Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.5% → 89.7%

Time: 11.1s

Alternatives: 12

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: 81.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{D\_m \cdot M\_m}{d\_m \cdot 2} \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{{\left(\frac{\frac{d\_m}{M\_m}}{D\_m} \cdot 2\right)}^{-2}}{\ell}, -h, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{0.5}{d\_m} \cdot M\_m\right) \cdot D\_m, \frac{\left(\left(0.5 \cdot D\_m\right) \cdot h\right) \cdot \frac{M\_m}{d\_m}}{-\ell}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

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 M_m) (* d_m 2.0)) 2e+104)
   (* (sqrt (fma (/ (pow (* (/ (/ d_m M_m) D_m) 2.0) -2.0) l) (- h) 1.0)) w0)
   (*
    (sqrt
     (fma
      (* (* (/ 0.5 d_m) M_m) D_m)
      (/ (* (* (* 0.5 D_m) h) (/ M_m d_m)) (- l))
      1.0))
    w0)))

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 * M_m) / (d_m * 2.0)) <= 2e+104) {
		tmp = sqrt(fma((pow((((d_m / M_m) / D_m) * 2.0), -2.0) / l), -h, 1.0)) * w0;
	} else {
		tmp = sqrt(fma((((0.5 / d_m) * M_m) * D_m), ((((0.5 * D_m) * h) * (M_m / d_m)) / -l), 1.0)) * w0;
	}
	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(D_m * M_m) / Float64(d_m * 2.0)) <= 2e+104)
		tmp = Float64(sqrt(fma(Float64((Float64(Float64(Float64(d_m / M_m) / D_m) * 2.0) ^ -2.0) / l), Float64(-h), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(0.5 / d_m) * M_m) * D_m), Float64(Float64(Float64(Float64(0.5 * D_m) * h) * Float64(M_m / d_m)) / Float64(-l)), 1.0)) * w0);
	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[(D$95$m * M$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2e+104], N[(N[Sqrt[N[(N[(N[Power[N[(N[(N[(d$95$m / M$95$m), $MachinePrecision] / D$95$m), $MachinePrecision] * 2.0), $MachinePrecision], -2.0], $MachinePrecision] / l), $MachinePrecision] * (-h) + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(0.5 / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(N[(0.5 * D$95$m), $MachinePrecision] * h), $MachinePrecision] * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 2e104

   1. Initial program 81.2%

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

      1. 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. frac-2neg N/A

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

      10. associate-/r/ N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

   4. Applied rewrites 91.3%

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

   if 2e104 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 49.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 60.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 60.2

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

   6. Applied rewrites 60.2%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{D \cdot M}{d \cdot 2} \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{{\left(\frac{\frac{d}{M}}{D} \cdot 2\right)}^{-2}}{\ell}, -h, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \frac{\left(\left(0.5 \cdot D\right) \cdot h\right) \cdot \frac{M}{d}}{-\ell}, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{d\_m \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{M\_m}{\ell \cdot d\_m} \cdot \frac{\left(D\_m \cdot D\_m\right) \cdot M\_m}{d\_m}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* D_m M_m) (* d_m 2.0)) 2.0) (/ h l)) -2e-7)
   (*
    (sqrt
     (fma (* -0.25 h) (* (/ M_m (* l d_m)) (/ (* (* D_m D_m) M_m) d_m)) 1.0))
    w0)
   (* 1.0 w0)))

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 ((pow(((D_m * M_m) / (d_m * 2.0)), 2.0) * (h / l)) <= -2e-7) {
		tmp = sqrt(fma((-0.25 * h), ((M_m / (l * d_m)) * (((D_m * D_m) * M_m) / d_m)), 1.0)) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	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(Float64(D_m * M_m) / Float64(d_m * 2.0)) ^ 2.0) * Float64(h / l)) <= -2e-7)
		tmp = Float64(sqrt(fma(Float64(-0.25 * h), Float64(Float64(M_m / Float64(l * d_m)) * Float64(Float64(Float64(D_m * D_m) * M_m) / d_m)), 1.0)) * w0);
	else
		tmp = Float64(1.0 * w0);
	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[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-7], N[(N[Sqrt[N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(M$95$m / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $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)) < -1.9999999999999999e-7

   1. Initial program 57.5%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 44.0%

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

   7. Applied rewrites 53.9%

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

   9. Applied rewrites 50.8%

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

   if -1.9999999999999999e-7 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 84.8%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 94.1%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{M}{\ell \cdot d} \cdot \frac{\left(D \cdot D\right) \cdot M}{d}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{d\_m \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \left(\frac{D\_m}{\left(d\_m \cdot d\_m\right) \cdot \ell} \cdot \left(D\_m \cdot M\_m\right)\right) \cdot M\_m, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* D_m M_m) (* d_m 2.0)) 2.0) (/ h l)) -2e-7)
   (*
    (sqrt
     (fma (* -0.25 h) (* (* (/ D_m (* (* d_m d_m) l)) (* D_m M_m)) M_m) 1.0))
    w0)
   (* 1.0 w0)))

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 ((pow(((D_m * M_m) / (d_m * 2.0)), 2.0) * (h / l)) <= -2e-7) {
		tmp = sqrt(fma((-0.25 * h), (((D_m / ((d_m * d_m) * l)) * (D_m * M_m)) * M_m), 1.0)) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	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(Float64(D_m * M_m) / Float64(d_m * 2.0)) ^ 2.0) * Float64(h / l)) <= -2e-7)
		tmp = Float64(sqrt(fma(Float64(-0.25 * h), Float64(Float64(Float64(D_m / Float64(Float64(d_m * d_m) * l)) * Float64(D_m * M_m)) * M_m), 1.0)) * w0);
	else
		tmp = Float64(1.0 * w0);
	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[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-7], N[(N[Sqrt[N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(D$95$m / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $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)) < -1.9999999999999999e-7

   1. Initial program 57.5%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 44.0%

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

   7. Applied rewrites 53.7%

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

   if -1.9999999999999999e-7 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 84.8%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 94.1%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \left(\frac{D}{\left(d \cdot d\right) \cdot \ell} \cdot \left(D \cdot M\right)\right) \cdot M, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{d\_m \cdot 2}\right)}^{2} \leq 2 \cdot 10^{-285}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{\left(\left(\frac{M\_m}{d\_m} \cdot M\_m\right) \cdot D\_m\right) \cdot D\_m}{\ell \cdot d\_m}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

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 (<= (pow (/ (* D_m M_m) (* d_m 2.0)) 2.0) 2e-285)
   (* 1.0 w0)
   (*
    (sqrt
     (fma (* -0.25 h) (/ (* (* (* (/ M_m d_m) M_m) D_m) D_m) (* l d_m)) 1.0))
    w0)))

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 (pow(((D_m * M_m) / (d_m * 2.0)), 2.0) <= 2e-285) {
		tmp = 1.0 * w0;
	} else {
		tmp = sqrt(fma((-0.25 * h), (((((M_m / d_m) * M_m) * D_m) * D_m) / (l * d_m)), 1.0)) * w0;
	}
	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(D_m * M_m) / Float64(d_m * 2.0)) ^ 2.0) <= 2e-285)
		tmp = Float64(1.0 * w0);
	else
		tmp = Float64(sqrt(fma(Float64(-0.25 * h), Float64(Float64(Float64(Float64(Float64(M_m / d_m) * M_m) * D_m) * D_m) / Float64(l * d_m)), 1.0)) * w0);
	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[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 2e-285], N[(1.0 * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 2.00000000000000015e-285

   1. Initial program 83.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 97.4%

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

   if 2.00000000000000015e-285 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 71.5%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 45.8%

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

   7. Applied rewrites 66.5%

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

   9. Applied rewrites 63.8%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq 2 \cdot 10^{-285}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot D}{\ell \cdot d}, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{d\_m \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D\_m \cdot D\_m\right), \left(\left(\frac{w0}{\left(d\_m \cdot d\_m\right) \cdot \ell} \cdot h\right) \cdot M\_m\right) \cdot M\_m, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* D_m M_m) (* d_m 2.0)) 2.0) (/ h l)) -4e+37)
   (fma
    (* -0.125 (* D_m D_m))
    (* (* (* (/ w0 (* (* d_m d_m) l)) h) M_m) M_m)
    w0)
   (* 1.0 w0)))

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 ((pow(((D_m * M_m) / (d_m * 2.0)), 2.0) * (h / l)) <= -4e+37) {
		tmp = fma((-0.125 * (D_m * D_m)), ((((w0 / ((d_m * d_m) * l)) * h) * M_m) * M_m), w0);
	} else {
		tmp = 1.0 * w0;
	}
	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(Float64(D_m * M_m) / Float64(d_m * 2.0)) ^ 2.0) * Float64(h / l)) <= -4e+37)
		tmp = fma(Float64(-0.125 * Float64(D_m * D_m)), Float64(Float64(Float64(Float64(w0 / Float64(Float64(d_m * d_m) * l)) * h) * M_m) * M_m), w0);
	else
		tmp = Float64(1.0 * w0);
	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[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e+37], N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(w0 / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $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)) < -3.99999999999999982e37

   1. Initial program 55.3%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   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. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 35.9%

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

   7. Applied rewrites 35.8%

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

   9. Applied rewrites 42.1%

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

   if -3.99999999999999982e37 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 92.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \left(\left(\frac{w0}{\left(d \cdot d\right) \cdot \ell} \cdot h\right) \cdot M\right) \cdot M, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m \cdot M\_m}{d\_m \cdot 2}\\ t_1 := \frac{M\_m}{d\_m} \cdot M\_m\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{\left(\frac{D\_m}{d\_m} \cdot t\_1\right) \cdot D\_m}{\ell}, 1\right)} \cdot w0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+211}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{-0.5 \cdot \left(D\_m \cdot M\_m\right)}{d\_m} \cdot \frac{h}{\ell}, \left(\frac{0.5}{d\_m} \cdot M\_m\right) \cdot D\_m, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(-0.25 \cdot h\right) \cdot D\_m\right) \cdot t\_1}{d\_m}, \frac{D\_m}{\ell}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

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
 (let* ((t_0 (/ (* D_m M_m) (* d_m 2.0))) (t_1 (* (/ M_m d_m) M_m)))
   (if (<= t_0 2e-87)
     (* (sqrt (fma (* -0.25 h) (/ (* (* (/ D_m d_m) t_1) D_m) l) 1.0)) w0)
     (if (<= t_0 2e+211)
       (*
        (sqrt
         (fma
          (* (/ (* -0.5 (* D_m M_m)) d_m) (/ h l))
          (* (* (/ 0.5 d_m) M_m) D_m)
          1.0))
        w0)
       (*
        (sqrt (fma (/ (* (* (* -0.25 h) D_m) t_1) d_m) (/ D_m l) 1.0))
        w0)))))

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 t_0 = (D_m * M_m) / (d_m * 2.0);
	double t_1 = (M_m / d_m) * M_m;
	double tmp;
	if (t_0 <= 2e-87) {
		tmp = sqrt(fma((-0.25 * h), ((((D_m / d_m) * t_1) * D_m) / l), 1.0)) * w0;
	} else if (t_0 <= 2e+211) {
		tmp = sqrt(fma((((-0.5 * (D_m * M_m)) / d_m) * (h / l)), (((0.5 / d_m) * M_m) * D_m), 1.0)) * w0;
	} else {
		tmp = sqrt(fma(((((-0.25 * h) * D_m) * t_1) / d_m), (D_m / l), 1.0)) * w0;
	}
	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)
	t_0 = Float64(Float64(D_m * M_m) / Float64(d_m * 2.0))
	t_1 = Float64(Float64(M_m / d_m) * M_m)
	tmp = 0.0
	if (t_0 <= 2e-87)
		tmp = Float64(sqrt(fma(Float64(-0.25 * h), Float64(Float64(Float64(Float64(D_m / d_m) * t_1) * D_m) / l), 1.0)) * w0);
	elseif (t_0 <= 2e+211)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(-0.5 * Float64(D_m * M_m)) / d_m) * Float64(h / l)), Float64(Float64(Float64(0.5 / d_m) * M_m) * D_m), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(-0.25 * h) * D_m) * t_1) / d_m), Float64(D_m / l), 1.0)) * w0);
	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_] := Block[{t$95$0 = N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M$95$m / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-87], N[(N[Sqrt[N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(N[(D$95$m / d$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], If[LessEqual[t$95$0, 2e+211], N[(N[Sqrt[N[(N[(N[(N[(-0.5 * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(-0.25 * h), $MachinePrecision] * D$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 2.00000000000000004e-87

   1. Initial program 78.9%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \sqrt{\color{blue}{1 + \frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   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. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 85.4%

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

   if 2.00000000000000004e-87 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1.9999999999999999e211

   1. Initial program 79.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. *-commutative N/A

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

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

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 79.8%

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

   if 1.9999999999999999e211 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 55.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 59.1%

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

   7. Applied rewrites 66.7%

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

   9. Applied rewrites 62.7%

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

   11. Applied rewrites 66.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{D \cdot M}{d \cdot 2} \leq 2 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{\left(\frac{D}{d} \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot D}{\ell}, 1\right)} \cdot w0\\ \mathbf{elif}\;\frac{D \cdot M}{d \cdot 2} \leq 2 \cdot 10^{+211}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{-0.5 \cdot \left(D \cdot M\right)}{d} \cdot \frac{h}{\ell}, \left(\frac{0.5}{d} \cdot M\right) \cdot D, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(-0.25 \cdot h\right) \cdot D\right) \cdot \left(\frac{M}{d} \cdot M\right)}{d}, \frac{D}{\ell}, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d\_m} \cdot M\_m\\ \mathbf{if}\;\frac{D\_m \cdot M\_m}{d\_m \cdot 2} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{\left(\frac{D\_m}{d\_m} \cdot t\_0\right) \cdot D\_m}{\ell}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(-0.25 \cdot h\right) \cdot D\_m\right) \cdot t\_0}{d\_m}, \frac{D\_m}{\ell}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (/ M_m d_m) M_m)))
   (if (<= (/ (* D_m M_m) (* d_m 2.0)) 2e+63)
     (* (sqrt (fma (* -0.25 h) (/ (* (* (/ D_m d_m) t_0) D_m) l) 1.0)) w0)
     (* (sqrt (fma (/ (* (* (* -0.25 h) D_m) t_0) d_m) (/ D_m l) 1.0)) w0))))

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 t_0 = (M_m / d_m) * M_m;
	double tmp;
	if (((D_m * M_m) / (d_m * 2.0)) <= 2e+63) {
		tmp = sqrt(fma((-0.25 * h), ((((D_m / d_m) * t_0) * D_m) / l), 1.0)) * w0;
	} else {
		tmp = sqrt(fma(((((-0.25 * h) * D_m) * t_0) / d_m), (D_m / l), 1.0)) * w0;
	}
	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)
	t_0 = Float64(Float64(M_m / d_m) * M_m)
	tmp = 0.0
	if (Float64(Float64(D_m * M_m) / Float64(d_m * 2.0)) <= 2e+63)
		tmp = Float64(sqrt(fma(Float64(-0.25 * h), Float64(Float64(Float64(Float64(D_m / d_m) * t_0) * D_m) / l), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(-0.25 * h) * D_m) * t_0) / d_m), Float64(D_m / l), 1.0)) * w0);
	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_] := Block[{t$95$0 = N[(N[(M$95$m / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2e+63], N[(N[Sqrt[N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(N[(D$95$m / d$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(-0.25 * h), $MachinePrecision] * D$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 2.00000000000000012e63

   1. Initial program 81.5%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 64.8%

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

   7. Applied rewrites 85.4%

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

   if 2.00000000000000012e63 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 49.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 44.7%

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

   7. Applied rewrites 57.5%

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

   9. Applied rewrites 54.8%

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

   11. Applied rewrites 57.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{D \cdot M}{d \cdot 2} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{\left(\frac{D}{d} \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot D}{\ell}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(-0.25 \cdot h\right) \cdot D\right) \cdot \left(\frac{M}{d} \cdot M\right)}{d}, \frac{D}{\ell}, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.2% accurate, 1.9× speedup

Math

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

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
 (*
  (sqrt
   (fma
    (* (* (/ 0.5 d_m) M_m) D_m)
    (/ (* (* (/ M_m d_m) h) (* 0.5 D_m)) (- l))
    1.0))
  w0))

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 sqrt(fma((((0.5 / d_m) * M_m) * D_m), ((((M_m / d_m) * h) * (0.5 * D_m)) / -l), 1.0)) * w0;
}

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(sqrt(fma(Float64(Float64(Float64(0.5 / d_m) * M_m) * D_m), Float64(Float64(Float64(Float64(M_m / d_m) * h) * Float64(0.5 * D_m)) / Float64(-l)), 1.0)) * w0)
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[(N[Sqrt[N[(N[(N[(N[(0.5 / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(N[(M$95$m / d$95$m), $MachinePrecision] * h), $MachinePrecision] * N[(0.5 * D$95$m), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]

Derivation

1. Initial program 76.5%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
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. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 83.7%

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

5. Final simplification 83.7%

   \[\leadsto \sqrt{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \frac{\left(\frac{M}{d} \cdot h\right) \cdot \left(0.5 \cdot D\right)}{-\ell}, 1\right)} \cdot w0 \]
6. Add Preprocessing

Alternative 9: 83.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{\left(\frac{D\_m}{d\_m} \cdot \left(\frac{M\_m}{d\_m} \cdot M\_m\right)\right) \cdot D\_m}{\ell}, 1\right)} \cdot w0 \end{array} \]

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
 (*
  (sqrt
   (fma (* -0.25 h) (/ (* (* (/ D_m d_m) (* (/ M_m d_m) M_m)) D_m) l) 1.0))
  w0))

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 sqrt(fma((-0.25 * h), ((((D_m / d_m) * ((M_m / d_m) * M_m)) * D_m) / l), 1.0)) * w0;
}

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(sqrt(fma(Float64(-0.25 * h), Float64(Float64(Float64(Float64(D_m / d_m) * Float64(Float64(M_m / d_m) * M_m)) * D_m) / l), 1.0)) * w0)
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[(N[Sqrt[N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(N[(M$95$m / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]

Derivation

1. Initial program 76.5%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
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. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. lft-mult-inverse N/A

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

   8. distribute-rgt-in N/A

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

   9. distribute-lft-in N/A

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

   10. associate-*r* N/A

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

   11. rgt-mult-inverse N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 61.7%

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

7. Applied rewrites 79.9%

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

8. Final simplification 79.9%

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

Alternative 10: 78.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \cdot M\_m \leq 4 \cdot 10^{-109}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(h \cdot M\_m\right) \cdot \left(\frac{w0}{\left(d\_m \cdot d\_m\right) \cdot \ell} \cdot M\_m\right)\right) \cdot D\_m, -0.125 \cdot D\_m, w0\right)\\ \end{array} \end{array} \]

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 M_m) 4e-109)
   (* 1.0 w0)
   (fma
    (* (* (* h M_m) (* (/ w0 (* (* d_m d_m) l)) M_m)) D_m)
    (* -0.125 D_m)
    w0)))

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 * M_m) <= 4e-109) {
		tmp = 1.0 * w0;
	} else {
		tmp = fma((((h * M_m) * ((w0 / ((d_m * d_m) * l)) * M_m)) * D_m), (-0.125 * D_m), w0);
	}
	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 * M_m) <= 4e-109)
		tmp = Float64(1.0 * w0);
	else
		tmp = fma(Float64(Float64(Float64(h * M_m) * Float64(Float64(w0 / Float64(Float64(d_m * d_m) * l)) * M_m)) * D_m), Float64(-0.125 * D_m), w0);
	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 * M$95$m), $MachinePrecision], 4e-109], N[(1.0 * w0), $MachinePrecision], N[(N[(N[(N[(h * M$95$m), $MachinePrecision] * N[(N[(w0 / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 4e-109

   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 77.8%

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

   if 4e-109 < (*.f64 M D)

   1. Initial program 71.3%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   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. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 38.2%

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

   7. Applied rewrites 38.4%

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

   9. Applied rewrites 65.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot M \leq 4 \cdot 10^{-109}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(h \cdot M\right) \cdot \left(\frac{w0}{\left(d \cdot d\right) \cdot \ell} \cdot M\right)\right) \cdot D, -0.125 \cdot D, w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \cdot M\_m \leq 10^{-72}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D\_m, \frac{\left(-0.125 \cdot w0\right) \cdot \left(\left(M\_m \cdot M\_m\right) \cdot h\right)}{\left(d\_m \cdot d\_m\right) \cdot \ell} \cdot D\_m, w0\right)\\ \end{array} \end{array} \]

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 M_m) 1e-72)
   (* 1.0 w0)
   (fma
    D_m
    (* (/ (* (* -0.125 w0) (* (* M_m M_m) h)) (* (* d_m d_m) l)) D_m)
    w0)))

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 * M_m) <= 1e-72) {
		tmp = 1.0 * w0;
	} else {
		tmp = fma(D_m, ((((-0.125 * w0) * ((M_m * M_m) * h)) / ((d_m * d_m) * l)) * D_m), w0);
	}
	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 * M_m) <= 1e-72)
		tmp = Float64(1.0 * w0);
	else
		tmp = fma(D_m, Float64(Float64(Float64(Float64(-0.125 * w0) * Float64(Float64(M_m * M_m) * h)) / Float64(Float64(d_m * d_m) * l)) * D_m), w0);
	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 * M$95$m), $MachinePrecision], 1e-72], N[(1.0 * w0), $MachinePrecision], N[(D$95$m * N[(N[(N[(N[(-0.125 * w0), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 9.9999999999999997e-73

   1. Initial program 80.6%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 78.1%

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

   if 9.9999999999999997e-73 < (*.f64 M D)

   1. Initial program 69.5%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   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. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 36.4%

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

   7. Applied rewrites 50.8%

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

   9. Applied rewrites 50.9%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot M \leq 10^{-72}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D, \frac{\left(-0.125 \cdot w0\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot D, w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.0% accurate, 26.2× speedup

Math

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

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 (* 1.0 w0))

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 1.0 * w0;
}

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 = 1.0d0 * w0
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 1.0 * w0;
}

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 1.0 * w0

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(1.0 * w0)
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 = 1.0 * w0;
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[(1.0 * w0), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 67.2%

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

6. Final simplification 67.2%

   \[\leadsto 1 \cdot w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024325 
(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))))))