Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 89.9%

Time: 13.8s

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: 81.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.0%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
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 88.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 88.9

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

6. Applied rewrites 88.9%

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

7. Final simplification 88.9%

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

Alternative 2: 88.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{\left(M\_m \cdot D\_m\right) \cdot -0.5}{d}\\ \mathbf{if}\;\sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}} \cdot w0 \leq 10^{+241}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(t\_0 \cdot \frac{h}{\ell}, \left(\frac{0.5}{d} \cdot M\_m\right) \cdot D\_m, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{t\_0}{\ell}, \left(\left(M\_m \cdot D\_m\right) \cdot h\right) \cdot \frac{0.5}{d}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (/ (* (* M_m D_m) -0.5) d)))
   (if (<=
        (* (sqrt (- 1.0 (* (/ h l) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)))) w0)
        1e+241)
     (* (sqrt (fma (* t_0 (/ h l)) (* (* (/ 0.5 d) M_m) D_m) 1.0)) w0)
     (* (sqrt (fma (/ t_0 l) (* (* (* M_m D_m) h) (/ 0.5 d)) 1.0)) w0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = ((M_m * D_m) * -0.5) / d;
	double tmp;
	if ((sqrt((1.0 - ((h / l) * pow(((M_m * D_m) / (2.0 * d)), 2.0)))) * w0) <= 1e+241) {
		tmp = sqrt(fma((t_0 * (h / l)), (((0.5 / d) * M_m) * D_m), 1.0)) * w0;
	} else {
		tmp = sqrt(fma((t_0 / l), (((M_m * D_m) * h) * (0.5 / d)), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(Float64(Float64(M_m * D_m) * -0.5) / d)
	tmp = 0.0
	if (Float64(sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)))) * w0) <= 1e+241)
		tmp = Float64(sqrt(fma(Float64(t_0 * Float64(h / l)), Float64(Float64(Float64(0.5 / d) * M_m) * D_m), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(t_0 / l), Float64(Float64(Float64(M_m * D_m) * h) * Float64(0.5 / d)), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], 1e+241], N[(N[Sqrt[N[(N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(t$95$0 / l), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.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))))) < 1.0000000000000001e241

   1. Initial program 92.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 91.7%

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

   if 1.0000000000000001e241 < (*.f64 w0 (sqrt.f64 (-.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 34.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 74.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. /-rgt-identity N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 74.3

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

   6. Applied rewrites 74.3%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot w0 \leq 10^{+241}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot -0.5}{d} \cdot \frac{h}{\ell}, \left(\frac{0.5}{d} \cdot M\right) \cdot D, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{\left(M \cdot D\right) \cdot -0.5}{d}}{\ell}, \left(\left(M \cdot D\right) \cdot h\right) \cdot \frac{0.5}{d}, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}} \cdot w0 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(M\_m \cdot D\_m\right) \cdot -0.5}{d} \cdot \frac{h}{\ell}, \left(\frac{0.5}{d} \cdot M\_m\right) \cdot D\_m, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{M\_m \cdot -0.5}{d} \cdot D\_m}{\ell}, \frac{\left(h \cdot D\_m\right) \cdot \left(0.5 \cdot M\_m\right)}{d}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<=
      (* (sqrt (- 1.0 (* (/ h l) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)))) w0)
      2e+301)
   (*
    (sqrt
     (fma
      (* (/ (* (* M_m D_m) -0.5) d) (/ h l))
      (* (* (/ 0.5 d) M_m) D_m)
      1.0))
    w0)
   (*
    (sqrt
     (fma
      (/ (* (/ (* M_m -0.5) d) D_m) l)
      (/ (* (* h D_m) (* 0.5 M_m)) d)
      1.0))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((sqrt((1.0 - ((h / l) * pow(((M_m * D_m) / (2.0 * d)), 2.0)))) * w0) <= 2e+301) {
		tmp = sqrt(fma(((((M_m * D_m) * -0.5) / d) * (h / l)), (((0.5 / d) * M_m) * D_m), 1.0)) * w0;
	} else {
		tmp = sqrt(fma(((((M_m * -0.5) / d) * D_m) / l), (((h * D_m) * (0.5 * M_m)) / d), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)))) * w0) <= 2e+301)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(M_m * D_m) * -0.5) / d) * Float64(h / l)), Float64(Float64(Float64(0.5 / d) * M_m) * D_m), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(M_m * -0.5) / d) * D_m) / l), Float64(Float64(Float64(h * D_m) * Float64(0.5 * M_m)) / d), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], 2e+301], N[(N[Sqrt[N[(N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / d), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(M$95$m * -0.5), $MachinePrecision] / d), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(h * D$95$m), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.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))))) < 2.00000000000000011e301

   1. Initial program 92.4%

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

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

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

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 91.7%

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

   if 2.00000000000000011e301 < (*.f64 w0 (sqrt.f64 (-.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 24.0%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 72.0

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

   6. Applied rewrites 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. div-inv N/A

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

      13. lift-/.f64 N/A

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

      14. remove-double-div N/A

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

      15. lower-*.f64 70.0

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

   8. Applied rewrites 70.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot w0 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot -0.5}{d} \cdot \frac{h}{\ell}, \left(\frac{0.5}{d} \cdot M\right) \cdot D, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{M \cdot -0.5}{d} \cdot D}{\ell}, \frac{\left(h \cdot D\right) \cdot \left(0.5 \cdot M\right)}{d}, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{0.5}{d} \cdot M\_m\\ \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+248}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(M\_m \cdot D\_m\right) \cdot -0.5}{d} \cdot \frac{h}{\ell}, t\_0 \cdot D\_m, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(D\_m, \left(t\_0 \cdot \left(h \cdot D\_m\right)\right) \cdot \frac{M\_m \cdot -0.5}{\ell \cdot d}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* (/ 0.5 d) M_m)))
   (if (<= (- 1.0 (* (/ h l) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))) 2e+248)
     (* (sqrt (fma (* (/ (* (* M_m D_m) -0.5) d) (/ h l)) (* t_0 D_m) 1.0)) w0)
     (*
      (sqrt (fma D_m (* (* t_0 (* h D_m)) (/ (* M_m -0.5) (* l d))) 1.0))
      w0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (0.5 / d) * M_m;
	double tmp;
	if ((1.0 - ((h / l) * pow(((M_m * D_m) / (2.0 * d)), 2.0))) <= 2e+248) {
		tmp = sqrt(fma(((((M_m * D_m) * -0.5) / d) * (h / l)), (t_0 * D_m), 1.0)) * w0;
	} else {
		tmp = sqrt(fma(D_m, ((t_0 * (h * D_m)) * ((M_m * -0.5) / (l * d))), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(Float64(0.5 / d) * M_m)
	tmp = 0.0
	if (Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0))) <= 2e+248)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(M_m * D_m) * -0.5) / d) * Float64(h / l)), Float64(t_0 * D_m), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(D_m, Float64(Float64(t_0 * Float64(h * D_m)) * Float64(Float64(M_m * -0.5) / Float64(l * d))), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+248], N[(N[Sqrt[N[(N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / d), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(D$95$m * N[(N[(t$95$0 * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * -0.5), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $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))) < 2.00000000000000009e248

   1. Initial program 99.9%

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

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

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

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 98.9%

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

   if 2.00000000000000009e248 < (-.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 34.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 70.2

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

   6. Applied rewrites 70.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 59.8%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+248}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot -0.5}{d} \cdot \frac{h}{\ell}, \left(\frac{0.5}{d} \cdot M\right) \cdot D, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(D, \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \left(h \cdot D\right)\right) \cdot \frac{M \cdot -0.5}{\ell \cdot d}, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq 10^{+249}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(0.25 \cdot M\_m\right) \cdot \left(\frac{D\_m}{d} \cdot M\_m\right)\right) \cdot \frac{-h}{\ell}, \frac{D\_m}{d}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(D\_m, \left(\left(\frac{0.5}{d} \cdot M\_m\right) \cdot \left(h \cdot D\_m\right)\right) \cdot \frac{M\_m \cdot -0.5}{\ell \cdot d}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (- 1.0 (* (/ h l) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))) 1e+249)
   (*
    (sqrt
     (fma (* (* (* 0.25 M_m) (* (/ D_m d) M_m)) (/ (- h) l)) (/ D_m d) 1.0))
    w0)
   (*
    (sqrt
     (fma
      D_m
      (* (* (* (/ 0.5 d) M_m) (* h D_m)) (/ (* M_m -0.5) (* l d)))
      1.0))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((1.0 - ((h / l) * pow(((M_m * D_m) / (2.0 * d)), 2.0))) <= 1e+249) {
		tmp = sqrt(fma((((0.25 * M_m) * ((D_m / d) * M_m)) * (-h / l)), (D_m / d), 1.0)) * w0;
	} else {
		tmp = sqrt(fma(D_m, ((((0.5 / d) * M_m) * (h * D_m)) * ((M_m * -0.5) / (l * d))), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0))) <= 1e+249)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(0.25 * M_m) * Float64(Float64(D_m / d) * M_m)) * Float64(Float64(-h) / l)), Float64(D_m / d), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(D_m, Float64(Float64(Float64(Float64(0.5 / d) * M_m) * Float64(h * D_m)) * Float64(Float64(M_m * -0.5) / Float64(l * d))), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+249], N[(N[Sqrt[N[(N[(N[(N[(0.25 * M$95$m), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * N[((-h) / l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(D$95$m * N[(N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * -0.5), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $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))) < 9.9999999999999992e248

   1. Initial program 99.9%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. Applied rewrites 84.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.0

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

   6. Applied rewrites 97.0%

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

   if 9.9999999999999992e248 < (-.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 33.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 68.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 69.9

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

   6. Applied rewrites 69.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 60.5%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 10^{+249}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(0.25 \cdot M\right) \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot \frac{-h}{\ell}, \frac{D}{d}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(D, \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \left(h \cdot D\right)\right) \cdot \frac{M \cdot -0.5}{\ell \cdot d}, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(-0.25 * N[(N[(h * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $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))) < 2

   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 h around 0

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

   5. Applied rewrites 99.8%

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

   if 2 < (-.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 45.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. Applied rewrites 37.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 51.3

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

   6. Applied rewrites 51.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. frac-times N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*r* N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

   8. Applied rewrites 53.3%

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

   9. Taylor expanded in h around 0

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 55.1

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

   11. Applied rewrites 55.1%

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

4. Final simplification 81.8%

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

Alternative 7: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D\_m \cdot D\_m\right), \left(\left(\frac{h}{\left(\ell \cdot d\right) \cdot d} \cdot M\_m\right) \cdot M\_m\right) \cdot w0, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) -1e+105)
   (fma (* -0.125 (* D_m D_m)) (* (* (* (/ h (* (* l d) d)) M_m) M_m) w0) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) <= -1e+105) {
		tmp = fma((-0.125 * (D_m * D_m)), ((((h / ((l * d) * d)) * M_m) * M_m) * w0), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) <= -1e+105)
		tmp = fma(Float64(-0.125 * Float64(D_m * D_m)), Float64(Float64(Float64(Float64(h / Float64(Float64(l * d) * d)) * M_m) * M_m) * w0), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e+105], N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(h / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * w0), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.9999999999999994e104

   1. Initial program 54.0%

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

   3. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 31.2%

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

   7. Applied rewrites 39.2%

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

   if -9.9999999999999994e104 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.7%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 90.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \left(\left(\frac{h}{\left(\ell \cdot d\right) \cdot d} \cdot M\right) \cdot M\right) \cdot w0, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+105}:\\ \;\;\;\;\left(\left(\left(\frac{h}{\left(d \cdot d\right) \cdot \ell} \cdot M\_m\right) \cdot M\_m\right) \cdot w0\right) \cdot \left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) -1e+105)
   (* (* (* (* (/ h (* (* d d) l)) M_m) M_m) w0) (* -0.125 (* D_m D_m)))
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) <= -1e+105) {
		tmp = ((((h / ((d * d) * l)) * M_m) * M_m) * w0) * (-0.125 * (D_m * D_m));
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((h / l) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) <= (-1d+105)) then
        tmp = ((((h / ((d * d) * l)) * m_m) * m_m) * w0) * ((-0.125d0) * (d_m * d_m))
    else
        tmp = 1.0d0 * w0
    end if
    code = tmp
end function

Java

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;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) <= -1e+105) {
		tmp = ((((h / ((d * d) * l)) * M_m) * M_m) * w0) * (-0.125 * (D_m * D_m));
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if ((h / l) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) <= -1e+105:
		tmp = ((((h / ((d * d) * l)) * M_m) * M_m) * w0) * (-0.125 * (D_m * D_m))
	else:
		tmp = 1.0 * w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) <= -1e+105)
		tmp = Float64(Float64(Float64(Float64(Float64(h / Float64(Float64(d * d) * l)) * M_m) * M_m) * w0) * Float64(-0.125 * Float64(D_m * D_m)));
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((h / l) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) <= -1e+105)
		tmp = ((((h / ((d * d) * l)) * M_m) * M_m) * w0) * (-0.125 * (D_m * D_m));
	else
		tmp = 1.0 * w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e+105], N[(N[(N[(N[(N[(h / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * w0), $MachinePrecision] * N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.9999999999999994e104

   1. Initial program 54.0%

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

   3. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in h 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 31.2%

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

   10. Applied rewrites 37.6%

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

   if -9.9999999999999994e104 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.7%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 90.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+105}:\\ \;\;\;\;\left(\left(\left(\frac{h}{\left(d \cdot d\right) \cdot \ell} \cdot M\right) \cdot M\right) \cdot w0\right) \cdot \left(-0.125 \cdot \left(D \cdot D\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d} \leq 4 \cdot 10^{-67}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(D\_m, \left(\left(\frac{0.5}{d} \cdot M\_m\right) \cdot \left(h \cdot D\_m\right)\right) \cdot \frac{M\_m \cdot -0.5}{\ell \cdot d}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ (* M_m D_m) (* 2.0 d)) 4e-67)
   (* 1.0 w0)
   (*
    (sqrt
     (fma
      D_m
      (* (* (* (/ 0.5 d) M_m) (* h D_m)) (/ (* M_m -0.5) (* l d)))
      1.0))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d)) <= 4e-67) {
		tmp = 1.0 * w0;
	} else {
		tmp = sqrt(fma(D_m, ((((0.5 / d) * M_m) * (h * D_m)) * ((M_m * -0.5) / (l * d))), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) <= 4e-67)
		tmp = Float64(1.0 * w0);
	else
		tmp = Float64(sqrt(fma(D_m, Float64(Float64(Float64(Float64(0.5 / d) * M_m) * Float64(h * D_m)) * Float64(Float64(M_m * -0.5) / Float64(l * d))), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 4e-67], N[(1.0 * w0), $MachinePrecision], N[(N[Sqrt[N[(D$95$m * N[(N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * -0.5), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 3.99999999999999977e-67

   1. Initial program 81.8%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 76.0%

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

   if 3.99999999999999977e-67 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 61.1%

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

      1. lift--.f64 N/A

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

      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 65.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 67.5

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

   6. Applied rewrites 67.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 63.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 4 \cdot 10^{-67}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(D, \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \left(h \cdot D\right)\right) \cdot \frac{M \cdot -0.5}{\ell \cdot d}, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 10^{-193}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D\_m \cdot D\_m\right), \frac{\frac{\frac{\left(M\_m \cdot M\_m\right) \cdot h}{\ell} \cdot w0}{d}}{d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(-0.25 \cdot D\_m\right) \cdot \left(\left(\frac{h}{\left(\ell \cdot d\right) \cdot d} \cdot M\_m\right) \cdot M\_m\right), D\_m, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 1e-193)
   (fma (* -0.125 (* D_m D_m)) (/ (/ (* (/ (* (* M_m M_m) h) l) w0) d) d) w0)
   (*
    (sqrt (fma (* (* -0.25 D_m) (* (* (/ h (* (* l d) d)) M_m) M_m)) D_m 1.0))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 1e-193) {
		tmp = fma((-0.125 * (D_m * D_m)), ((((((M_m * M_m) * h) / l) * w0) / d) / d), w0);
	} else {
		tmp = sqrt(fma(((-0.25 * D_m) * (((h / ((l * d) * d)) * M_m) * M_m)), D_m, 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 1e-193)
		tmp = fma(Float64(-0.125 * Float64(D_m * D_m)), Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) / l) * w0) / d) / d), w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(-0.25 * D_m) * Float64(Float64(Float64(h / Float64(Float64(l * d) * d)) * M_m) * M_m)), D_m, 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e-193], N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] * w0), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] + w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(-0.25 * D$95$m), $MachinePrecision] * N[(N[(N[(h / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1e-193

   1. Initial program 76.4%

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

   3. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 46.7%

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

   7. Applied rewrites 57.5%

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

   if 1e-193 < (*.f64 M D)

   1. Initial program 81.7%

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

   3. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 48.1

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

   5. Applied rewrites 48.1%

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

   7. Applied rewrites 52.1%

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

   9. Applied rewrites 74.8%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-193}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{\frac{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot w0}{d}}{d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(-0.25 \cdot D\right) \cdot \left(\left(\frac{h}{\left(\ell \cdot d\right) \cdot d} \cdot M\right) \cdot M\right), D, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 9.5 \cdot 10^{-208}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(\left(h \cdot M\_m\right) \cdot M\_m\right) \cdot -0.25\right) \cdot D\_m}{\ell \cdot d}, \frac{D\_m}{d}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 9.5e-208)
   (* 1.0 w0)
   (*
    (sqrt (fma (/ (* (* (* (* h M_m) M_m) -0.25) D_m) (* l d)) (/ D_m d) 1.0))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 9.5e-208) {
		tmp = 1.0 * w0;
	} else {
		tmp = sqrt(fma((((((h * M_m) * M_m) * -0.25) * D_m) / (l * d)), (D_m / d), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 9.5e-208)
		tmp = Float64(1.0 * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(Float64(h * M_m) * M_m) * -0.25) * D_m) / Float64(l * d)), Float64(D_m / d), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 9.5e-208], N[(1.0 * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(N[(h * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * -0.25), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 9.5000000000000001e-208

   1. Initial program 79.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 72.0%

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

   if 9.5000000000000001e-208 < M

   1. Initial program 75.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. Applied rewrites 63.8%

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

   5. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 66.2

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

   7. Applied rewrites 66.2%

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

   9. Applied rewrites 73.7%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9.5 \cdot 10^{-208}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(\left(h \cdot M\right) \cdot M\right) \cdot -0.25\right) \cdot D}{\ell \cdot d}, \frac{D}{d}, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 10^{-193}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(-0.25 \cdot D\_m\right) \cdot \left(\left(\frac{h}{\left(\ell \cdot d\right) \cdot d} \cdot M\_m\right) \cdot M\_m\right), D\_m, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 1e-193)
   (* 1.0 w0)
   (*
    (sqrt (fma (* (* -0.25 D_m) (* (* (/ h (* (* l d) d)) M_m) M_m)) D_m 1.0))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 1e-193) {
		tmp = 1.0 * w0;
	} else {
		tmp = sqrt(fma(((-0.25 * D_m) * (((h / ((l * d) * d)) * M_m) * M_m)), D_m, 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 1e-193)
		tmp = Float64(1.0 * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(-0.25 * D_m) * Float64(Float64(Float64(h / Float64(Float64(l * d) * d)) * M_m) * M_m)), D_m, 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e-193], N[(1.0 * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(-0.25 * D$95$m), $MachinePrecision] * N[(N[(N[(h / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1e-193

   1. Initial program 76.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 72.8%

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

   if 1e-193 < (*.f64 M D)

   1. Initial program 81.7%

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

   3. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 48.1

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

   5. Applied rewrites 48.1%

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

   7. Applied rewrites 52.1%

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

   9. Applied rewrites 74.8%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-193}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(-0.25 \cdot D\right) \cdot \left(\left(\frac{h}{\left(\ell \cdot d\right) \cdot d} \cdot M\right) \cdot M\right), D, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 10^{-193}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\frac{h}{\left(d \cdot d\right) \cdot \ell} \cdot M\_m\right) \cdot M\_m\right) \cdot \left(-0.25 \cdot D\_m\right), D\_m, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 1e-193)
   (* 1.0 w0)
   (*
    (sqrt (fma (* (* (* (/ h (* (* d d) l)) M_m) M_m) (* -0.25 D_m)) D_m 1.0))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 1e-193) {
		tmp = 1.0 * w0;
	} else {
		tmp = sqrt(fma(((((h / ((d * d) * l)) * M_m) * M_m) * (-0.25 * D_m)), D_m, 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 1e-193)
		tmp = Float64(1.0 * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(h / Float64(Float64(d * d) * l)) * M_m) * M_m) * Float64(-0.25 * D_m)), D_m, 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e-193], N[(1.0 * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(h / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(-0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1e-193

   1. Initial program 76.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 72.8%

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

   if 1e-193 < (*.f64 M D)

   1. Initial program 81.7%

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

   3. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 48.1

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

   5. Applied rewrites 48.1%

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

   7. Applied rewrites 70.9%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-193}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\frac{h}{\left(d \cdot d\right) \cdot \ell} \cdot M\right) \cdot M\right) \cdot \left(-0.25 \cdot D\right), D, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.8% accurate, 26.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return 1.0 * w0;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in h around 0

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

5. Applied rewrites 67.8%

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

6. Final simplification 67.8%

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

Reproduce

herbie shell --seed 2024240 
(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))))))