Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 87.2%

Time: 15.8s

Alternatives: 13

Speedup: 2.1×

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: 87.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;d\_m \cdot 2 \leq 10^{-120}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}, \frac{M\_m \cdot D\_m}{d\_m \cdot -2} \cdot \frac{h}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D\_m}{d\_m}, \frac{D\_m}{d\_m} \cdot \left(\left(M\_m \cdot -0.25\right) \cdot \frac{M\_m \cdot h}{\ell}\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* d_m 2.0) 1e-120)
   (*
    w0
    (sqrt
     (fma
      (/ (* M_m D_m) (* d_m 2.0))
      (* (/ (* M_m D_m) (* d_m -2.0)) (/ h l))
      1.0)))
   (*
    w0
    (sqrt
     (fma
      (/ D_m d_m)
      (* (/ D_m d_m) (* (* M_m -0.25) (/ (* M_m h) l)))
      1.0)))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((d_m * 2.0) <= 1e-120) {
		tmp = w0 * sqrt(fma(((M_m * D_m) / (d_m * 2.0)), (((M_m * D_m) / (d_m * -2.0)) * (h / l)), 1.0));
	} else {
		tmp = w0 * sqrt(fma((D_m / d_m), ((D_m / d_m) * ((M_m * -0.25) * ((M_m * h) / l))), 1.0));
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(d_m * 2.0) <= 1e-120)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M_m * D_m) / Float64(d_m * 2.0)), Float64(Float64(Float64(M_m * D_m) / Float64(d_m * -2.0)) * Float64(h / l)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(D_m / d_m), Float64(Float64(D_m / d_m) * Float64(Float64(M_m * -0.25) * Float64(Float64(M_m * h) / l))), 1.0)));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(d$95$m * 2.0), $MachinePrecision], 1e-120], N[(w0 * N[Sqrt[N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(N[(M$95$m * -0.25), $MachinePrecision] * N[(N[(M$95$m * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) d) < 9.99999999999999979e-121

   1. Initial program 77.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 80.9%

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

   if 9.99999999999999979e-121 < (*.f64 #s(literal 2 binary64) d)

   1. Initial program 83.5%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left({D}^{2}, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right), 1\right)}} \]

   5. Applied rewrites 62.4%

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

   7. Applied rewrites 64.5%

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

   9. Applied rewrites 81.5%

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

   11. Applied rewrites 86.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot 2 \leq 10^{-120}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot -2} \cdot \frac{h}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D}{d}, \frac{D}{d} \cdot \left(\left(M \cdot -0.25\right) \cdot \frac{M \cdot h}{\ell}\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}\right)}^{2} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D\_m \cdot \left(M\_m \cdot \frac{M\_m \cdot D\_m}{d\_m \cdot \left(\ell \cdot \left(d\_m \cdot -4\right)\right)}\right), h, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* M_m D_m) (* d_m 2.0)) 2.0)) -1e-12)
   (*
    w0
    (sqrt
     (fma (* D_m (* M_m (/ (* M_m D_m) (* d_m (* l (* d_m -4.0)))))) h 1.0)))
   (* w0 1.0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((M_m * D_m) / (d_m * 2.0)), 2.0)) <= -1e-12) {
		tmp = w0 * sqrt(fma((D_m * (M_m * ((M_m * D_m) / (d_m * (l * (d_m * -4.0)))))), h, 1.0));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(d_m * 2.0)) ^ 2.0)) <= -1e-12)
		tmp = Float64(w0 * sqrt(fma(Float64(D_m * Float64(M_m * Float64(Float64(M_m * D_m) / Float64(d_m * Float64(l * Float64(d_m * -4.0)))))), h, 1.0)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e-12], N[(w0 * N[Sqrt[N[(N[(D$95$m * N[(M$95$m * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * N[(l * N[(d$95$m * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.9999999999999998e-13

   1. Initial program 59.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 66.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.0

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

   6. Applied rewrites 46.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 50.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 50.2

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. associate-*l* N/A

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

   8. Applied rewrites 50.2%

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{D \cdot \left(M \cdot \frac{D \cdot M}{d \cdot \left(\ell \cdot \left(-4 \cdot d\right)\right)}\right)}, h, 1\right)} \cdot w0 \]

   if -9.9999999999999998e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.0%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.2%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D \cdot \left(M \cdot \frac{M \cdot D}{d \cdot \left(\ell \cdot \left(d \cdot -4\right)\right)}\right), h, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}\right)}^{2} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D\_m \cdot \left(-0.25 \cdot \left(M\_m \cdot \left(M\_m \cdot h\right)\right)\right)}{d\_m \cdot \left(d\_m \cdot \ell\right)}, D\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* M_m D_m) (* d_m 2.0)) 2.0)) -1e-12)
   (*
    w0
    (sqrt
     (fma (/ (* D_m (* -0.25 (* M_m (* M_m h)))) (* d_m (* d_m l))) D_m 1.0)))
   (* w0 1.0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((M_m * D_m) / (d_m * 2.0)), 2.0)) <= -1e-12) {
		tmp = w0 * sqrt(fma(((D_m * (-0.25 * (M_m * (M_m * h)))) / (d_m * (d_m * l))), D_m, 1.0));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(d_m * 2.0)) ^ 2.0)) <= -1e-12)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(D_m * Float64(-0.25 * Float64(M_m * Float64(M_m * h)))) / Float64(d_m * Float64(d_m * l))), D_m, 1.0)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e-12], N[(w0 * N[Sqrt[N[(N[(N[(D$95$m * N[(-0.25 * N[(M$95$m * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * D$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.9999999999999998e-13

   1. Initial program 59.4%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left({D}^{2}, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right), 1\right)}} \]

   5. Applied rewrites 39.5%

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

   7. Applied rewrites 39.6%

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

   9. Applied rewrites 46.8%

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

   if -9.9999999999999998e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.0%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.2%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D \cdot \left(-0.25 \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, D, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}\right)}^{2} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D\_m \cdot D\_m, \left(-0.25 \cdot \left(M\_m \cdot M\_m\right)\right) \cdot \frac{h}{d\_m \cdot \left(d\_m \cdot \ell\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* M_m D_m) (* d_m 2.0)) 2.0)) -1e-12)
   (*
    w0
    (sqrt
     (fma (* D_m D_m) (* (* -0.25 (* M_m M_m)) (/ h (* d_m (* d_m l)))) 1.0)))
   (* w0 1.0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((M_m * D_m) / (d_m * 2.0)), 2.0)) <= -1e-12) {
		tmp = w0 * sqrt(fma((D_m * D_m), ((-0.25 * (M_m * M_m)) * (h / (d_m * (d_m * l)))), 1.0));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(d_m * 2.0)) ^ 2.0)) <= -1e-12)
		tmp = Float64(w0 * sqrt(fma(Float64(D_m * D_m), Float64(Float64(-0.25 * Float64(M_m * M_m)) * Float64(h / Float64(d_m * Float64(d_m * l)))), 1.0)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e-12], N[(w0 * N[Sqrt[N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(-0.25 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.9999999999999998e-13

   1. Initial program 59.4%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left({D}^{2}, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right), 1\right)}} \]

   5. Applied rewrites 39.5%

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

   7. Applied rewrites 41.2%

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

   if -9.9999999999999998e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.0%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.2%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D \cdot D, \left(-0.25 \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot \left(d \cdot \ell\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}\right)}^{2} \leq -1 \cdot 10^{+243}:\\ \;\;\;\;\frac{\left(M\_m \cdot \left(M\_m \cdot \left(w0 \cdot h\right)\right)\right) \cdot \left(D\_m \cdot -0.125\right)}{d\_m} \cdot \frac{D\_m}{d\_m \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* M_m D_m) (* d_m 2.0)) 2.0)) -1e+243)
   (* (/ (* (* M_m (* M_m (* w0 h))) (* D_m -0.125)) d_m) (/ D_m (* d_m l)))
   (* w0 1.0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((M_m * D_m) / (d_m * 2.0)), 2.0)) <= -1e+243) {
		tmp = (((M_m * (M_m * (w0 * h))) * (D_m * -0.125)) / d_m) * (D_m / (d_m * l));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: tmp
    if (((h / l) * (((m_m * d_m) / (d_m_1 * 2.0d0)) ** 2.0d0)) <= (-1d+243)) then
        tmp = (((m_m * (m_m * (w0 * h))) * (d_m * (-0.125d0))) / d_m_1) * (d_m / (d_m_1 * l))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * Math.pow(((M_m * D_m) / (d_m * 2.0)), 2.0)) <= -1e+243) {
		tmp = (((M_m * (M_m * (w0 * h))) * (D_m * -0.125)) / d_m) * (D_m / (d_m * l));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((h / l) * math.pow(((M_m * D_m) / (d_m * 2.0)), 2.0)) <= -1e+243:
		tmp = (((M_m * (M_m * (w0 * h))) * (D_m * -0.125)) / d_m) * (D_m / (d_m * l))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(d_m * 2.0)) ^ 2.0)) <= -1e+243)
		tmp = Float64(Float64(Float64(Float64(M_m * Float64(M_m * Float64(w0 * h))) * Float64(D_m * -0.125)) / d_m) * Float64(D_m / Float64(d_m * l)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((h / l) * (((M_m * D_m) / (d_m * 2.0)) ^ 2.0)) <= -1e+243)
		tmp = (((M_m * (M_m * (w0 * h))) * (D_m * -0.125)) / d_m) * (D_m / (d_m * l));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e+243], N[(N[(N[(N[(M$95$m * N[(M$95$m * N[(w0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * -0.125), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(D$95$m / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.0000000000000001e243

   1. Initial program 52.2%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right)} \]

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in D around inf

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

   8. Applied rewrites 41.5%

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

   10. Applied rewrites 47.2%

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

   if -1.0000000000000001e243 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.7%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 91.6%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \leq -1 \cdot 10^{+243}:\\ \;\;\;\;\frac{\left(M \cdot \left(M \cdot \left(w0 \cdot h\right)\right)\right) \cdot \left(D \cdot -0.125\right)}{d} \cdot \frac{D}{d \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}\right)}^{2} \leq -1 \cdot 10^{+243}:\\ \;\;\;\;\frac{D\_m \cdot D\_m}{d\_m} \cdot \frac{h \cdot \left(-0.125 \cdot \left(M\_m \cdot \left(w0 \cdot M\_m\right)\right)\right)}{d\_m \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* M_m D_m) (* d_m 2.0)) 2.0)) -1e+243)
   (* (/ (* D_m D_m) d_m) (/ (* h (* -0.125 (* M_m (* w0 M_m)))) (* d_m l)))
   (* w0 1.0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((M_m * D_m) / (d_m * 2.0)), 2.0)) <= -1e+243) {
		tmp = ((D_m * D_m) / d_m) * ((h * (-0.125 * (M_m * (w0 * M_m)))) / (d_m * l));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: tmp
    if (((h / l) * (((m_m * d_m) / (d_m_1 * 2.0d0)) ** 2.0d0)) <= (-1d+243)) then
        tmp = ((d_m * d_m) / d_m_1) * ((h * ((-0.125d0) * (m_m * (w0 * m_m)))) / (d_m_1 * l))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * Math.pow(((M_m * D_m) / (d_m * 2.0)), 2.0)) <= -1e+243) {
		tmp = ((D_m * D_m) / d_m) * ((h * (-0.125 * (M_m * (w0 * M_m)))) / (d_m * l));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((h / l) * math.pow(((M_m * D_m) / (d_m * 2.0)), 2.0)) <= -1e+243:
		tmp = ((D_m * D_m) / d_m) * ((h * (-0.125 * (M_m * (w0 * M_m)))) / (d_m * l))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(d_m * 2.0)) ^ 2.0)) <= -1e+243)
		tmp = Float64(Float64(Float64(D_m * D_m) / d_m) * Float64(Float64(h * Float64(-0.125 * Float64(M_m * Float64(w0 * M_m)))) / Float64(d_m * l)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((h / l) * (((M_m * D_m) / (d_m * 2.0)) ^ 2.0)) <= -1e+243)
		tmp = ((D_m * D_m) / d_m) * ((h * (-0.125 * (M_m * (w0 * M_m)))) / (d_m * l));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e+243], N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(N[(h * N[(-0.125 * N[(M$95$m * N[(w0 * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.0000000000000001e243

   1. Initial program 52.2%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right)} \]

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in D around inf

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

   8. Applied rewrites 41.5%

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

   10. Applied rewrites 45.3%

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

   if -1.0000000000000001e243 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.7%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 91.6%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \leq -1 \cdot 10^{+243}:\\ \;\;\;\;\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(-0.125 \cdot \left(M \cdot \left(w0 \cdot M\right)\right)\right)}{d \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}\right)}^{2} \leq -1 \cdot 10^{+292}:\\ \;\;\;\;\frac{\left(M\_m \cdot \left(w0 \cdot M\_m\right)\right) \cdot \left(\left(D\_m \cdot D\_m\right) \cdot \left(h \cdot -0.125\right)\right)}{\ell \cdot \left(d\_m \cdot d\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* M_m D_m) (* d_m 2.0)) 2.0)) -1e+292)
   (/ (* (* M_m (* w0 M_m)) (* (* D_m D_m) (* h -0.125))) (* l (* d_m d_m)))
   (* w0 1.0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((M_m * D_m) / (d_m * 2.0)), 2.0)) <= -1e+292) {
		tmp = ((M_m * (w0 * M_m)) * ((D_m * D_m) * (h * -0.125))) / (l * (d_m * d_m));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: tmp
    if (((h / l) * (((m_m * d_m) / (d_m_1 * 2.0d0)) ** 2.0d0)) <= (-1d+292)) then
        tmp = ((m_m * (w0 * m_m)) * ((d_m * d_m) * (h * (-0.125d0)))) / (l * (d_m_1 * d_m_1))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * Math.pow(((M_m * D_m) / (d_m * 2.0)), 2.0)) <= -1e+292) {
		tmp = ((M_m * (w0 * M_m)) * ((D_m * D_m) * (h * -0.125))) / (l * (d_m * d_m));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((h / l) * math.pow(((M_m * D_m) / (d_m * 2.0)), 2.0)) <= -1e+292:
		tmp = ((M_m * (w0 * M_m)) * ((D_m * D_m) * (h * -0.125))) / (l * (d_m * d_m))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(d_m * 2.0)) ^ 2.0)) <= -1e+292)
		tmp = Float64(Float64(Float64(M_m * Float64(w0 * M_m)) * Float64(Float64(D_m * D_m) * Float64(h * -0.125))) / Float64(l * Float64(d_m * d_m)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((h / l) * (((M_m * D_m) / (d_m * 2.0)) ^ 2.0)) <= -1e+292)
		tmp = ((M_m * (w0 * M_m)) * ((D_m * D_m) * (h * -0.125))) / (l * (d_m * d_m));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e+292], N[(N[(N[(M$95$m * N[(w0 * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(h * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1e292

   1. Initial program 51.4%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right)} \]

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in D around inf

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

   8. Applied rewrites 42.2%

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

   10. Applied rewrites 44.0%

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

   if -1e292 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.7%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 91.2%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \leq -1 \cdot 10^{+292}:\\ \;\;\;\;\frac{\left(M \cdot \left(w0 \cdot M\right)\right) \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot -0.125\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}\right)}^{2} \leq -1 \cdot 10^{+292}:\\ \;\;\;\;D\_m \cdot \frac{\left(M\_m \cdot \left(M\_m \cdot \left(w0 \cdot h\right)\right)\right) \cdot \left(D\_m \cdot -0.125\right)}{\ell \cdot \left(d\_m \cdot d\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* M_m D_m) (* d_m 2.0)) 2.0)) -1e+292)
   (* D_m (/ (* (* M_m (* M_m (* w0 h))) (* D_m -0.125)) (* l (* d_m d_m))))
   (* w0 1.0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((M_m * D_m) / (d_m * 2.0)), 2.0)) <= -1e+292) {
		tmp = D_m * (((M_m * (M_m * (w0 * h))) * (D_m * -0.125)) / (l * (d_m * d_m)));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: tmp
    if (((h / l) * (((m_m * d_m) / (d_m_1 * 2.0d0)) ** 2.0d0)) <= (-1d+292)) then
        tmp = d_m * (((m_m * (m_m * (w0 * h))) * (d_m * (-0.125d0))) / (l * (d_m_1 * d_m_1)))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * Math.pow(((M_m * D_m) / (d_m * 2.0)), 2.0)) <= -1e+292) {
		tmp = D_m * (((M_m * (M_m * (w0 * h))) * (D_m * -0.125)) / (l * (d_m * d_m)));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((h / l) * math.pow(((M_m * D_m) / (d_m * 2.0)), 2.0)) <= -1e+292:
		tmp = D_m * (((M_m * (M_m * (w0 * h))) * (D_m * -0.125)) / (l * (d_m * d_m)))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(d_m * 2.0)) ^ 2.0)) <= -1e+292)
		tmp = Float64(D_m * Float64(Float64(Float64(M_m * Float64(M_m * Float64(w0 * h))) * Float64(D_m * -0.125)) / Float64(l * Float64(d_m * d_m))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((h / l) * (((M_m * D_m) / (d_m * 2.0)) ^ 2.0)) <= -1e+292)
		tmp = D_m * (((M_m * (M_m * (w0 * h))) * (D_m * -0.125)) / (l * (d_m * d_m)));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e+292], N[(D$95$m * N[(N[(N[(M$95$m * N[(M$95$m * N[(w0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * -0.125), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1e292

   1. Initial program 51.4%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right)} \]

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in D around inf

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

   8. Applied rewrites 42.2%

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

   10. Applied rewrites 46.0%

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

   if -1e292 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.7%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 91.2%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \leq -1 \cdot 10^{+292}:\\ \;\;\;\;D \cdot \frac{\left(M \cdot \left(M \cdot \left(w0 \cdot h\right)\right)\right) \cdot \left(D \cdot -0.125\right)}{\ell \cdot \left(d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;d\_m \cdot 2 \leq 10^{-120}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}, h \cdot \frac{M\_m \cdot D\_m}{-2 \cdot \left(d\_m \cdot \ell\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D\_m}{d\_m}, \frac{D\_m}{d\_m} \cdot \left(\left(M\_m \cdot -0.25\right) \cdot \frac{M\_m \cdot h}{\ell}\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* d_m 2.0) 1e-120)
   (*
    w0
    (sqrt
     (fma
      (/ (* M_m D_m) (* d_m 2.0))
      (* h (/ (* M_m D_m) (* -2.0 (* d_m l))))
      1.0)))
   (*
    w0
    (sqrt
     (fma
      (/ D_m d_m)
      (* (/ D_m d_m) (* (* M_m -0.25) (/ (* M_m h) l)))
      1.0)))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((d_m * 2.0) <= 1e-120) {
		tmp = w0 * sqrt(fma(((M_m * D_m) / (d_m * 2.0)), (h * ((M_m * D_m) / (-2.0 * (d_m * l)))), 1.0));
	} else {
		tmp = w0 * sqrt(fma((D_m / d_m), ((D_m / d_m) * ((M_m * -0.25) * ((M_m * h) / l))), 1.0));
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(d_m * 2.0) <= 1e-120)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M_m * D_m) / Float64(d_m * 2.0)), Float64(h * Float64(Float64(M_m * D_m) / Float64(-2.0 * Float64(d_m * l)))), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(D_m / d_m), Float64(Float64(D_m / d_m) * Float64(Float64(M_m * -0.25) * Float64(Float64(M_m * h) / l))), 1.0)));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(d$95$m * 2.0), $MachinePrecision], 1e-120], N[(w0 * N[Sqrt[N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(-2.0 * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(N[(M$95$m * -0.25), $MachinePrecision] * N[(N[(M$95$m * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) d) < 9.99999999999999979e-121

   1. Initial program 77.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. times-frac N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. pow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. div-inv N/A

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

      21. metadata-eval N/A

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

      22. unpow-prod-down N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

      25. lower-*.f64 N/A

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

      26. pow2 N/A

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

      27. lower-*.f64 N/A

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

   4. Applied rewrites 69.1%

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

   6. Applied rewrites 82.0%

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

   if 9.99999999999999979e-121 < (*.f64 #s(literal 2 binary64) d)

   1. Initial program 83.5%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left({D}^{2}, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right), 1\right)}} \]

   5. Applied rewrites 62.4%

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

   7. Applied rewrites 64.5%

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

   9. Applied rewrites 81.5%

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

   11. Applied rewrites 86.7%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot 2 \leq 10^{-120}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, h \cdot \frac{M \cdot D}{-2 \cdot \left(d \cdot \ell\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D}{d}, \frac{D}{d} \cdot \left(\left(M \cdot -0.25\right) \cdot \frac{M \cdot h}{\ell}\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.9%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-neg-in N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

   11. distribute-lft-neg-in N/A

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

   12. div-inv N/A

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

   13. times-frac N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 88.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. /-rgt-identity N/A

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

   5. lower-*.f64 88.0

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

6. Applied rewrites 88.0%

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

7. Final simplification 88.0%

   \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{M \cdot D}{d \cdot -2}}{\ell}, \frac{M \cdot D}{d \cdot 2} \cdot h, 1\right)} \]
8. Add Preprocessing

Alternative 11: 83.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2.8 \cdot 10^{-184}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D\_m}{d\_m}, D\_m \cdot \frac{-0.25 \cdot \left(M\_m \cdot \left(M\_m \cdot h\right)\right)}{d\_m \cdot \ell}, 1\right)}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 2.8e-184)
   (* w0 1.0)
   (*
    w0
    (sqrt
     (fma
      (/ D_m d_m)
      (* D_m (/ (* -0.25 (* M_m (* M_m h))) (* d_m l)))
      1.0)))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 2.8e-184) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * sqrt(fma((D_m / d_m), (D_m * ((-0.25 * (M_m * (M_m * h))) / (d_m * l))), 1.0));
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 2.8e-184)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * sqrt(fma(Float64(D_m / d_m), Float64(D_m * Float64(Float64(-0.25 * Float64(M_m * Float64(M_m * h))) / Float64(d_m * l))), 1.0)));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 2.8e-184], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(D$95$m * N[(N[(-0.25 * N[(M$95$m * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.7999999999999998e-184

   1. Initial program 79.6%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 76.6%

      \[\leadsto w0 \cdot \color{blue}{1} \]

   if 2.7999999999999998e-184 < M

   1. Initial program 80.3%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left({D}^{2}, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right), 1\right)}} \]

   5. Applied rewrites 50.5%

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

   7. Applied rewrites 54.6%

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

   9. Applied rewrites 77.7%

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

   11. Applied rewrites 76.2%

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

Alternative 12: 86.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.9%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. times-frac N/A

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

   15. *-commutative N/A

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

   16. associate-*l* N/A

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

   17. pow2 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. div-inv N/A

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

   21. metadata-eval N/A

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

   22. unpow-prod-down N/A

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

   23. metadata-eval N/A

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

   24. metadata-eval N/A

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

   25. lower-*.f64 N/A

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

   26. pow2 N/A

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

   27. lower-*.f64 N/A

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

4. Applied rewrites 72.1%

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

6. Applied rewrites 84.5%

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

7. Final simplification 84.5%

   \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, h \cdot \frac{M \cdot D}{-2 \cdot \left(d \cdot \ell\right)}, 1\right)} \]
8. Add Preprocessing

Alternative 13: 68.2% accurate, 26.2× speedup

Math

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

FPCore

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

C

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

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    code = w0 * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.9%

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

3. Taylor expanded in M around 0

   \[\leadsto w0 \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 71.0%

   \[\leadsto w0 \cdot \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024232 
(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))))))