Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 88.2%

Time: 16.4s

Alternatives: 3

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: 81.4% 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: 88.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. associate-*r/ 86.4%

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

   2. pow1 86.4%

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

   3. pow1 86.4%

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

   4. times-frac 86.0%

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

   5. div-inv 86.0%

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

   6. metadata-eval 86.0%

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

3. Applied egg-rr 86.0%

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

   1. associate-/l* 82.3%

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

   2. unpow2 82.3%

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

   3. *-un-lft-identity 82.3%

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

   4. times-frac 84.7%

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

   5. associate-*l* 84.7%

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

   6. associate-*l* 84.7%

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

5. Applied egg-rr 84.7%

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

   1. associate-*r* 84.7%

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

   2. associate-*r/ 84.3%

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

7. Applied egg-rr 84.3%

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

   1. expm1-log1p-u 59.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{1} \cdot \frac{\frac{\left(M \cdot 0.5\right) \cdot D}{d}}{\frac{\ell}{h}}}\right)\right)} \]

   2. expm1-udef 27.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{1} \cdot \frac{\frac{\left(M \cdot 0.5\right) \cdot D}{d}}{\frac{\ell}{h}}}\right)} - 1} \]

   3. /-rgt-identity 27.8%

      \[\leadsto e^{\mathsf{log1p}\left(w0 \cdot \sqrt{1 - \color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)} \cdot \frac{\frac{\left(M \cdot 0.5\right) \cdot D}{d}}{\frac{\ell}{h}}}\right)} - 1 \]

   4. associate-/l* 28.1%

      \[\leadsto e^{\mathsf{log1p}\left(w0 \cdot \sqrt{1 - \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{\frac{M \cdot 0.5}{\frac{d}{D}}}}{\frac{\ell}{h}}}\right)} - 1 \]

9. Applied egg-rr 28.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(w0 \cdot \sqrt{1 - \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \frac{\frac{M \cdot 0.5}{\frac{d}{D}}}{\frac{\ell}{h}}}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 59.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0 \cdot \sqrt{1 - \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \frac{\frac{M \cdot 0.5}{\frac{d}{D}}}{\frac{\ell}{h}}}\right)\right)} \]

    2. expm1-log1p 84.7%

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

    3. associate-*l* 84.7%

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

    4. associate-/r/ 89.7%

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

    5. *-commutative 89.7%

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

    6. associate-/l* 89.7%

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

11. Simplified 89.7%

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

12. Final simplification 89.7%

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

Alternative 2: 70.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 3.7 \cdot 10^{+209}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(w0 \cdot h\right)}{\ell \cdot {d}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* M D) 3.7e+209)
   w0
   (* -0.125 (/ (* (* (* M D) (* M D)) (* w0 h)) (* l (pow d 2.0))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 3.7e+209) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((((M * D) * (M * D)) * (w0 * h)) / (l * pow(d, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((m * d) <= 3.7d+209) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((((m * d) * (m * d)) * (w0 * h)) / (l * (d_1 ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 3.7e+209) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((((M * D) * (M * D)) * (w0 * h)) / (l * Math.pow(d, 2.0)));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (M * D) <= 3.7e+209:
		tmp = w0
	else:
		tmp = -0.125 * ((((M * D) * (M * D)) * (w0 * h)) / (l * math.pow(d, 2.0)))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= 3.7e+209)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(w0 * h)) / Float64(l * (d ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((M * D) <= 3.7e+209)
		tmp = w0;
	else
		tmp = -0.125 * ((((M * D) * (M * D)) * (w0 * h)) / (l * (d ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(M * D), $MachinePrecision], 3.7e+209], w0, N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(w0 * h), $MachinePrecision]), $MachinePrecision] / N[(l * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 3.7e209

   1. Initial program 82.9%

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

   2. Taylor expanded in M around 0 73.7%

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

   if 3.7e209 < (*.f64 M D)

   1. Initial program 62.8%

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

   2. Taylor expanded in M around 0 44.7%

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

      1. +-commutative 44.7%

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

      2. *-commutative 44.7%

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

      3. fma-def 44.7%

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

      4. times-frac 44.7%

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

      5. associate-/l* 39.9%

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

   4. Simplified 39.9%

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

   5. Taylor expanded in D around inf 43.9%

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

      1. associate-*r* 43.9%

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

      2. unpow2 43.9%

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

      3. unpow2 43.9%

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

      4. swap-sqr 43.9%

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

      5. unpow2 43.9%

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

   7. Simplified 43.9%

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

      1. unpow2 43.9%

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

      2. *-commutative 43.9%

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

      3. *-commutative 43.9%

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

   9. Applied egg-rr 43.9%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 3.7 \cdot 10^{+209}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(w0 \cdot h\right)}{\ell \cdot {d}^{2}}\\ \end{array} \]

Alternative 3: 68.3% 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 81.2%

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

2. Taylor expanded in M around 0 67.7%

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

3. Final simplification 67.7%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023335 
(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))))))