Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 86.6%

Time: 17.5s

Alternatives: 5

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.1% 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: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D\_m \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* h (/ (pow (* D_m (* M (/ 0.5 d))) 2.0) l))))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	return w0 * sqrt((1.0 - (h * (pow((D_m * (M * (0.5 / d))), 2.0) / l))));
}

Fortran

D_m = abs(d)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0 * sqrt((1.0d0 - (h * (((d_m * (m * (0.5d0 / d))) ** 2.0d0) / l))))
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (h * (Math.pow((D_m * (M * (0.5 / d))), 2.0) / l))));
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	return w0 * math.sqrt((1.0 - (h * (math.pow((D_m * (M * (0.5 / d))), 2.0) / l))))

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(D_m * Float64(M * Float64(0.5 / d))) ^ 2.0) / l)))))
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp = code(w0, M, D_m, h, l, d)
	tmp = w0 * sqrt((1.0 - (h * (((D_m * (M * (0.5 / d))) ^ 2.0) / l))));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(D$95$m * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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 84.1%

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

   1. clear-num 84.0%

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

   2. un-div-inv 84.0%

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

   3. *-commutative 84.0%

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

   4. associate-*l/ 84.4%

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

   5. associate-*r/ 83.5%

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

   6. div-inv 83.5%

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

   7. metadata-eval 83.5%

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

6. Applied egg-rr 83.5%

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

   1. associate-/r/ 86.5%

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

   2. associate-*l/ 86.5%

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

   3. *-commutative 86.5%

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

   4. associate-/l* 86.5%

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

   5. associate-*r/ 87.4%

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

   6. *-commutative 87.4%

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

   7. associate-/l* 87.0%

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

   8. associate-*r/ 87.0%

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

8. Simplified 87.0%

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

Alternative 2: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \mathsf{fma}\left(-0.125, \frac{h \cdot \left(w0 \cdot {\left(D\_m \cdot \frac{M}{d}\right)}^{2}\right)}{\ell}, w0\right) \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (fma -0.125 (/ (* h (* w0 (pow (* D_m (/ M d)) 2.0))) l) w0))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	return fma(-0.125, ((h * (w0 * pow((D_m * (M / d)), 2.0))) / l), w0);
}

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	return fma(-0.125, Float64(Float64(h * Float64(w0 * (Float64(D_m * Float64(M / d)) ^ 2.0))) / l), w0)
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := N[(-0.125 * N[(N[(h * N[(w0 * N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + w0), $MachinePrecision]

Derivation

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 84.1%

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

5. Taylor expanded in D around 0 57.6%

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

   1. +-commutative 57.6%

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

   2. fma-define 57.6%

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

   3. associate-*r* 57.5%

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

   4. unpow2 57.5%

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

   5. unpow2 57.5%

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

   6. swap-sqr 64.3%

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

   7. unpow2 64.3%

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

7. Simplified 64.3%

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

   1. times-frac 65.3%

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

9. Applied egg-rr 65.3%

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

    1. associate-*r/ 68.4%

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

    2. div-inv 67.7%

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

    3. pow-flip 67.7%

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

    4. metadata-eval 67.7%

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

11. Applied egg-rr 67.7%

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

    1. pow1 67.7%

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

    2. *-commutative 67.7%

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

    3. metadata-eval 67.7%

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

    4. pow-flip 67.7%

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

    5. div-inv 68.4%

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

    6. add-sqr-sqrt 68.4%

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

    7. pow2 68.4%

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

    8. sqrt-div 68.4%

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

    9. sqrt-pow1 70.9%

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

    10. metadata-eval 70.9%

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

    11. pow1 70.9%

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

    12. sqrt-pow1 78.3%

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

    13. metadata-eval 78.3%

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

    14. pow1 78.3%

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

13. Applied egg-rr 78.3%

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

    1. unpow1 78.3%

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

    2. associate-*l* 81.9%

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

    3. associate-/l* 81.5%

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

15. Simplified 81.5%

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

Alternative 3: 71.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.3 \cdot 10^{-194}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.5 \cdot \left(\frac{1}{\ell \cdot 4} \cdot \left(\left(w0 \cdot h\right) \cdot {\left(\frac{D\_m \cdot M}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= M 1.3e-194)
   w0
   (+
    w0
    (* -0.5 (* (/ 1.0 (* l 4.0)) (* (* w0 h) (pow (/ (* D_m M) d) 2.0)))))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (M <= 1.3e-194) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.5 * ((1.0 / (l * 4.0)) * ((w0 * h) * pow(((D_m * M) / d), 2.0))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m <= 1.3d-194) then
        tmp = w0
    else
        tmp = w0 + ((-0.5d0) * ((1.0d0 / (l * 4.0d0)) * ((w0 * h) * (((d_m * m) / d) ** 2.0d0))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (M <= 1.3e-194) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.5 * ((1.0 / (l * 4.0)) * ((w0 * h) * Math.pow(((D_m * M) / d), 2.0))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	tmp = 0
	if M <= 1.3e-194:
		tmp = w0
	else:
		tmp = w0 + (-0.5 * ((1.0 / (l * 4.0)) * ((w0 * h) * math.pow(((D_m * M) / d), 2.0))))
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (M <= 1.3e-194)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.5 * Float64(Float64(1.0 / Float64(l * 4.0)) * Float64(Float64(w0 * h) * (Float64(Float64(D_m * M) / d) ^ 2.0)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	tmp = 0.0;
	if (M <= 1.3e-194)
		tmp = w0;
	else
		tmp = w0 + (-0.5 * ((1.0 / (l * 4.0)) * ((w0 * h) * (((D_m * M) / d) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[M, 1.3e-194], w0, N[(w0 + N[(-0.5 * N[(N[(1.0 / N[(l * 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * h), $MachinePrecision] * N[Power[N[(N[(D$95$m * M), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.30000000000000001e-194

   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 85.7%

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

   5. Taylor expanded in D around 0 72.3%

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

   if 1.30000000000000001e-194 < M

   1. Initial program 82.3%

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

      1. add-sqr-sqrt 39.1%

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

      2. times-frac 40.2%

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

   4. Applied egg-rr 40.2%

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

   5. Taylor expanded in M around 0 58.0%

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

      1. associate-*r* 56.9%

         \[\leadsto w0 + -0.5 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

      2. unpow2 56.9%

         \[\leadsto w0 + -0.5 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

      3. unpow2 56.9%

         \[\leadsto w0 + -0.5 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

      4. swap-sqr 57.0%

         \[\leadsto w0 + -0.5 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

      5. unpow2 57.0%

         \[\leadsto w0 + -0.5 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

      6. *-commutative 57.0%

         \[\leadsto w0 + -0.5 \cdot \frac{\color{blue}{\left(h \cdot w0\right) \cdot {\left(D \cdot M\right)}^{2}}}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

   7. Simplified 57.0%

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

      1. unpow2 57.0%

         \[\leadsto w0 + -0.5 \cdot \frac{\left(h \cdot w0\right) \cdot \color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

   9. Applied egg-rr 57.0%

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

       1. pow2 57.0%

          \[\leadsto w0 + -0.5 \cdot \frac{\left(h \cdot w0\right) \cdot \color{blue}{{\left(D \cdot M\right)}^{2}}}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

       2. *-commutative 57.0%

          \[\leadsto w0 + -0.5 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

       3. rem-exp-log 38.1%

          \[\leadsto w0 + -0.5 \cdot \frac{\color{blue}{e^{\log \left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)\right)}}}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

       4. *-un-lft-identity 38.1%

          \[\leadsto w0 + -0.5 \cdot \frac{\color{blue}{1 \cdot e^{\log \left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)\right)}}}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

       5. *-commutative 38.1%

          \[\leadsto w0 + -0.5 \cdot \frac{1 \cdot e^{\log \left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)\right)}}{\color{blue}{\left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right) \cdot {d}^{2}}} \]

       6. times-frac 39.3%

          \[\leadsto w0 + -0.5 \cdot \color{blue}{\left(\frac{1}{\ell \cdot {\left(\sqrt{2}\right)}^{4}} \cdot \frac{e^{\log \left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)\right)}}{{d}^{2}}\right)} \]

       7. sqrt-pow2 39.3%

          \[\leadsto w0 + -0.5 \cdot \left(\frac{1}{\ell \cdot \color{blue}{{2}^{\left(\frac{4}{2}\right)}}} \cdot \frac{e^{\log \left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)\right)}}{{d}^{2}}\right) \]

       8. metadata-eval 39.3%

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

       9. metadata-eval 39.3%

          \[\leadsto w0 + -0.5 \cdot \left(\frac{1}{\ell \cdot \color{blue}{4}} \cdot \frac{e^{\log \left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)\right)}}{{d}^{2}}\right) \]

       10. rem-exp-log 59.6%

           \[\leadsto w0 + -0.5 \cdot \left(\frac{1}{\ell \cdot 4} \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}}{{d}^{2}}\right) \]

       11. *-commutative 59.6%

           \[\leadsto w0 + -0.5 \cdot \left(\frac{1}{\ell \cdot 4} \cdot \frac{\color{blue}{\left(h \cdot w0\right) \cdot {\left(D \cdot M\right)}^{2}}}{{d}^{2}}\right) \]

       12. *-un-lft-identity 59.6%

           \[\leadsto w0 + -0.5 \cdot \left(\frac{1}{\ell \cdot 4} \cdot \frac{\left(h \cdot w0\right) \cdot {\left(D \cdot M\right)}^{2}}{\color{blue}{1 \cdot {d}^{2}}}\right) \]

       13. times-frac 62.7%

           \[\leadsto w0 + -0.5 \cdot \left(\frac{1}{\ell \cdot 4} \cdot \color{blue}{\left(\frac{h \cdot w0}{1} \cdot \frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}}\right)}\right) \]

       14. add-sqr-sqrt 62.7%

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

       15. pow2 62.7%

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

   11. Applied egg-rr 74.8%

       \[\leadsto w0 + -0.5 \cdot \color{blue}{\left(\frac{1}{\ell \cdot 4} \cdot \left(\frac{h \cdot w0}{1} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.3 \cdot 10^{-194}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.5 \cdot \left(\frac{1}{\ell \cdot 4} \cdot \left(\left(w0 \cdot h\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.7 \cdot 10^{-183}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.5 \cdot \left({\left(D\_m \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot \frac{w0}{\ell \cdot 4}\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= M 3.7e-183)
   w0
   (+ w0 (* -0.5 (* (pow (* D_m (/ M d)) 2.0) (* h (/ w0 (* l 4.0))))))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (M <= 3.7e-183) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.5 * (pow((D_m * (M / d)), 2.0) * (h * (w0 / (l * 4.0)))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m <= 3.7d-183) then
        tmp = w0
    else
        tmp = w0 + ((-0.5d0) * (((d_m * (m / d)) ** 2.0d0) * (h * (w0 / (l * 4.0d0)))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (M <= 3.7e-183) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.5 * (Math.pow((D_m * (M / d)), 2.0) * (h * (w0 / (l * 4.0)))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	tmp = 0
	if M <= 3.7e-183:
		tmp = w0
	else:
		tmp = w0 + (-0.5 * (math.pow((D_m * (M / d)), 2.0) * (h * (w0 / (l * 4.0)))))
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (M <= 3.7e-183)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.5 * Float64((Float64(D_m * Float64(M / d)) ^ 2.0) * Float64(h * Float64(w0 / Float64(l * 4.0))))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	tmp = 0.0;
	if (M <= 3.7e-183)
		tmp = w0;
	else
		tmp = w0 + (-0.5 * (((D_m * (M / d)) ^ 2.0) * (h * (w0 / (l * 4.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[M, 3.7e-183], w0, N[(w0 + N[(-0.5 * N[(N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * N[(w0 / N[(l * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.6999999999999999e-183

   1. Initial program 85.6%

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

   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 73.3%

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

   if 3.6999999999999999e-183 < M

   1. Initial program 82.2%

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

      1. add-sqr-sqrt 38.4%

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

      2. times-frac 39.5%

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

   4. Applied egg-rr 39.5%

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

   5. Taylor expanded in M around 0 56.2%

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

      1. associate-*r* 55.0%

         \[\leadsto w0 + -0.5 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

      2. unpow2 55.0%

         \[\leadsto w0 + -0.5 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

      3. unpow2 55.0%

         \[\leadsto w0 + -0.5 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

      4. swap-sqr 55.2%

         \[\leadsto w0 + -0.5 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

      5. unpow2 55.2%

         \[\leadsto w0 + -0.5 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

      6. *-commutative 55.2%

         \[\leadsto w0 + -0.5 \cdot \frac{\color{blue}{\left(h \cdot w0\right) \cdot {\left(D \cdot M\right)}^{2}}}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

   7. Simplified 55.2%

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

      1. unpow2 55.2%

         \[\leadsto w0 + -0.5 \cdot \frac{\left(h \cdot w0\right) \cdot \color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}}{{d}^{2} \cdot \left(\ell \cdot {\left(\sqrt{2}\right)}^{4}\right)} \]

   9. Applied egg-rr 55.2%

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

       1. *-un-lft-identity 55.2%

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

       2. pow2 55.2%

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

       3. *-commutative 55.2%

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

       4. times-frac 57.8%

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

       5. add-sqr-sqrt 57.8%

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

       6. pow2 57.8%

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

       7. sqrt-div 57.8%

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

       8. sqrt-pow1 61.5%

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

       9. metadata-eval 61.5%

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

       10. pow1 61.5%

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

       11. sqrt-pow1 70.5%

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

       12. metadata-eval 70.5%

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

       13. pow1 70.5%

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

       14. sqrt-pow2 70.5%

           \[\leadsto w0 + -0.5 \cdot \left(1 \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell \cdot \color{blue}{{2}^{\left(\frac{4}{2}\right)}}}\right)\right) \]

       15. metadata-eval 70.5%

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

       16. metadata-eval 70.5%

           \[\leadsto w0 + -0.5 \cdot \left(1 \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell \cdot \color{blue}{4}}\right)\right) \]

   11. Applied egg-rr 70.5%

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

       1. *-lft-identity 70.5%

          \[\leadsto w0 + -0.5 \cdot \color{blue}{\left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell \cdot 4}\right)} \]

       2. associate-/l* 70.4%

          \[\leadsto w0 + -0.5 \cdot \left({\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{h \cdot w0}{\ell \cdot 4}\right) \]

       3. associate-/l* 71.5%

          \[\leadsto w0 + -0.5 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{w0}{\ell \cdot 4}\right)}\right) \]

   13. Simplified 71.5%

       \[\leadsto w0 + -0.5 \cdot \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot \frac{w0}{\ell \cdot 4}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 67.6% accurate, 216.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ w0 \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d) :precision binary64 w0)

C

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

Fortran

D_m = abs(d)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	return w0;
}

Python

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

Julia

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

MATLAB

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

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := w0

Derivation

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 84.1%

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

5. Taylor expanded in D around 0 68.5%

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

Reproduce

herbie shell --seed 2024108 
(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))))))