Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.2% → 87.9%

Time: 16.8s

Alternatives: 14

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: 80.2% 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: 87.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;d\_m \cdot 2 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}, \frac{\frac{\left(M\_m \cdot D\_m\right) \cdot h}{d\_m \cdot 2}}{-\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{D\_m}{d\_m} \cdot \left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right)}{\ell} \cdot \frac{\frac{D\_m}{d\_m}}{\frac{-1}{h}}}\\ \end{array} \end{array} \]

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) 5e+153)
   (*
    w0
    (sqrt
     (fma
      (/ (* M_m D_m) (* d_m 2.0))
      (/ (/ (* (* M_m D_m) h) (* d_m 2.0)) (- l))
      1.0)))
   (*
    w0
    (sqrt
     (+
      1.0
      (*
       (/ (* (/ D_m d_m) (* (* M_m M_m) 0.25)) l)
       (/ (/ D_m d_m) (/ -1.0 h))))))))

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) <= 5e+153) {
		tmp = w0 * sqrt(fma(((M_m * D_m) / (d_m * 2.0)), ((((M_m * D_m) * h) / (d_m * 2.0)) / -l), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 + ((((D_m / d_m) * ((M_m * M_m) * 0.25)) / l) * ((D_m / d_m) / (-1.0 / h)))));
	}
	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) <= 5e+153)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M_m * D_m) / Float64(d_m * 2.0)), Float64(Float64(Float64(Float64(M_m * D_m) * h) / Float64(d_m * 2.0)) / Float64(-l)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(D_m / d_m) * Float64(Float64(M_m * M_m) * 0.25)) / l) * Float64(Float64(D_m / d_m) / Float64(-1.0 / h))))));
	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], 5e+153], 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[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(D$95$m / d$95$m), $MachinePrecision] / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) d) < 5.00000000000000018e153

   1. Initial program 83.7%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. 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 90.9%

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

   if 5.00000000000000018e153 < (*.f64 #s(literal 2 binary64) d)

   1. Initial program 81.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*r* N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 73.3%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot 2 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{\frac{\left(M \cdot D\right) \cdot h}{d \cdot 2}}{-\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\ell} \cdot \frac{\frac{D}{d}}{\frac{-1}{h}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{d\_m \cdot 2}\\ \mathbf{if}\;{t\_0}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-118}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{M\_m \cdot D\_m}{d\_m \cdot -2} \cdot \frac{h}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D\_m \cdot \frac{M\_m}{\ell}}{d\_m \cdot -2}, \left(D\_m \cdot 0.5\right) \cdot \frac{M\_m \cdot h}{d\_m}, 1\right)}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (/ (* M_m D_m) (* d_m 2.0))))
   (if (<= (* (pow t_0 2.0) (/ h l)) -1e-118)
     (* w0 (sqrt (fma t_0 (* (/ (* M_m D_m) (* d_m -2.0)) (/ h l)) 1.0)))
     (*
      w0
      (sqrt
       (fma
        (/ (* D_m (/ M_m l)) (* d_m -2.0))
        (* (* D_m 0.5) (/ (* M_m h) d_m))
        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 t_0 = (M_m * D_m) / (d_m * 2.0);
	double tmp;
	if ((pow(t_0, 2.0) * (h / l)) <= -1e-118) {
		tmp = w0 * sqrt(fma(t_0, (((M_m * D_m) / (d_m * -2.0)) * (h / l)), 1.0));
	} else {
		tmp = w0 * sqrt(fma(((D_m * (M_m / l)) / (d_m * -2.0)), ((D_m * 0.5) * ((M_m * h) / d_m)), 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)
	t_0 = Float64(Float64(M_m * D_m) / Float64(d_m * 2.0))
	tmp = 0.0
	if (Float64((t_0 ^ 2.0) * Float64(h / l)) <= -1e-118)
		tmp = Float64(w0 * sqrt(fma(t_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(Float64(D_m * Float64(M_m / l)) / Float64(d_m * -2.0)), Float64(Float64(D_m * 0.5) * Float64(Float64(M_m * h) / d_m)), 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_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e-118], N[(w0 * N[Sqrt[N[(t$95$0 * 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[(N[(D$95$m * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * 0.5), $MachinePrecision] * N[(N[(M$95$m * h), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $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)) < -9.99999999999999985e-119

   1. Initial program 70.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 77.1%

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

   if -9.99999999999999985e-119 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.9%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 98.8%

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

   5. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 94.7

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

   7. Applied rewrites 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 92.3

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

   9. Applied rewrites 92.3%

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

   11. Applied rewrites 91.8%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-118}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot -2} \cdot \frac{h}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D \cdot \frac{M}{\ell}}{d \cdot -2}, \left(D \cdot 0.5\right) \cdot \frac{M \cdot h}{d}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{d\_m \cdot 2}\\ \mathbf{if}\;{t\_0}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{M\_m \cdot D\_m}{d\_m \cdot -2} \cdot \frac{h}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(D\_m \cdot -0.25\right) \cdot \frac{h \cdot \left(M\_m \cdot M\_m\right)}{d\_m \cdot \left(d\_m \cdot \ell\right)}, D\_m, 1\right)}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (/ (* M_m D_m) (* d_m 2.0))))
   (if (<= (* (pow t_0 2.0) (/ h l)) 2e-54)
     (* w0 (sqrt (fma t_0 (* (/ (* M_m D_m) (* d_m -2.0)) (/ h l)) 1.0)))
     (*
      w0
      (sqrt
       (fma
        (* (* D_m -0.25) (/ (* h (* M_m M_m)) (* d_m (* d_m l))))
        D_m
        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 t_0 = (M_m * D_m) / (d_m * 2.0);
	double tmp;
	if ((pow(t_0, 2.0) * (h / l)) <= 2e-54) {
		tmp = w0 * sqrt(fma(t_0, (((M_m * D_m) / (d_m * -2.0)) * (h / l)), 1.0));
	} else {
		tmp = w0 * sqrt(fma(((D_m * -0.25) * ((h * (M_m * M_m)) / (d_m * (d_m * l)))), D_m, 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)
	t_0 = Float64(Float64(M_m * D_m) / Float64(d_m * 2.0))
	tmp = 0.0
	if (Float64((t_0 ^ 2.0) * Float64(h / l)) <= 2e-54)
		tmp = Float64(w0 * sqrt(fma(t_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(Float64(D_m * -0.25) * Float64(Float64(h * Float64(M_m * M_m)) / Float64(d_m * Float64(d_m * l)))), D_m, 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_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 2e-54], N[(w0 * N[Sqrt[N[(t$95$0 * 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[(N[(D$95$m * -0.25), $MachinePrecision] * N[(N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * D$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $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)) < 2.0000000000000001e-54

   1. Initial program 89.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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-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 91.6%

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

   if 2.0000000000000001e-54 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 10.5%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 47.8%

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

   7. Applied rewrites 58.7%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot -2} \cdot \frac{h}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(D \cdot -0.25\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot \left(d \cdot \ell\right)}, D, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+30}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M\_m \cdot D\_m}{\left(d\_m \cdot -2\right) \cdot \ell}, h \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot 0.5}{d\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* d_m 2.0)) 2.0) (/ h l)) -4e+30)
   (*
    w0
    (sqrt
     (fma
      (/ (* M_m D_m) (* (* d_m -2.0) l))
      (* h (/ (* (* M_m D_m) 0.5) 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 ((pow(((M_m * D_m) / (d_m * 2.0)), 2.0) * (h / l)) <= -4e+30) {
		tmp = w0 * sqrt(fma(((M_m * D_m) / ((d_m * -2.0) * l)), (h * (((M_m * D_m) * 0.5) / 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(Float64(M_m * D_m) / Float64(d_m * 2.0)) ^ 2.0) * Float64(h / l)) <= -4e+30)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M_m * D_m) / Float64(Float64(d_m * -2.0) * l)), Float64(h * Float64(Float64(Float64(M_m * D_m) * 0.5) / 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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e+30], N[(w0 * N[Sqrt[N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(N[(d$95$m * -2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / d$95$m), $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)) < -4.0000000000000001e30

   1. Initial program 68.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. 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 77.0%

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

   6. Applied rewrites 74.1%

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

   if -4.0000000000000001e30 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 97.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+30}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{\left(d \cdot -2\right) \cdot \ell}, h \cdot \frac{\left(M \cdot D\right) \cdot 0.5}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 2000000000000:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(M\_m \cdot D\_m, h \cdot \frac{M\_m \cdot D\_m}{d\_m \cdot \left(\left(d\_m \cdot \ell\right) \cdot -4\right)}, 1\right)}\\ \end{array} \end{array} \]

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 (<=
      (- 1.0 (* (pow (/ (* M_m D_m) (* d_m 2.0)) 2.0) (/ h l)))
      2000000000000.0)
   (* w0 1.0)
   (*
    w0
    (sqrt
     (fma (* M_m D_m) (* h (/ (* M_m D_m) (* d_m (* (* d_m l) -4.0)))) 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 ((1.0 - (pow(((M_m * D_m) / (d_m * 2.0)), 2.0) * (h / l))) <= 2000000000000.0) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * sqrt(fma((M_m * D_m), (h * ((M_m * D_m) / (d_m * ((d_m * l) * -4.0)))), 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(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(d_m * 2.0)) ^ 2.0) * Float64(h / l))) <= 2000000000000.0)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * sqrt(fma(Float64(M_m * D_m), Float64(h * Float64(Float64(M_m * D_m) / Float64(d_m * Float64(Float64(d_m * l) * -4.0)))), 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[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2000000000000.0], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(h * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * N[(N[(d$95$m * l), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 2e12

   1. Initial program 100.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.4%

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

   if 2e12 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 57.0%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 79.0%

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

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

   6. Applied rewrites 59.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 70.0%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 2000000000000:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(M \cdot D, h \cdot \frac{M \cdot D}{d \cdot \left(\left(d \cdot \ell\right) \cdot -4\right)}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+30}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(M\_m \cdot D\_m\right) \cdot h}{\ell \cdot \left(d\_m \cdot \left(d\_m \cdot 4\right)\right)}, -M\_m \cdot D\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* d_m 2.0)) 2.0) (/ h l)) -4e+30)
   (*
    w0
    (sqrt
     (fma
      (/ (* (* M_m D_m) h) (* l (* d_m (* d_m 4.0))))
      (- (* M_m 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 ((pow(((M_m * D_m) / (d_m * 2.0)), 2.0) * (h / l)) <= -4e+30) {
		tmp = w0 * sqrt(fma((((M_m * D_m) * h) / (l * (d_m * (d_m * 4.0)))), -(M_m * 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(Float64(M_m * D_m) / Float64(d_m * 2.0)) ^ 2.0) * Float64(h / l)) <= -4e+30)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(M_m * D_m) * h) / Float64(l * Float64(d_m * Float64(d_m * 4.0)))), Float64(-Float64(M_m * 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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e+30], N[(w0 * N[Sqrt[N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(l * N[(d$95$m * N[(d$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[(M$95$m * D$95$m), $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)) < -4.0000000000000001e30

   1. Initial program 68.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lower-/.f64 N/A

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

   4. Applied rewrites 67.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.9

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

   6. Applied rewrites 68.0%

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

   if -4.0000000000000001e30 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 97.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+30}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot h}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, -M \cdot D, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+51}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{-0.25 \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot \left(D\_m \cdot h\right)\right)\right)}{\ell \cdot \left(d\_m \cdot d\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* d_m 2.0)) 2.0) (/ h l)) -2e+51)
   (*
    w0
    (sqrt (/ (* -0.25 (* (* M_m D_m) (* M_m (* D_m h)))) (* 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 ((pow(((M_m * D_m) / (d_m * 2.0)), 2.0) * (h / l)) <= -2e+51) {
		tmp = w0 * sqrt(((-0.25 * ((M_m * D_m) * (M_m * (D_m * h)))) / (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 (((((m_m * d_m) / (d_m_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= (-2d+51)) then
        tmp = w0 * sqrt((((-0.25d0) * ((m_m * d_m) * (m_m * (d_m * h)))) / (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 ((Math.pow(((M_m * D_m) / (d_m * 2.0)), 2.0) * (h / l)) <= -2e+51) {
		tmp = w0 * Math.sqrt(((-0.25 * ((M_m * D_m) * (M_m * (D_m * h)))) / (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 (math.pow(((M_m * D_m) / (d_m * 2.0)), 2.0) * (h / l)) <= -2e+51:
		tmp = w0 * math.sqrt(((-0.25 * ((M_m * D_m) * (M_m * (D_m * h)))) / (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(Float64(M_m * D_m) / Float64(d_m * 2.0)) ^ 2.0) * Float64(h / l)) <= -2e+51)
		tmp = Float64(w0 * sqrt(Float64(Float64(-0.25 * Float64(Float64(M_m * D_m) * Float64(M_m * Float64(D_m * h)))) / 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 (((((M_m * D_m) / (d_m * 2.0)) ^ 2.0) * (h / l)) <= -2e+51)
		tmp = w0 * sqrt(((-0.25 * ((M_m * D_m) * (M_m * (D_m * h)))) / (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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+51], N[(w0 * N[Sqrt[N[(N[(-0.25 * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * N[(D$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m * d$95$m), $MachinePrecision]), $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)) < -2e51

   1. Initial program 68.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lower-/.f64 N/A

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

   4. Applied rewrites 68.4%

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

   5. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 53.7

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

   7. Applied rewrites 53.7%

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

   9. Applied rewrites 62.1%

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

   if -2e51 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.9%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+51}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{-0.25 \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot \left(D \cdot h\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\left(D\_m \cdot D\_m\right) \cdot \left(\frac{M\_m \cdot \left(\left(M\_m \cdot h\right) \cdot -0.125\right)}{d\_m \cdot \ell} \cdot \frac{w0}{d\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* d_m 2.0)) 2.0) (/ h l)) -2e+85)
   (* (* D_m D_m) (* (/ (* M_m (* (* M_m h) -0.125)) (* d_m l)) (/ w0 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 ((pow(((M_m * D_m) / (d_m * 2.0)), 2.0) * (h / l)) <= -2e+85) {
		tmp = (D_m * D_m) * (((M_m * ((M_m * h) * -0.125)) / (d_m * l)) * (w0 / 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 (((((m_m * d_m) / (d_m_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= (-2d+85)) then
        tmp = (d_m * d_m) * (((m_m * ((m_m * h) * (-0.125d0))) / (d_m_1 * l)) * (w0 / 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 ((Math.pow(((M_m * D_m) / (d_m * 2.0)), 2.0) * (h / l)) <= -2e+85) {
		tmp = (D_m * D_m) * (((M_m * ((M_m * h) * -0.125)) / (d_m * l)) * (w0 / 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 (math.pow(((M_m * D_m) / (d_m * 2.0)), 2.0) * (h / l)) <= -2e+85:
		tmp = (D_m * D_m) * (((M_m * ((M_m * h) * -0.125)) / (d_m * l)) * (w0 / 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(Float64(M_m * D_m) / Float64(d_m * 2.0)) ^ 2.0) * Float64(h / l)) <= -2e+85)
		tmp = Float64(Float64(D_m * D_m) * Float64(Float64(Float64(M_m * Float64(Float64(M_m * h) * -0.125)) / Float64(d_m * l)) * Float64(w0 / 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 (((((M_m * D_m) / (d_m * 2.0)) ^ 2.0) * (h / l)) <= -2e+85)
		tmp = (D_m * D_m) * (((M_m * ((M_m * h) * -0.125)) / (d_m * l)) * (w0 / 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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+85], N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * N[(N[(M$95$m * h), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision] * N[(w0 / 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)) < -2e85

   1. Initial program 67.3%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. 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 47.9%

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

   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 46.5%

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

   10. Applied rewrites 51.8%

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

   if -2e85 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.9%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \left(\frac{M \cdot \left(\left(M \cdot h\right) \cdot -0.125\right)}{d \cdot \ell} \cdot \frac{w0}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(D\_m \cdot D\_m, \frac{-0.125 \cdot \left(M\_m \cdot \left(h \cdot \left(w0 \cdot M\_m\right)\right)\right)}{\ell \cdot \left(d\_m \cdot d\_m\right)}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* d_m 2.0)) 2.0) (/ h l)) -1e+33)
   (fma
    (* D_m D_m)
    (/ (* -0.125 (* M_m (* h (* w0 M_m)))) (* l (* d_m d_m)))
    w0)
   (* 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 ((pow(((M_m * D_m) / (d_m * 2.0)), 2.0) * (h / l)) <= -1e+33) {
		tmp = fma((D_m * D_m), ((-0.125 * (M_m * (h * (w0 * M_m)))) / (l * (d_m * d_m))), w0);
	} 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(Float64(M_m * D_m) / Float64(d_m * 2.0)) ^ 2.0) * Float64(h / l)) <= -1e+33)
		tmp = fma(Float64(D_m * D_m), Float64(Float64(-0.125 * Float64(M_m * Float64(h * Float64(w0 * M_m)))) / Float64(l * Float64(d_m * d_m))), w0);
	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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+33], N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(-0.125 * N[(M$95$m * N[(h * N[(w0 * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $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)) < -9.9999999999999995e32

   1. Initial program 68.5%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. 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 47.4%

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

   7. Applied rewrites 48.6%

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

   if -9.9999999999999995e32 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.3%

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

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{-0.125 \cdot \left(M \cdot \left(h \cdot \left(w0 \cdot M\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{d\_m \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\left(D\_m \cdot D\_m\right) \cdot \left(w0 \cdot \frac{M\_m \cdot \left(\left(M\_m \cdot h\right) \cdot -0.125\right)}{\ell \cdot \left(d\_m \cdot d\_m\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* d_m 2.0)) 2.0) (/ h l)) -2e+85)
   (* (* D_m D_m) (* w0 (/ (* M_m (* (* M_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 ((pow(((M_m * D_m) / (d_m * 2.0)), 2.0) * (h / l)) <= -2e+85) {
		tmp = (D_m * D_m) * (w0 * ((M_m * ((M_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 (((((m_m * d_m) / (d_m_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= (-2d+85)) then
        tmp = (d_m * d_m) * (w0 * ((m_m * ((m_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 ((Math.pow(((M_m * D_m) / (d_m * 2.0)), 2.0) * (h / l)) <= -2e+85) {
		tmp = (D_m * D_m) * (w0 * ((M_m * ((M_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 (math.pow(((M_m * D_m) / (d_m * 2.0)), 2.0) * (h / l)) <= -2e+85:
		tmp = (D_m * D_m) * (w0 * ((M_m * ((M_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(Float64(M_m * D_m) / Float64(d_m * 2.0)) ^ 2.0) * Float64(h / l)) <= -2e+85)
		tmp = Float64(Float64(D_m * D_m) * Float64(w0 * Float64(Float64(M_m * Float64(Float64(M_m * 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 (((((M_m * D_m) / (d_m * 2.0)) ^ 2.0) * (h / l)) <= -2e+85)
		tmp = (D_m * D_m) * (w0 * ((M_m * ((M_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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+85], N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(w0 * N[(N[(M$95$m * N[(N[(M$95$m * h), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m * d$95$m), $MachinePrecision]), $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)) < -2e85

   1. Initial program 67.3%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. 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 47.9%

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

   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 46.5%

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

   10. Applied rewrites 49.2%

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

   if -2e85 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \left(w0 \cdot \frac{M \cdot \left(\left(M \cdot h\right) \cdot -0.125\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.2% accurate, 1.6× speedup

Math

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

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)) (/ 1.0 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)) / (1.0 / 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)) / Float64(1.0 / 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] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.4%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
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 91.9%

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

5. Final simplification 91.9%

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

Alternative 12: 86.5% accurate, 1.9× speedup

Math

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

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))
    (/ (/ (* (* M_m D_m) h) (* d_m 2.0)) (- 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)), ((((M_m * D_m) * h) / (d_m * 2.0)) / -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(Float64(Float64(Float64(M_m * D_m) * h) / Float64(d_m * 2.0)) / Float64(-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[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(d$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.4%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
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. 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 90.4%

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

5. Final simplification 90.4%

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

Alternative 13: 84.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 5.1 \cdot 10^{+20}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M\_m \cdot D\_m}{\left(d\_m \cdot \ell\right) \cdot -4} \cdot \frac{M\_m \cdot D\_m}{d\_m}, h, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(D\_m \cdot h\right) \cdot \frac{M\_m \cdot M\_m}{\left(d\_m \cdot \left(d\_m \cdot \ell\right)\right) \cdot -4}, D\_m, 1\right)}\\ \end{array} \end{array} \]

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 5.1e+20)
   (*
    w0
    (sqrt
     (fma (* (/ (* M_m D_m) (* (* d_m l) -4.0)) (/ (* M_m D_m) d_m)) h 1.0)))
   (*
    w0
    (sqrt
     (fma (* (* D_m h) (/ (* M_m M_m) (* (* d_m (* d_m l)) -4.0))) D_m 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 <= 5.1e+20) {
		tmp = w0 * sqrt(fma((((M_m * D_m) / ((d_m * l) * -4.0)) * ((M_m * D_m) / d_m)), h, 1.0));
	} else {
		tmp = w0 * sqrt(fma(((D_m * h) * ((M_m * M_m) / ((d_m * (d_m * l)) * -4.0))), D_m, 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 <= 5.1e+20)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(M_m * D_m) / Float64(Float64(d_m * l) * -4.0)) * Float64(Float64(M_m * D_m) / d_m)), h, 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(D_m * h) * Float64(Float64(M_m * M_m) / Float64(Float64(d_m * Float64(d_m * l)) * -4.0))), D_m, 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, 5.1e+20], N[(w0 * N[Sqrt[N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(N[(d$95$m * l), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(D$95$m * h), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * D$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.1e20

   1. Initial program 84.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 91.9%

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

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

   6. Applied rewrites 78.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 87.9

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

   8. Applied rewrites 87.9%

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

   if 5.1e20 < M

   1. Initial program 77.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. *-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 46.8%

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

   7. Applied rewrites 46.8%

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

   9. Applied rewrites 57.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.1 \cdot 10^{+20}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{\left(d \cdot \ell\right) \cdot -4} \cdot \frac{M \cdot D}{d}, h, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(D \cdot h\right) \cdot \frac{M \cdot M}{\left(d \cdot \left(d \cdot \ell\right)\right) \cdot -4}, D, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.3% 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 83.4%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 68.5%

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

Reproduce

herbie shell --seed 2024231 
(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))))))