M) * (D * M)) * ((h / l) * (w0 / (d * d))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 7e+24)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * M) * Float64(D * M)) * Float64(Float64(h / l) * Float64(w0 / Float64(d * d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 7e+24)
		tmp = w0;
	else
		tmp = -0.125 * (((D * M) * (D * M)) * ((h / l) * (w0 / (d * d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 7e+24], w0, N[(-0.125 * N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(w0 / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.0000000000000004e24

   1. Initial program 87.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in D around 0 73.2%

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

   if 7.0000000000000004e24 < M

   1. Initial program 63.4%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in D around 0 37.8%

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

      1. +-commutative 37.8%

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

      2. fma-define 37.8%

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

      3. associate-*r* 40.3%

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

      4. unpow2 40.3%

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

      5. unpow2 40.3%

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

      6. swap-sqr 55.3%

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

      7. unpow2 55.3%

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

      8. *-commutative 55.3%

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

   7. Simplified 55.3%

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

   8. Taylor expanded in D around inf 16.0%

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

      1. associate-*r* 16.1%

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

      2. unpow2 16.1%

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

      3. unpow2 16.1%

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

      4. swap-sqr 16.6%

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

      5. unpow2 16.6%

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

      6. associate-*r/ 16.7%

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

      7. *-commutative 16.7%

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

      8. times-frac 19.2%

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

   10. Simplified 19.2%

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

       1. unpow2 55.3%

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

   12. Applied egg-rr 19.2%

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

       1. unpow2 55.3%

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

   14. Applied egg-rr 19.2%

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

Alternative 5: 67.2% 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 83.9%

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

3. Simplified 85.1%

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

5. Taylor expanded in D around 0 70.0%

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

Reproduce

herbie shell --seed 2024154 
(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))))))