Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 89.0%

Time: 14.4s

Alternatives: 8

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: 89.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(D_m \cdot \frac{M_m}{d}\right)\\ w0 \cdot \sqrt{1 - h \cdot \left(t_0 \cdot \frac{t_0}{\ell}\right)} \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* 0.5 (* D_m (/ M_m d)))))
   (* w0 (sqrt (- 1.0 (* h (* t_0 (/ t_0 l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = 0.5 * (D_m * (M_m / d));
	return w0 * sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    t_0 = 0.5d0 * (d_m * (m_m / d))
    code = w0 * sqrt((1.0d0 - (h * (t_0 * (t_0 / l)))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = 0.5 * (D_m * (M_m / d));
	return w0 * Math.sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = 0.5 * (D_m * (M_m / d))
	return w0 * math.sqrt((1.0 - (h * (t_0 * (t_0 / l)))))

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(0.5 * Float64(D_m * Float64(M_m / d)))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(t_0 * Float64(t_0 / l))))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp = code(w0, M_m, D_m, h, l, d)
	t_0 = 0.5 * (D_m * (M_m / d));
	tmp = w0 * sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(0.5 * N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.3%

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

3. Simplified 77.7%

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

   1. expm1-log1p-u 55.5%

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

   2. expm1-udef 55.5%

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

   3. log1p-udef 55.5%

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

   4. add-exp-log 77.7%

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

   5. +-commutative 77.7%

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

   6. associate-*l/ 78.3%

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

   7. div-inv 78.3%

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

   8. associate-*l* 78.2%

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

   9. associate-/r* 78.2%

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

   10. metadata-eval 78.2%

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

5. Applied egg-rr 78.2%

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

   1. associate--l+ 78.2%

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

   2. metadata-eval 78.2%

      \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

   3. +-rgt-identity 78.2%

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

   4. associate-*r/ 85.4%

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

   5. associate-*l/ 86.9%

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

   6. *-commutative 86.9%

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

   7. associate-*r* 86.9%

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

   8. *-commutative 86.9%

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

   9. associate-*l/ 86.9%

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

   10. *-commutative 86.9%

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

   11. associate-*r/ 86.9%

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

   12. associate-*r/ 86.3%

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

7. Simplified 86.3%

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

8. Taylor expanded in D around 0 86.9%

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

   1. unpow2 86.9%

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

   2. *-un-lft-identity 86.9%

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

   3. times-frac 89.1%

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

   4. associate-*r/ 87.8%

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

   5. associate-*r/ 88.2%

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

10. Applied egg-rr 88.2%

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

11. Final simplification 88.2%

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

Alternative 2: 87.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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 * ((0.5d0 * (d_m * (m_m / d))) * ((m_m * (d_m / d)) * (0.5d0 / l))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Simplified 77.7%

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

   1. expm1-log1p-u 55.5%

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

   2. expm1-udef 55.5%

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

   3. log1p-udef 55.5%

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

   4. add-exp-log 77.7%

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

   5. +-commutative 77.7%

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

   6. associate-*l/ 78.3%

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

   7. div-inv 78.3%

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

   8. associate-*l* 78.2%

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

   9. associate-/r* 78.2%

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

   10. metadata-eval 78.2%

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

5. Applied egg-rr 78.2%

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

   1. associate--l+ 78.2%

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

   2. metadata-eval 78.2%

      \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

   3. +-rgt-identity 78.2%

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

   4. associate-*r/ 85.4%

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

   5. associate-*l/ 86.9%

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

   6. *-commutative 86.9%

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

   7. associate-*r* 86.9%

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

   8. *-commutative 86.9%

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

   9. associate-*l/ 86.9%

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

   10. *-commutative 86.9%

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

   11. associate-*r/ 86.9%

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

   12. associate-*r/ 86.3%

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

7. Simplified 86.3%

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

8. Taylor expanded in D around 0 86.9%

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

   1. unpow2 86.9%

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

   2. *-un-lft-identity 86.9%

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

   3. times-frac 89.1%

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

   4. associate-*r/ 87.8%

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

   5. associate-*r/ 88.2%

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

10. Applied egg-rr 88.2%

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

11. Taylor expanded in D around 0 83.5%

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

    1. associate-/r* 87.8%

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

    2. associate-*r/ 88.2%

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

    3. associate-*r/ 88.2%

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

    4. *-commutative 88.2%

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

    5. associate-*r/ 88.2%

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

    6. associate-*r/ 87.8%

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

    7. associate-*l/ 87.4%

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

13. Simplified 87.4%

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

14. Final simplification 87.4%

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

Alternative 3: 82.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D_m \cdot \frac{M_m}{d}\\ w0 \cdot \mathsf{fma}\left(h \cdot \left(t_0 \cdot \frac{t_0}{\ell}\right), -0.125, 1\right) \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d))))
   (* w0 (fma (* h (* t_0 (/ t_0 l))) -0.125 1.0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M_m / d);
	return w0 * fma((h * (t_0 * (t_0 / l))), -0.125, 1.0);
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m * Float64(M_m / d))
	return Float64(w0 * fma(Float64(h * Float64(t_0 * Float64(t_0 / l))), -0.125, 1.0))
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[(N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.3%

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

3. Simplified 77.7%

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

   1. expm1-log1p-u 55.5%

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

   2. expm1-udef 55.5%

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

   3. log1p-udef 55.5%

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

   4. add-exp-log 77.7%

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

   5. +-commutative 77.7%

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

   6. associate-*l/ 78.3%

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

   7. div-inv 78.3%

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

   8. associate-*l* 78.2%

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

   9. associate-/r* 78.2%

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

   10. metadata-eval 78.2%

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

5. Applied egg-rr 78.2%

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

   1. associate--l+ 78.2%

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

   2. metadata-eval 78.2%

      \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

   3. +-rgt-identity 78.2%

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

   4. associate-*r/ 85.4%

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

   5. associate-*l/ 86.9%

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

   6. *-commutative 86.9%

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

   7. associate-*r* 86.9%

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

   8. *-commutative 86.9%

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

   9. associate-*l/ 86.9%

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

   10. *-commutative 86.9%

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

   11. associate-*r/ 86.9%

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

   12. associate-*r/ 86.3%

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

7. Simplified 86.3%

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

8. Taylor expanded in h around 0 54.2%

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

   1. +-commutative 54.2%

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

   2. *-commutative 54.2%

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

   3. fma-def 54.2%

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

10. Simplified 80.1%

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

    1. associate-*r/ 80.9%

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

    2. pow2 80.9%

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

    3. *-un-lft-identity 80.9%

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

    4. times-frac 82.1%

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

    5. associate-*r/ 81.3%

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

    6. associate-*r/ 81.3%

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

12. Applied egg-rr 81.3%

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

13. Final simplification 81.3%

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

Alternative 4: 72.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M_m \leq 1.42:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(h \cdot \left(\frac{w0}{\ell} \cdot {\left(M_m \cdot \frac{D_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 1.42)
   w0
   (* -0.125 (* h (* (/ w0 l) (pow (* M_m (/ D_m d)) 2.0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 1.42) {
		tmp = w0;
	} else {
		tmp = -0.125 * (h * ((w0 / l) * pow((M_m * (D_m / d)), 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m <= 1.42d0) then
        tmp = w0
    else
        tmp = (-0.125d0) * (h * ((w0 / l) * ((m_m * (d_m / d)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 1.42) {
		tmp = w0;
	} else {
		tmp = -0.125 * (h * ((w0 / l) * Math.pow((M_m * (D_m / d)), 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 1.42:
		tmp = w0
	else:
		tmp = -0.125 * (h * ((w0 / l) * math.pow((M_m * (D_m / d)), 2.0)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 1.42)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(h * Float64(Float64(w0 / l) * (Float64(M_m * Float64(D_m / d)) ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 1.42)
		tmp = w0;
	else
		tmp = -0.125 * (h * ((w0 / l) * ((M_m * (D_m / d)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 1.42], w0, N[(-0.125 * N[(h * N[(N[(w0 / l), $MachinePrecision] * N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.4199999999999999

   1. Initial program 80.9%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in M around 0 74.0%

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

   if 1.4199999999999999 < M

   1. Initial program 68.6%

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

   3. Simplified 66.7%

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

      1. expm1-log1p-u 29.1%

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

      2. expm1-udef 29.1%

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

      3. log1p-udef 29.1%

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

      4. add-exp-log 66.7%

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

      5. +-commutative 66.7%

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

      6. associate-*l/ 68.6%

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

      7. div-inv 68.6%

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

      8. associate-*l* 70.3%

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

      9. associate-/r* 70.3%

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

      10. metadata-eval 70.3%

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

   5. Applied egg-rr 70.3%

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

      1. associate--l+ 70.3%

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

      2. metadata-eval 70.3%

         \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

      3. +-rgt-identity 70.3%

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

      4. associate-*r/ 70.5%

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

      5. associate-*l/ 75.7%

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

      6. *-commutative 75.7%

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

      7. associate-*r* 74.0%

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

      8. *-commutative 74.0%

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

      9. associate-*l/ 74.0%

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

      10. *-commutative 74.0%

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

      11. associate-*r/ 74.0%

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

      12. associate-*r/ 72.1%

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

   7. Simplified 72.1%

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

   8. Taylor expanded in h around 0 44.6%

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

      1. +-commutative 44.6%

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

      2. *-commutative 44.6%

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

      3. fma-def 44.6%

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

   10. Simplified 62.0%

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

   11. Taylor expanded in D around inf 25.0%

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

       1. associate-*r* 25.0%

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

       2. times-frac 25.3%

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

       3. unpow2 25.3%

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

       4. unpow2 25.3%

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

       5. unpow2 25.3%

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

       6. swap-sqr 27.5%

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

       7. times-frac 30.0%

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

       8. associate-*r/ 29.9%

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

       9. associate-*r/ 30.0%

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

       10. unpow2 30.0%

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

       11. *-commutative 30.0%

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

       12. associate-*l/ 30.0%

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

       13. associate-*r/ 30.1%

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

       14. associate-*r/ 30.2%

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

   13. Simplified 30.2%

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

       1. expm1-log1p-u 14.9%

          \[\leadsto -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\right)} \]

       2. expm1-udef 14.9%

          \[\leadsto -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)} - 1\right)} \]

       3. *-commutative 14.9%

          \[\leadsto -0.125 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(h \cdot \frac{w0}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}\right)} - 1\right) \]

       4. associate-*r/ 14.8%

          \[\leadsto -0.125 \cdot \left(e^{\mathsf{log1p}\left(\left(h \cdot \frac{w0}{\ell}\right) \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}\right)} - 1\right) \]

       5. *-commutative 14.8%

          \[\leadsto -0.125 \cdot \left(e^{\mathsf{log1p}\left(\left(h \cdot \frac{w0}{\ell}\right) \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}\right)} - 1\right) \]

       6. associate-*r/ 14.7%

          \[\leadsto -0.125 \cdot \left(e^{\mathsf{log1p}\left(\left(h \cdot \frac{w0}{\ell}\right) \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}\right)} - 1\right) \]

   15. Applied egg-rr 14.7%

       \[\leadsto -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(h \cdot \frac{w0}{\ell}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)} - 1\right)} \]
   16. Step-by-step derivation

       1. expm1-def 14.7%

          \[\leadsto -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(h \cdot \frac{w0}{\ell}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 30.0%

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

       3. associate-*l* 30.2%

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

       4. associate-*r/ 30.2%

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

       5. associate-*l/ 30.3%

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

   17. Simplified 30.3%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.42:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(h \cdot \left(\frac{w0}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 5: 72.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M_m \leq 0.83:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{w0 \cdot h}{\ell} \cdot {\left(M_m \cdot \frac{D_m}{d}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 0.83)
   w0
   (* -0.125 (* (/ (* w0 h) l) (pow (* M_m (/ D_m d)) 2.0)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 0.83) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((w0 * h) / l) * pow((M_m * (D_m / d)), 2.0));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m <= 0.83d0) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((w0 * h) / l) * ((m_m * (d_m / d)) ** 2.0d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 0.83) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((w0 * h) / l) * Math.pow((M_m * (D_m / d)), 2.0));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 0.83:
		tmp = w0
	else:
		tmp = -0.125 * (((w0 * h) / l) * math.pow((M_m * (D_m / d)), 2.0))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 0.83)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(w0 * h) / l) * (Float64(M_m * Float64(D_m / d)) ^ 2.0)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 0.83)
		tmp = w0;
	else
		tmp = -0.125 * (((w0 * h) / l) * ((M_m * (D_m / d)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 0.83], w0, N[(-0.125 * N[(N[(N[(w0 * h), $MachinePrecision] / l), $MachinePrecision] * N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 0.82999999999999996

   1. Initial program 80.9%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in M around 0 74.0%

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

   if 0.82999999999999996 < M

   1. Initial program 68.6%

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

   3. Simplified 66.7%

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

      1. expm1-log1p-u 29.1%

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

      2. expm1-udef 29.1%

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

      3. log1p-udef 29.1%

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

      4. add-exp-log 66.7%

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

      5. +-commutative 66.7%

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

      6. associate-*l/ 68.6%

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

      7. div-inv 68.6%

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

      8. associate-*l* 70.3%

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

      9. associate-/r* 70.3%

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

      10. metadata-eval 70.3%

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

   5. Applied egg-rr 70.3%

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

      1. associate--l+ 70.3%

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

      2. metadata-eval 70.3%

         \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

      3. +-rgt-identity 70.3%

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

      4. associate-*r/ 70.5%

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

      5. associate-*l/ 75.7%

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

      6. *-commutative 75.7%

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

      7. associate-*r* 74.0%

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

      8. *-commutative 74.0%

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

      9. associate-*l/ 74.0%

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

      10. *-commutative 74.0%

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

      11. associate-*r/ 74.0%

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

      12. associate-*r/ 72.1%

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

   7. Simplified 72.1%

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

   8. Taylor expanded in h around 0 44.6%

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

      1. +-commutative 44.6%

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

      2. *-commutative 44.6%

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

      3. fma-def 44.6%

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

   10. Simplified 62.0%

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

   11. Taylor expanded in D around inf 25.0%

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

       1. associate-*r* 25.0%

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

       2. times-frac 25.3%

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

       3. unpow2 25.3%

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

       4. unpow2 25.3%

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

       5. unpow2 25.3%

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

       6. swap-sqr 27.5%

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

       7. times-frac 30.0%

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

       8. associate-*r/ 29.9%

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

       9. associate-*r/ 30.0%

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

       10. unpow2 30.0%

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

       11. *-commutative 30.0%

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

       12. associate-*l/ 30.0%

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

       13. associate-*r/ 30.1%

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

       14. associate-*r/ 30.2%

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

   13. Simplified 30.2%

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

   14. Taylor expanded in h around 0 30.1%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 0.83:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{w0 \cdot h}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\\ \end{array} \]

Alternative 6: 81.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0 + (w0 * ((-0.125d0) * (h * (((d_m * (m_m / d)) ** 2.0d0) / l))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0 + (w0 * (-0.125 * (h * (Math.pow((D_m * (M_m / d)), 2.0) / l))));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	return w0 + (w0 * (-0.125 * (h * (math.pow((D_m * (M_m / d)), 2.0) / l))))

Julia

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

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp = code(w0, M_m, D_m, h, l, d)
	tmp = w0 + (w0 * (-0.125 * (h * (((D_m * (M_m / d)) ^ 2.0) / l))));
end

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Simplified 77.7%

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

   1. expm1-log1p-u 55.5%

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

   2. expm1-udef 55.5%

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

   3. log1p-udef 55.5%

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

   4. add-exp-log 77.7%

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

   5. +-commutative 77.7%

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

   6. associate-*l/ 78.3%

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

   7. div-inv 78.3%

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

   8. associate-*l* 78.2%

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

   9. associate-/r* 78.2%

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

   10. metadata-eval 78.2%

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

5. Applied egg-rr 78.2%

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

   1. associate--l+ 78.2%

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

   2. metadata-eval 78.2%

      \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

   3. +-rgt-identity 78.2%

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

   4. associate-*r/ 85.4%

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

   5. associate-*l/ 86.9%

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

   6. *-commutative 86.9%

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

   7. associate-*r* 86.9%

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

   8. *-commutative 86.9%

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

   9. associate-*l/ 86.9%

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

   10. *-commutative 86.9%

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

   11. associate-*r/ 86.9%

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

   12. associate-*r/ 86.3%

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

7. Simplified 86.3%

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

8. Taylor expanded in h around 0 54.2%

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

   1. +-commutative 54.2%

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

   2. *-commutative 54.2%

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

   3. fma-def 54.2%

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

10. Simplified 80.1%

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

    1. fma-udef 80.1%

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

    2. distribute-rgt-in 80.1%

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

    3. *-commutative 80.1%

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

    4. *-un-lft-identity 80.1%

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

12. Applied egg-rr 80.1%

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

13. Final simplification 80.1%

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

Alternative 7: 72.6% accurate, 10.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D_m \cdot \frac{M_m}{d}\\ \mathbf{if}\;M_m \leq 1.55:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(t_0 \cdot t_0\right) \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d))))
   (if (<= M_m 1.55) w0 (* -0.125 (* (* t_0 t_0) (* h (/ w0 l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M_m / d);
	double tmp;
	if (M_m <= 1.55) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((t_0 * t_0) * (h * (w0 / l)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_m * (m_m / d)
    if (m_m <= 1.55d0) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((t_0 * t_0) * (h * (w0 / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M_m / d);
	double tmp;
	if (M_m <= 1.55) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((t_0 * t_0) * (h * (w0 / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = D_m * (M_m / d)
	tmp = 0
	if M_m <= 1.55:
		tmp = w0
	else:
		tmp = -0.125 * ((t_0 * t_0) * (h * (w0 / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m * Float64(M_m / d))
	tmp = 0.0
	if (M_m <= 1.55)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(t_0 * t_0) * Float64(h * Float64(w0 / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = D_m * (M_m / d);
	tmp = 0.0;
	if (M_m <= 1.55)
		tmp = w0;
	else
		tmp = -0.125 * ((t_0 * t_0) * (h * (w0 / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 1.55], w0, N[(-0.125 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 1.55000000000000004

   1. Initial program 80.9%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in M around 0 74.0%

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

   if 1.55000000000000004 < M

   1. Initial program 68.6%

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

   3. Simplified 66.7%

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

      1. expm1-log1p-u 29.1%

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

      2. expm1-udef 29.1%

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

      3. log1p-udef 29.1%

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

      4. add-exp-log 66.7%

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

      5. +-commutative 66.7%

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

      6. associate-*l/ 68.6%

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

      7. div-inv 68.6%

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

      8. associate-*l* 70.3%

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

      9. associate-/r* 70.3%

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

      10. metadata-eval 70.3%

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

   5. Applied egg-rr 70.3%

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

      1. associate--l+ 70.3%

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

      2. metadata-eval 70.3%

         \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

      3. +-rgt-identity 70.3%

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

      4. associate-*r/ 70.5%

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

      5. associate-*l/ 75.7%

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

      6. *-commutative 75.7%

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

      7. associate-*r* 74.0%

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

      8. *-commutative 74.0%

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

      9. associate-*l/ 74.0%

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

      10. *-commutative 74.0%

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

      11. associate-*r/ 74.0%

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

      12. associate-*r/ 72.1%

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

   7. Simplified 72.1%

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

   8. Taylor expanded in h around 0 44.6%

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

      1. +-commutative 44.6%

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

      2. *-commutative 44.6%

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

      3. fma-def 44.6%

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

   10. Simplified 62.0%

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

   11. Taylor expanded in D around inf 25.0%

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

       1. associate-*r* 25.0%

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

       2. times-frac 25.3%

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

       3. unpow2 25.3%

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

       4. unpow2 25.3%

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

       5. unpow2 25.3%

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

       6. swap-sqr 27.5%

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

       7. times-frac 30.0%

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

       8. associate-*r/ 29.9%

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

       9. associate-*r/ 30.0%

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

       10. unpow2 30.0%

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

       11. *-commutative 30.0%

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

       12. associate-*l/ 30.0%

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

       13. associate-*r/ 30.1%

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

       14. associate-*r/ 30.2%

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

   13. Simplified 30.2%

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

       1. associate-*r/ 30.1%

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

       2. *-commutative 30.1%

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

       3. associate-*r/ 30.0%

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

       4. unpow2 30.0%

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

   15. Applied egg-rr 30.0%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.55:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \]

Alternative 8: 68.1% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Simplified 77.7%

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

4. Taylor expanded in M around 0 66.5%

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

5. Final simplification 66.5%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023314 
(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))))))