Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 89.2%

Time: 13.4s

Alternatives: 18

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.8% 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: 89.2% accurate, 2.0× speedup

Math

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

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

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

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

Derivation

1. Initial program 83.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 88.5%

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

5. Taylor expanded in h around 0

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

   1. associate-/r* N/A

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

   2. associate-*r/ N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 83.0

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

7. Applied rewrites 83.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 83.0

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

9. Applied rewrites 88.5%

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

10. Final simplification 88.5%

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

Alternative 2: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(M\_m \cdot D\_m\right) \cdot -0.5}{d}, \left(\frac{0.5}{d} \cdot M\_m\right) \cdot \left(\frac{h}{\ell} \cdot D\_m\right), 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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)) -2e-11)
   (*
    (sqrt
     (fma
      (/ (* (* M_m D_m) -0.5) d)
      (* (* (/ 0.5 d) M_m) (* (/ h l) D_m))
      1.0))
    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)) <= -2e-11) {
		tmp = sqrt(fma((((M_m * D_m) * -0.5) / d), (((0.5 / d) * M_m) * ((h / l) * D_m)), 1.0)) * 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)) <= -2e-11)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(M_m * D_m) * -0.5) / d), Float64(Float64(Float64(0.5 / d) * M_m) * Float64(Float64(h / l) * D_m)), 1.0)) * 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], -2e-11], N[(N[Sqrt[N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.99999999999999988e-11

   1. Initial program 65.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 65.2%

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

   if -1.99999999999999988e-11 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.2%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 97.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot -0.5}{d}, \left(\frac{0.5}{d} \cdot M\right) \cdot \left(\frac{h}{\ell} \cdot D\right), 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{D\_m}{d}, \left(\frac{-h}{\ell} \cdot 0.25\right) \cdot \left(\left(\frac{D\_m}{d} \cdot M\_m\right) \cdot M\_m\right), 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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)) -2e-11)
   (*
    (sqrt
     (fma (/ D_m d) (* (* (/ (- h) l) 0.25) (* (* (/ D_m d) M_m) M_m)) 1.0))
    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)) <= -2e-11) {
		tmp = sqrt(fma((D_m / d), (((-h / l) * 0.25) * (((D_m / d) * M_m) * M_m)), 1.0)) * 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)) <= -2e-11)
		tmp = Float64(sqrt(fma(Float64(D_m / d), Float64(Float64(Float64(Float64(-h) / l) * 0.25) * Float64(Float64(Float64(D_m / d) * M_m) * M_m)), 1.0)) * 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], -2e-11], N[(N[Sqrt[N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(N[((-h) / l), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.99999999999999988e-11

   1. Initial program 65.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 69.4%

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

   6. Applied rewrites 66.8%

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

   if -1.99999999999999988e-11 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.2%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 97.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{D}{d}, \left(\frac{-h}{\ell} \cdot 0.25\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right), 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -200:\\ \;\;\;\;\sqrt{\left(-0.25 \cdot h\right) \cdot \left(\left(\left(\frac{M\_m}{\ell} \cdot D\_m\right) \cdot \frac{M\_m}{d}\right) \cdot \frac{D\_m}{d}\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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)) -200.0)
   (* (sqrt (* (* -0.25 h) (* (* (* (/ M_m l) D_m) (/ M_m d)) (/ D_m d)))) 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)) <= -200.0) {
		tmp = sqrt(((-0.25 * h) * ((((M_m / l) * D_m) * (M_m / d)) * (D_m / d)))) * w0;
	} 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)) <= (-200.0d0)) then
        tmp = sqrt((((-0.25d0) * h) * ((((m_m / l) * d_m) * (m_m / d)) * (d_m / d)))) * w0
    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)) <= -200.0) {
		tmp = Math.sqrt(((-0.25 * h) * ((((M_m / l) * D_m) * (M_m / d)) * (D_m / d)))) * w0;
	} 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)) <= -200.0:
		tmp = math.sqrt(((-0.25 * h) * ((((M_m / l) * D_m) * (M_m / d)) * (D_m / d)))) * 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)) <= -200.0)
		tmp = Float64(sqrt(Float64(Float64(-0.25 * h) * Float64(Float64(Float64(Float64(M_m / l) * D_m) * Float64(M_m / d)) * Float64(D_m / d)))) * w0);
	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)) <= -200.0)
		tmp = sqrt(((-0.25 * h) * ((((M_m / l) * D_m) * (M_m / d)) * (D_m / d)))) * w0;
	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], -200.0], N[(N[Sqrt[N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(N[(M$95$m / l), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -200

   1. Initial program 65.0%

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

   3. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 50.0%

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

   9. Applied rewrites 49.6%

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

   11. Applied rewrites 62.4%

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

   if -200 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.3%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 97.1%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -200:\\ \;\;\;\;\sqrt{\left(-0.25 \cdot h\right) \cdot \left(\left(\left(\frac{M}{\ell} \cdot D\right) \cdot \frac{M}{d}\right) \cdot \frac{D}{d}\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -200:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{\frac{D\_m}{d}}{\ell} \cdot M\_m\right) \cdot M\_m\right) \cdot \frac{D\_m}{d}\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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)) -200.0)
   (* (sqrt (* (* (* (* (/ (/ D_m d) l) M_m) M_m) (/ D_m d)) (* -0.25 h))) 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)) <= -200.0) {
		tmp = sqrt(((((((D_m / d) / l) * M_m) * M_m) * (D_m / d)) * (-0.25 * h))) * w0;
	} 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)) <= (-200.0d0)) then
        tmp = sqrt(((((((d_m / d) / l) * m_m) * m_m) * (d_m / d)) * ((-0.25d0) * h))) * w0
    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)) <= -200.0) {
		tmp = Math.sqrt(((((((D_m / d) / l) * M_m) * M_m) * (D_m / d)) * (-0.25 * h))) * w0;
	} 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)) <= -200.0:
		tmp = math.sqrt(((((((D_m / d) / l) * M_m) * M_m) * (D_m / d)) * (-0.25 * h))) * 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)) <= -200.0)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(D_m / d) / l) * M_m) * M_m) * Float64(D_m / d)) * Float64(-0.25 * h))) * w0);
	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)) <= -200.0)
		tmp = sqrt(((((((D_m / d) / l) * M_m) * M_m) * (D_m / d)) * (-0.25 * h))) * w0;
	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], -200.0], N[(N[Sqrt[N[(N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] / l), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -200

   1. Initial program 65.0%

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

   3. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 50.0%

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

   9. Applied rewrites 60.3%

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

   if -200 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.3%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 97.1%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -200:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{\frac{D}{d}}{\ell} \cdot M\right) \cdot M\right) \cdot \frac{D}{d}\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -200:\\ \;\;\;\;\sqrt{\left(\frac{D\_m}{\ell \cdot d} \cdot \left(\left(\frac{D\_m}{d} \cdot M\_m\right) \cdot M\_m\right)\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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)) -200.0)
   (* (sqrt (* (* (/ D_m (* l d)) (* (* (/ D_m d) M_m) M_m)) (* -0.25 h))) 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)) <= -200.0) {
		tmp = sqrt((((D_m / (l * d)) * (((D_m / d) * M_m) * M_m)) * (-0.25 * h))) * w0;
	} 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)) <= (-200.0d0)) then
        tmp = sqrt((((d_m / (l * d)) * (((d_m / d) * m_m) * m_m)) * ((-0.25d0) * h))) * w0
    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)) <= -200.0) {
		tmp = Math.sqrt((((D_m / (l * d)) * (((D_m / d) * M_m) * M_m)) * (-0.25 * h))) * w0;
	} 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)) <= -200.0:
		tmp = math.sqrt((((D_m / (l * d)) * (((D_m / d) * M_m) * M_m)) * (-0.25 * h))) * 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)) <= -200.0)
		tmp = Float64(sqrt(Float64(Float64(Float64(D_m / Float64(l * d)) * Float64(Float64(Float64(D_m / d) * M_m) * M_m)) * Float64(-0.25 * h))) * w0);
	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)) <= -200.0)
		tmp = sqrt((((D_m / (l * d)) * (((D_m / d) * M_m) * M_m)) * (-0.25 * h))) * w0;
	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], -200.0], N[(N[Sqrt[N[(N[(N[(D$95$m / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -200

   1. Initial program 65.0%

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

   3. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 55.5%

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

   if -200 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.3%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 97.1%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -200:\\ \;\;\;\;\sqrt{\left(\frac{D}{\ell \cdot d} \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\left(\left(\left(\frac{\frac{M\_m}{\ell}}{d} \cdot h\right) \cdot \frac{M\_m}{d}\right) \cdot \left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right)\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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)) -5e+24)
   (* (* (* (* (/ (/ M_m l) d) h) (/ M_m d)) (* (* D_m D_m) -0.125)) 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)) <= -5e+24) {
		tmp = (((((M_m / l) / d) * h) * (M_m / d)) * ((D_m * D_m) * -0.125)) * w0;
	} 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)) <= (-5d+24)) then
        tmp = (((((m_m / l) / d) * h) * (m_m / d)) * ((d_m * d_m) * (-0.125d0))) * w0
    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)) <= -5e+24) {
		tmp = (((((M_m / l) / d) * h) * (M_m / d)) * ((D_m * D_m) * -0.125)) * w0;
	} 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)) <= -5e+24:
		tmp = (((((M_m / l) / d) * h) * (M_m / d)) * ((D_m * D_m) * -0.125)) * 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)) <= -5e+24)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M_m / l) / d) * h) * Float64(M_m / d)) * Float64(Float64(D_m * D_m) * -0.125)) * w0);
	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)) <= -5e+24)
		tmp = (((((M_m / l) / d) * h) * (M_m / d)) * ((D_m * D_m) * -0.125)) * w0;
	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], -5e+24], N[(N[(N[(N[(N[(N[(M$95$m / l), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.00000000000000045e24

   1. Initial program 63.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 67.7%

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

   5. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 40.7

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in h around inf

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

   10. Applied rewrites 42.0%

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

   12. Applied rewrites 51.4%

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

   if -5.00000000000000045e24 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 95.8%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\left(\left(\left(\frac{\frac{M}{\ell}}{d} \cdot h\right) \cdot \frac{M}{d}\right) \cdot \left(\left(D \cdot D\right) \cdot -0.125\right)\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -200:\\ \;\;\;\;\sqrt{\frac{\left(\left(M\_m \cdot D\_m\right) \cdot M\_m\right) \cdot D\_m}{\left(\ell \cdot d\right) \cdot d} \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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)) -200.0)
   (* (sqrt (* (/ (* (* (* M_m D_m) M_m) D_m) (* (* l d) d)) (* -0.25 h))) 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)) <= -200.0) {
		tmp = sqrt((((((M_m * D_m) * M_m) * D_m) / ((l * d) * d)) * (-0.25 * h))) * w0;
	} 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)) <= (-200.0d0)) then
        tmp = sqrt((((((m_m * d_m) * m_m) * d_m) / ((l * d) * d)) * ((-0.25d0) * h))) * w0
    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)) <= -200.0) {
		tmp = Math.sqrt((((((M_m * D_m) * M_m) * D_m) / ((l * d) * d)) * (-0.25 * h))) * w0;
	} 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)) <= -200.0:
		tmp = math.sqrt((((((M_m * D_m) * M_m) * D_m) / ((l * d) * d)) * (-0.25 * h))) * 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)) <= -200.0)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(M_m * D_m) * M_m) * D_m) / Float64(Float64(l * d) * d)) * Float64(-0.25 * h))) * w0);
	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)) <= -200.0)
		tmp = sqrt((((((M_m * D_m) * M_m) * D_m) / ((l * d) * d)) * (-0.25 * h))) * w0;
	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], -200.0], N[(N[Sqrt[N[(N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -200

   1. Initial program 65.0%

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

   3. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 46.2%

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

   9. Applied rewrites 50.3%

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

   if -200 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.3%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 97.1%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -200:\\ \;\;\;\;\sqrt{\frac{\left(\left(M \cdot D\right) \cdot M\right) \cdot D}{\left(\ell \cdot d\right) \cdot d} \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -200:\\ \;\;\;\;\sqrt{\left(\frac{M\_m \cdot D\_m}{\left(d \cdot d\right) \cdot \ell} \cdot \left(M\_m \cdot D\_m\right)\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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)) -200.0)
   (* (sqrt (* (* (/ (* M_m D_m) (* (* d d) l)) (* M_m D_m)) (* -0.25 h))) 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)) <= -200.0) {
		tmp = sqrt(((((M_m * D_m) / ((d * d) * l)) * (M_m * D_m)) * (-0.25 * h))) * w0;
	} 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)) <= (-200.0d0)) then
        tmp = sqrt(((((m_m * d_m) / ((d * d) * l)) * (m_m * d_m)) * ((-0.25d0) * h))) * w0
    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)) <= -200.0) {
		tmp = Math.sqrt(((((M_m * D_m) / ((d * d) * l)) * (M_m * D_m)) * (-0.25 * h))) * w0;
	} 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)) <= -200.0:
		tmp = math.sqrt(((((M_m * D_m) / ((d * d) * l)) * (M_m * D_m)) * (-0.25 * h))) * 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)) <= -200.0)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(M_m * D_m) / Float64(Float64(d * d) * l)) * Float64(M_m * D_m)) * Float64(-0.25 * h))) * w0);
	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)) <= -200.0)
		tmp = sqrt(((((M_m * D_m) / ((d * d) * l)) * (M_m * D_m)) * (-0.25 * h))) * w0;
	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], -200.0], N[(N[Sqrt[N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -200

   1. Initial program 65.0%

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

   3. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 46.2%

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

   9. Applied rewrites 50.0%

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

   if -200 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.3%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 97.1%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -200:\\ \;\;\;\;\sqrt{\left(\frac{M \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \left(M \cdot D\right)\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -200:\\ \;\;\;\;\sqrt{\left(\frac{\left(M\_m \cdot D\_m\right) \cdot M\_m}{\left(d \cdot d\right) \cdot \ell} \cdot D\_m\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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)) -200.0)
   (* (sqrt (* (* (/ (* (* M_m D_m) M_m) (* (* d d) l)) D_m) (* -0.25 h))) 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)) <= -200.0) {
		tmp = sqrt((((((M_m * D_m) * M_m) / ((d * d) * l)) * D_m) * (-0.25 * h))) * w0;
	} 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)) <= (-200.0d0)) then
        tmp = sqrt((((((m_m * d_m) * m_m) / ((d * d) * l)) * d_m) * ((-0.25d0) * h))) * w0
    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)) <= -200.0) {
		tmp = Math.sqrt((((((M_m * D_m) * M_m) / ((d * d) * l)) * D_m) * (-0.25 * h))) * w0;
	} 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)) <= -200.0:
		tmp = math.sqrt((((((M_m * D_m) * M_m) / ((d * d) * l)) * D_m) * (-0.25 * h))) * 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)) <= -200.0)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(M_m * D_m) * M_m) / Float64(Float64(d * d) * l)) * D_m) * Float64(-0.25 * h))) * w0);
	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)) <= -200.0)
		tmp = sqrt((((((M_m * D_m) * M_m) / ((d * d) * l)) * D_m) * (-0.25 * h))) * w0;
	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], -200.0], N[(N[Sqrt[N[(N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.25 * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -200

   1. Initial program 65.0%

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

   3. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 46.2%

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

   9. Applied rewrites 49.0%

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

   if -200 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.3%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 97.1%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -200:\\ \;\;\;\;\sqrt{\left(\frac{\left(M \cdot D\right) \cdot M}{\left(d \cdot d\right) \cdot \ell} \cdot D\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot h, \frac{\left(D\_m \cdot D\_m\right) \cdot M\_m}{\ell \cdot d} \cdot \frac{M\_m}{d}, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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)) -2e-11)
   (* (fma (* -0.125 h) (* (/ (* (* D_m D_m) M_m) (* l d)) (/ M_m d)) 1.0) 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)) <= -2e-11) {
		tmp = fma((-0.125 * h), ((((D_m * D_m) * M_m) / (l * d)) * (M_m / d)), 1.0) * 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)) <= -2e-11)
		tmp = Float64(fma(Float64(-0.125 * h), Float64(Float64(Float64(Float64(D_m * D_m) * M_m) / Float64(l * d)) * Float64(M_m / d)), 1.0) * 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], -2e-11], N[(N[(N[(-0.125 * h), $MachinePrecision] * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.99999999999999988e-11

   1. Initial program 65.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 38.9

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

   7. Applied rewrites 38.9%

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

   8. Taylor expanded in h around inf

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

   10. Applied rewrites 40.5%

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

   12. Applied rewrites 45.8%

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

   if -1.99999999999999988e-11 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.2%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 97.3%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot h, \frac{\left(D \cdot D\right) \cdot M}{\ell \cdot d} \cdot \frac{M}{d}, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\left(\left(\left(\frac{h}{d \cdot d} \cdot M\_m\right) \cdot \frac{M\_m}{\ell}\right) \cdot \left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right)\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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)) -5e+24)
   (* (* (* (* (/ h (* d d)) M_m) (/ M_m l)) (* (* D_m D_m) -0.125)) 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)) <= -5e+24) {
		tmp = ((((h / (d * d)) * M_m) * (M_m / l)) * ((D_m * D_m) * -0.125)) * w0;
	} 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)) <= (-5d+24)) then
        tmp = ((((h / (d * d)) * m_m) * (m_m / l)) * ((d_m * d_m) * (-0.125d0))) * w0
    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)) <= -5e+24) {
		tmp = ((((h / (d * d)) * M_m) * (M_m / l)) * ((D_m * D_m) * -0.125)) * w0;
	} 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)) <= -5e+24:
		tmp = ((((h / (d * d)) * M_m) * (M_m / l)) * ((D_m * D_m) * -0.125)) * 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)) <= -5e+24)
		tmp = Float64(Float64(Float64(Float64(Float64(h / Float64(d * d)) * M_m) * Float64(M_m / l)) * Float64(Float64(D_m * D_m) * -0.125)) * w0);
	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)) <= -5e+24)
		tmp = ((((h / (d * d)) * M_m) * (M_m / l)) * ((D_m * D_m) * -0.125)) * w0;
	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], -5e+24], N[(N[(N[(N[(N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.00000000000000045e24

   1. Initial program 63.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 67.7%

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

   5. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 40.7

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in h around inf

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

   10. Applied rewrites 42.0%

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

   12. Applied rewrites 45.0%

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

   if -5.00000000000000045e24 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 95.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\left(\left(\left(\frac{h}{d \cdot d} \cdot M\right) \cdot \frac{M}{\ell}\right) \cdot \left(\left(D \cdot D\right) \cdot -0.125\right)\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot h, \frac{\left(\left(M\_m \cdot D\_m\right) \cdot M\_m\right) \cdot D\_m}{\left(\ell \cdot d\right) \cdot d}, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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)) -2e+41)
   (* (fma (* -0.125 h) (/ (* (* (* M_m D_m) M_m) D_m) (* (* l d) d)) 1.0) 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)) <= -2e+41) {
		tmp = fma((-0.125 * h), ((((M_m * D_m) * M_m) * D_m) / ((l * d) * d)), 1.0) * 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)) <= -2e+41)
		tmp = Float64(fma(Float64(-0.125 * h), Float64(Float64(Float64(Float64(M_m * D_m) * M_m) * D_m) / Float64(Float64(l * d) * d)), 1.0) * 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], -2e+41], N[(N[(N[(-0.125 * h), $MachinePrecision] * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] + 1.0), $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)) < -2.00000000000000001e41

   1. Initial program 61.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 65.8%

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

   5. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 43.1

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

   7. Applied rewrites 43.1%

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

   8. Taylor expanded in h around inf

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

   10. Applied rewrites 44.8%

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

   12. Applied rewrites 48.0%

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

   if -2.00000000000000001e41 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.5%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 94.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot h, \frac{\left(\left(M \cdot D\right) \cdot M\right) \cdot D}{\left(\ell \cdot d\right) \cdot d}, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot h, \frac{\left(\left(D\_m \cdot D\_m\right) \cdot M\_m\right) \cdot M\_m}{\left(\ell \cdot d\right) \cdot d}, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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)) -2e+41)
   (* (fma (* -0.125 h) (/ (* (* (* D_m D_m) M_m) M_m) (* (* l d) d)) 1.0) 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)) <= -2e+41) {
		tmp = fma((-0.125 * h), ((((D_m * D_m) * M_m) * M_m) / ((l * d) * d)), 1.0) * 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)) <= -2e+41)
		tmp = Float64(fma(Float64(-0.125 * h), Float64(Float64(Float64(Float64(D_m * D_m) * M_m) * M_m) / Float64(Float64(l * d) * d)), 1.0) * 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], -2e+41], N[(N[(N[(-0.125 * h), $MachinePrecision] * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] + 1.0), $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)) < -2.00000000000000001e41

   1. Initial program 61.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 65.8%

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

   5. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 43.1

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

   7. Applied rewrites 43.1%

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

   8. Taylor expanded in h around inf

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

   10. Applied rewrites 44.8%

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

   12. Applied rewrites 48.0%

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

   if -2.00000000000000001e41 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.5%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 94.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}{\left(\ell \cdot d\right) \cdot d}, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot h, \frac{\left(D\_m \cdot D\_m\right) \cdot M\_m}{\left(d \cdot d\right) \cdot \ell} \cdot M\_m, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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)) -5e+24)
   (* (fma (* -0.125 h) (* (/ (* (* D_m D_m) M_m) (* (* d d) l)) M_m) 1.0) 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)) <= -5e+24) {
		tmp = fma((-0.125 * h), ((((D_m * D_m) * M_m) / ((d * d) * l)) * M_m), 1.0) * 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)) <= -5e+24)
		tmp = Float64(fma(Float64(-0.125 * h), Float64(Float64(Float64(Float64(D_m * D_m) * M_m) / Float64(Float64(d * d) * l)) * M_m), 1.0) * 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], -5e+24], N[(N[(N[(-0.125 * h), $MachinePrecision] * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] + 1.0), $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)) < -5.00000000000000045e24

   1. Initial program 63.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 67.7%

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

   5. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 40.7

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in h around inf

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

   10. Applied rewrites 42.4%

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

   12. Applied rewrites 44.0%

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

   if -5.00000000000000045e24 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 95.8%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot h, \frac{\left(D \cdot D\right) \cdot M}{\left(d \cdot d\right) \cdot \ell} \cdot M, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+274}:\\ \;\;\;\;\left(\left(\frac{h}{\left(d \cdot d\right) \cdot \ell} \cdot \left(M\_m \cdot M\_m\right)\right) \cdot \left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right)\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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)) -2e+274)
   (* (* (* (/ h (* (* d d) l)) (* M_m M_m)) (* (* D_m D_m) -0.125)) 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)) <= -2e+274) {
		tmp = (((h / ((d * d) * l)) * (M_m * M_m)) * ((D_m * D_m) * -0.125)) * w0;
	} 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)) <= (-2d+274)) then
        tmp = (((h / ((d * d) * l)) * (m_m * m_m)) * ((d_m * d_m) * (-0.125d0))) * w0
    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)) <= -2e+274) {
		tmp = (((h / ((d * d) * l)) * (M_m * M_m)) * ((D_m * D_m) * -0.125)) * w0;
	} 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)) <= -2e+274:
		tmp = (((h / ((d * d) * l)) * (M_m * M_m)) * ((D_m * D_m) * -0.125)) * 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)) <= -2e+274)
		tmp = Float64(Float64(Float64(Float64(h / Float64(Float64(d * d) * l)) * Float64(M_m * M_m)) * Float64(Float64(D_m * D_m) * -0.125)) * w0);
	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)) <= -2e+274)
		tmp = (((h / ((d * d) * l)) * (M_m * M_m)) * ((D_m * D_m) * -0.125)) * w0;
	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], -2e+274], N[(N[(N[(N[(h / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.99999999999999984e274

   1. Initial program 58.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 63.1%

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

   5. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 46.3

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

   7. Applied rewrites 46.3%

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

   8. Taylor expanded in h around inf

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

   10. Applied rewrites 48.0%

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

   12. Applied rewrites 46.3%

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

   if -1.99999999999999984e274 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.8%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 91.8%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+274}:\\ \;\;\;\;\left(\left(\frac{h}{\left(d \cdot d\right) \cdot \ell} \cdot \left(M \cdot M\right)\right) \cdot \left(\left(D \cdot D\right) \cdot -0.125\right)\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 86.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 10^{+93}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\_m\right) \cdot D\_m, \frac{\frac{\left(h \cdot M\_m\right) \cdot D\_m}{d} \cdot -0.5}{\ell}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{D\_m}{d}, \left(\frac{-h}{\ell} \cdot 0.25\right) \cdot \left(\left(\frac{D\_m}{d} \cdot M\_m\right) \cdot M\_m\right), 1\right)} \cdot w0\\ \end{array} \end{array} \]

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+93)
   (*
    (sqrt
     (fma
      (* (* (/ 0.5 d) M_m) D_m)
      (/ (* (/ (* (* h M_m) D_m) d) -0.5) l)
      1.0))
    w0)
   (*
    (sqrt
     (fma (/ D_m d) (* (* (/ (- h) l) 0.25) (* (* (/ D_m d) M_m) M_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+93) {
		tmp = sqrt(fma((((0.5 / d) * M_m) * D_m), (((((h * M_m) * D_m) / d) * -0.5) / l), 1.0)) * w0;
	} else {
		tmp = sqrt(fma((D_m / d), (((-h / l) * 0.25) * (((D_m / d) * M_m) * M_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+93)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(0.5 / d) * M_m) * D_m), Float64(Float64(Float64(Float64(Float64(h * M_m) * D_m) / d) * -0.5) / l), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(D_m / d), Float64(Float64(Float64(Float64(-h) / l) * 0.25) * Float64(Float64(Float64(D_m / d) * M_m) * M_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+93], N[(N[Sqrt[N[(N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(N[(N[(h * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / d), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(N[((-h) / l), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1.00000000000000004e93

   1. Initial program 86.6%

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

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 91.5%

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

   5. Taylor expanded in h around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 87.1

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

   7. Applied rewrites 87.1%

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

   if 1.00000000000000004e93 < (*.f64 M D)

   1. Initial program 69.6%

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

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 74.1%

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

   6. Applied rewrites 71.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{+93}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \frac{\frac{\left(h \cdot M\right) \cdot D}{d} \cdot -0.5}{\ell}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{D}{d}, \left(\frac{-h}{\ell} \cdot 0.25\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right), 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 68.1% 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 83.7%

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

3. Taylor expanded in h around 0

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

5. Applied rewrites 70.7%

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

6. Final simplification 70.7%

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

Reproduce

herbie shell --seed 2024242 
(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))))))