Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.2% → 86.8%

Time: 15.0s

Alternatives: 13

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M\_m \cdot D\_m}{d \cdot -2}, \left(M\_m \cdot D\_m\right) \cdot \left(\frac{h}{\ell} \cdot \frac{0.5}{d}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, \left(D\_m \cdot -0.125\right) \cdot \frac{\frac{M\_m \cdot \frac{M\_m \cdot h}{\ell}}{d}}{d}, 1\right)\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e-16)
   (*
    w0
    (sqrt
     (fma
      (/ (* M_m D_m) (* d -2.0))
      (* (* M_m D_m) (* (/ h l) (/ 0.5 d)))
      1.0)))
   (*
    w0
    (fma D_m (* (* D_m -0.125) (/ (/ (* M_m (/ (* M_m h) l)) d) d)) 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) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e-16) {
		tmp = w0 * sqrt(fma(((M_m * D_m) / (d * -2.0)), ((M_m * D_m) * ((h / l) * (0.5 / d))), 1.0));
	} else {
		tmp = w0 * fma(D_m, ((D_m * -0.125) * (((M_m * ((M_m * h) / l)) / d) / d)), 1.0);
	}
	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(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e-16)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M_m * D_m) / Float64(d * -2.0)), Float64(Float64(M_m * D_m) * Float64(Float64(h / l) * Float64(0.5 / d))), 1.0)));
	else
		tmp = Float64(w0 * fma(D_m, Float64(Float64(D_m * -0.125) * Float64(Float64(Float64(M_m * Float64(Float64(M_m * h) / l)) / d) / d)), 1.0));
	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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-16], N[(w0 * N[Sqrt[N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(D$95$m * N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(N[(N[(M$95$m * N[(N[(M$95$m * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2e-16

   1. Initial program 66.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. 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 67.6%

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

   if -2e-16 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 59.3

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

   5. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.3

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

   7. Applied rewrites 70.0%

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

   9. Applied rewrites 89.8%

      \[\leadsto \mathsf{fma}\left(D, \left(D \cdot -0.125\right) \cdot \frac{\frac{M \cdot \frac{M \cdot h}{\ell}}{d}}{\color{blue}{d}}, 1\right) \cdot w0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot -2}, \left(M \cdot D\right) \cdot \left(\frac{h}{\ell} \cdot \frac{0.5}{d}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D, \left(D \cdot -0.125\right) \cdot \frac{\frac{M \cdot \frac{M \cdot h}{\ell}}{d}}{d}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, \frac{D\_m \cdot -0.125}{d} \cdot \left(M\_m \cdot \left(M\_m \cdot \frac{h}{d \cdot \ell}\right)\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+48}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{D\_m \cdot \left(\left(D\_m \cdot -0.25\right) \cdot \left(M\_m \cdot \left(M\_m \cdot h\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 (- INFINITY))
     (*
      w0
      (fma D_m (* (/ (* D_m -0.125) d) (* M_m (* M_m (/ h (* d l))))) 1.0))
     (if (<= t_0 -1e+48)
       (*
        w0
        (sqrt (/ (* D_m (* (* D_m -0.25) (* M_m (* M_m h)))) (* l (* d d)))))
       (* w0 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) {
	double t_0 = pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = w0 * fma(D_m, (((D_m * -0.125) / d) * (M_m * (M_m * (h / (d * l))))), 1.0);
	} else if (t_0 <= -1e+48) {
		tmp = w0 * sqrt(((D_m * ((D_m * -0.25) * (M_m * (M_m * h)))) / (l * (d * d))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(w0 * fma(D_m, Float64(Float64(Float64(D_m * -0.125) / d) * Float64(M_m * Float64(M_m * Float64(h / Float64(d * l))))), 1.0));
	elseif (t_0 <= -1e+48)
		tmp = Float64(w0 * sqrt(Float64(Float64(D_m * Float64(Float64(D_m * -0.25) * Float64(M_m * Float64(M_m * h)))) / Float64(l * Float64(d * d)))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(w0 * N[(D$95$m * N[(N[(N[(D$95$m * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[(M$95$m * N[(M$95$m * N[(h / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1e+48], N[(w0 * N[Sqrt[N[(N[(D$95$m * N[(N[(D$95$m * -0.25), $MachinePrecision] * N[(M$95$m * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -inf.0

   1. Initial program 55.6%

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

   3. Taylor expanded in M around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 40.2

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

   5. Applied rewrites 40.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 40.2

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

   7. Applied rewrites 45.4%

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

   9. Applied rewrites 50.5%

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

   11. Applied rewrites 54.0%

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

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.00000000000000004e48

   1. Initial program 99.5%

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

   3. Taylor expanded in M around 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. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 38.7

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

   5. Applied rewrites 38.7%

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

   7. Applied rewrites 54.8%

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

   if -1.00000000000000004e48 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.7%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D, \frac{D \cdot -0.125}{d} \cdot \left(M \cdot \left(M \cdot \frac{h}{d \cdot \ell}\right)\right), 1\right)\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+48}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{D \cdot \left(\left(D \cdot -0.25\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+178}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, \frac{D\_m \cdot -0.125}{d} \cdot \left(M\_m \cdot \left(M\_m \cdot \frac{h}{d \cdot \ell}\right)\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+48}:\\ \;\;\;\;w0 \cdot \sqrt{\left(D\_m \cdot -0.25\right) \cdot \left(D\_m \cdot \frac{M\_m \cdot \left(M\_m \cdot h\right)}{d \cdot \left(d \cdot \ell\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 -2e+178)
     (*
      w0
      (fma D_m (* (/ (* D_m -0.125) d) (* M_m (* M_m (/ h (* d l))))) 1.0))
     (if (<= t_0 -1e+48)
       (*
        w0
        (sqrt (* (* D_m -0.25) (* D_m (/ (* M_m (* M_m h)) (* d (* d l)))))))
       (* w0 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) {
	double t_0 = pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -2e+178) {
		tmp = w0 * fma(D_m, (((D_m * -0.125) / d) * (M_m * (M_m * (h / (d * l))))), 1.0);
	} else if (t_0 <= -1e+48) {
		tmp = w0 * sqrt(((D_m * -0.25) * (D_m * ((M_m * (M_m * h)) / (d * (d * l))))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= -2e+178)
		tmp = Float64(w0 * fma(D_m, Float64(Float64(Float64(D_m * -0.125) / d) * Float64(M_m * Float64(M_m * Float64(h / Float64(d * l))))), 1.0));
	elseif (t_0 <= -1e+48)
		tmp = Float64(w0 * sqrt(Float64(Float64(D_m * -0.25) * Float64(D_m * Float64(Float64(M_m * Float64(M_m * h)) / Float64(d * Float64(d * l)))))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+178], N[(w0 * N[(D$95$m * N[(N[(N[(D$95$m * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[(M$95$m * N[(M$95$m * N[(h / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1e+48], N[(w0 * N[Sqrt[N[(N[(D$95$m * -0.25), $MachinePrecision] * N[(D$95$m * N[(N[(M$95$m * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e178

   1. Initial program 58.2%

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

   3. Taylor expanded in M around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 37.9

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

   5. Applied rewrites 37.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 37.9

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

   7. Applied rewrites 44.4%

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

   9. Applied rewrites 49.2%

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

   11. Applied rewrites 52.5%

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

   if -2.0000000000000001e178 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.00000000000000004e48

   1. Initial program 99.3%

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

   3. Taylor expanded in M around 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. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 44.5

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

   5. Applied rewrites 44.5%

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

   7. Applied rewrites 43.8%

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

   if -1.00000000000000004e48 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+178}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D, \frac{D \cdot -0.125}{d} \cdot \left(M \cdot \left(M \cdot \frac{h}{d \cdot \ell}\right)\right), 1\right)\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+48}:\\ \;\;\;\;w0 \cdot \sqrt{\left(D \cdot -0.25\right) \cdot \left(D \cdot \frac{M \cdot \left(M \cdot h\right)}{d \cdot \left(d \cdot \ell\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M\_m}{d}, \left(D\_m \cdot -0.5\right) \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \frac{h \cdot 0.5}{d \cdot \ell}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, \left(D\_m \cdot -0.125\right) \cdot \frac{\frac{M\_m \cdot \frac{M\_m \cdot h}{\ell}}{d}}{d}, 1\right)\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e-16)
   (*
    w0
    (sqrt
     (fma
      (/ M_m d)
      (* (* D_m -0.5) (* (* M_m D_m) (/ (* h 0.5) (* d l))))
      1.0)))
   (*
    w0
    (fma D_m (* (* D_m -0.125) (/ (/ (* M_m (/ (* M_m h) l)) d) d)) 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) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e-16) {
		tmp = w0 * sqrt(fma((M_m / d), ((D_m * -0.5) * ((M_m * D_m) * ((h * 0.5) / (d * l)))), 1.0));
	} else {
		tmp = w0 * fma(D_m, ((D_m * -0.125) * (((M_m * ((M_m * h) / l)) / d) / d)), 1.0);
	}
	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(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e-16)
		tmp = Float64(w0 * sqrt(fma(Float64(M_m / d), Float64(Float64(D_m * -0.5) * Float64(Float64(M_m * D_m) * Float64(Float64(h * 0.5) / Float64(d * l)))), 1.0)));
	else
		tmp = Float64(w0 * fma(D_m, Float64(Float64(D_m * -0.125) * Float64(Float64(Float64(M_m * Float64(Float64(M_m * h) / l)) / d) / d)), 1.0));
	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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-16], N[(w0 * N[Sqrt[N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(D$95$m * -0.5), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(h * 0.5), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(D$95$m * N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(N[(N[(M$95$m * N[(N[(M$95$m * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2e-16

   1. Initial program 66.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. 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 67.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 65.3%

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

   if -2e-16 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 59.3

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

   5. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.3

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

   7. Applied rewrites 70.0%

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

   9. Applied rewrites 89.8%

      \[\leadsto \mathsf{fma}\left(D, \left(D \cdot -0.125\right) \cdot \frac{\frac{M \cdot \frac{M \cdot h}{\ell}}{d}}{\color{blue}{d}}, 1\right) \cdot w0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d}, \left(D \cdot -0.5\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{h \cdot 0.5}{d \cdot \ell}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D, \left(D \cdot -0.125\right) \cdot \frac{\frac{M \cdot \frac{M \cdot h}{\ell}}{d}}{d}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+48}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\left(M\_m \cdot h\right) \cdot \left(M\_m \cdot \left(D\_m \cdot -0.25\right)\right)}{d \cdot \ell} \cdot \frac{D\_m}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, \left(D\_m \cdot -0.125\right) \cdot \frac{\frac{M\_m \cdot \frac{M\_m \cdot h}{\ell}}{d}}{d}, 1\right)\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -1e+48)
   (* w0 (sqrt (* (/ (* (* M_m h) (* M_m (* D_m -0.25))) (* d l)) (/ D_m d))))
   (*
    w0
    (fma D_m (* (* D_m -0.125) (/ (/ (* M_m (/ (* M_m h) l)) d) d)) 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) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+48) {
		tmp = w0 * sqrt(((((M_m * h) * (M_m * (D_m * -0.25))) / (d * l)) * (D_m / d)));
	} else {
		tmp = w0 * fma(D_m, ((D_m * -0.125) * (((M_m * ((M_m * h) / l)) / d) / d)), 1.0);
	}
	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(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+48)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(Float64(M_m * h) * Float64(M_m * Float64(D_m * -0.25))) / Float64(d * l)) * Float64(D_m / d))));
	else
		tmp = Float64(w0 * fma(D_m, Float64(Float64(D_m * -0.125) * Float64(Float64(Float64(M_m * Float64(Float64(M_m * h) / l)) / d) / d)), 1.0));
	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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+48], N[(w0 * N[Sqrt[N[(N[(N[(N[(M$95$m * h), $MachinePrecision] * N[(M$95$m * N[(D$95$m * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(D$95$m * N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(N[(N[(M$95$m * N[(N[(M$95$m * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.00000000000000004e48

   1. Initial program 63.0%

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

   3. Taylor expanded in M around 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. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 39.9

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

   5. Applied rewrites 39.9%

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

   7. Applied rewrites 48.2%

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

   9. Applied rewrites 59.0%

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

   if -1.00000000000000004e48 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.4%

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

   3. Taylor expanded in M around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.0

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

   7. Applied rewrites 67.3%

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

   9. Applied rewrites 86.4%

      \[\leadsto \mathsf{fma}\left(D, \left(D \cdot -0.125\right) \cdot \frac{\frac{M \cdot \frac{M \cdot h}{\ell}}{d}}{\color{blue}{d}}, 1\right) \cdot w0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+48}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\left(M \cdot h\right) \cdot \left(M \cdot \left(D \cdot -0.25\right)\right)}{d \cdot \ell} \cdot \frac{D}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D, \left(D \cdot -0.125\right) \cdot \frac{\frac{M \cdot \frac{M \cdot h}{\ell}}{d}}{d}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.0005:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot D\_m\right) \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, \left(D\_m \cdot -0.125\right) \cdot \frac{\frac{M\_m \cdot \frac{M\_m \cdot h}{\ell}}{d}}{d}, 1\right)\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -0.0005)
   (*
    w0
    (sqrt (- 1.0 (* (* M_m D_m) (/ (* (* M_m D_m) h) (* l (* (* d d) 4.0)))))))
   (*
    w0
    (fma D_m (* (* D_m -0.125) (/ (/ (* M_m (/ (* M_m h) l)) d) d)) 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) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.0005) {
		tmp = w0 * sqrt((1.0 - ((M_m * D_m) * (((M_m * D_m) * h) / (l * ((d * d) * 4.0))))));
	} else {
		tmp = w0 * fma(D_m, ((D_m * -0.125) * (((M_m * ((M_m * h) / l)) / d) / d)), 1.0);
	}
	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(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.0005)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * D_m) * Float64(Float64(Float64(M_m * D_m) * h) / Float64(l * Float64(Float64(d * d) * 4.0)))))));
	else
		tmp = Float64(w0 * fma(D_m, Float64(Float64(D_m * -0.125) * Float64(Float64(Float64(M_m * Float64(Float64(M_m * h) / l)) / d) / d)), 1.0));
	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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.0005], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(l * N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(D$95$m * N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(N[(N[(M$95$m * N[(N[(M$95$m * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e-4

   1. Initial program 65.7%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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-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 67.2%

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

   6. Applied rewrites 52.6%

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

   if -5.0000000000000001e-4 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.0

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

   7. Applied rewrites 69.6%

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

   9. Applied rewrites 89.3%

      \[\leadsto \mathsf{fma}\left(D, \left(D \cdot -0.125\right) \cdot \frac{\frac{M \cdot \frac{M \cdot h}{\ell}}{d}}{\color{blue}{d}}, 1\right) \cdot w0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.0005:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\left(M \cdot D\right) \cdot h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D, \left(D \cdot -0.125\right) \cdot \frac{\frac{M \cdot \frac{M \cdot h}{\ell}}{d}}{d}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.0005:\\ \;\;\;\;w0 \cdot \sqrt{1 - M\_m \cdot \left(D\_m \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, \left(D\_m \cdot -0.125\right) \cdot \frac{\frac{M\_m \cdot \frac{M\_m \cdot h}{\ell}}{d}}{d}, 1\right)\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -0.0005)
   (*
    w0
    (sqrt (- 1.0 (* M_m (* D_m (/ (* (* M_m D_m) h) (* l (* (* d d) 4.0))))))))
   (*
    w0
    (fma D_m (* (* D_m -0.125) (/ (/ (* M_m (/ (* M_m h) l)) d) d)) 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) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.0005) {
		tmp = w0 * sqrt((1.0 - (M_m * (D_m * (((M_m * D_m) * h) / (l * ((d * d) * 4.0)))))));
	} else {
		tmp = w0 * fma(D_m, ((D_m * -0.125) * (((M_m * ((M_m * h) / l)) / d) / d)), 1.0);
	}
	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(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.0005)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(M_m * Float64(D_m * Float64(Float64(Float64(M_m * D_m) * h) / Float64(l * Float64(Float64(d * d) * 4.0))))))));
	else
		tmp = Float64(w0 * fma(D_m, Float64(Float64(D_m * -0.125) * Float64(Float64(Float64(M_m * Float64(Float64(M_m * h) / l)) / d) / d)), 1.0));
	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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.0005], N[(w0 * N[Sqrt[N[(1.0 - N[(M$95$m * N[(D$95$m * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(l * N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(D$95$m * N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(N[(N[(M$95$m * N[(N[(M$95$m * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e-4

   1. Initial program 65.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. swap-sqr N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 54.0

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

   4. Applied rewrites 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 52.6

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

   6. Applied rewrites 52.6%

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

   if -5.0000000000000001e-4 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.0

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

   7. Applied rewrites 69.6%

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

   9. Applied rewrites 89.3%

      \[\leadsto \mathsf{fma}\left(D, \left(D \cdot -0.125\right) \cdot \frac{\frac{M \cdot \frac{M \cdot h}{\ell}}{d}}{\color{blue}{d}}, 1\right) \cdot w0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.0005:\\ \;\;\;\;w0 \cdot \sqrt{1 - M \cdot \left(D \cdot \frac{\left(M \cdot D\right) \cdot h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D, \left(D \cdot -0.125\right) \cdot \frac{\frac{M \cdot \frac{M \cdot h}{\ell}}{d}}{d}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.0005:\\ \;\;\;\;w0 \cdot \sqrt{1 - M\_m \cdot \left(D\_m \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -0.0005)
   (*
    w0
    (sqrt (- 1.0 (* M_m (* D_m (/ (* (* M_m D_m) h) (* l (* (* d d) 4.0))))))))
   (* w0 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) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.0005) {
		tmp = w0 * sqrt((1.0 - (M_m * (D_m * (((M_m * D_m) * h) / (l * ((d * d) * 4.0)))))));
	} else {
		tmp = w0 * 1.0;
	}
	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 (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-0.0005d0)) then
        tmp = w0 * sqrt((1.0d0 - (m_m * (d_m * (((m_m * d_m) * h) / (l * ((d * d) * 4.0d0)))))))
    else
        tmp = w0 * 1.0d0
    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 ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.0005) {
		tmp = w0 * Math.sqrt((1.0 - (M_m * (D_m * (((M_m * D_m) * h) / (l * ((d * d) * 4.0)))))));
	} else {
		tmp = w0 * 1.0;
	}
	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 (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.0005:
		tmp = w0 * math.sqrt((1.0 - (M_m * (D_m * (((M_m * D_m) * h) / (l * ((d * d) * 4.0)))))))
	else:
		tmp = w0 * 1.0
	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(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.0005)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(M_m * Float64(D_m * Float64(Float64(Float64(M_m * D_m) * h) / Float64(l * Float64(Float64(d * d) * 4.0))))))));
	else
		tmp = Float64(w0 * 1.0);
	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 (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -0.0005)
		tmp = w0 * sqrt((1.0 - (M_m * (D_m * (((M_m * D_m) * h) / (l * ((d * d) * 4.0)))))));
	else
		tmp = w0 * 1.0;
	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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.0005], N[(w0 * N[Sqrt[N[(1.0 - N[(M$95$m * N[(D$95$m * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(l * N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e-4

   1. Initial program 65.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. swap-sqr N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 54.0

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

   4. Applied rewrites 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 52.6

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

   6. Applied rewrites 52.6%

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

   if -5.0000000000000001e-4 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.4%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.0005:\\ \;\;\;\;w0 \cdot \sqrt{1 - M \cdot \left(D \cdot \frac{\left(M \cdot D\right) \cdot h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.9% accurate, 0.8× speedup

Math

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

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e+63)
   (* w0 (sqrt (/ (* D_m (* M_m (* M_m (* h (* D_m -0.25))))) (* l (* d d)))))
   (* w0 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) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+63) {
		tmp = w0 * sqrt(((D_m * (M_m * (M_m * (h * (D_m * -0.25))))) / (l * (d * d))));
	} else {
		tmp = w0 * 1.0;
	}
	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 (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-2d+63)) then
        tmp = w0 * sqrt(((d_m * (m_m * (m_m * (h * (d_m * (-0.25d0)))))) / (l * (d * d))))
    else
        tmp = w0 * 1.0d0
    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 ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+63) {
		tmp = w0 * Math.sqrt(((D_m * (M_m * (M_m * (h * (D_m * -0.25))))) / (l * (d * d))));
	} else {
		tmp = w0 * 1.0;
	}
	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 (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+63:
		tmp = w0 * math.sqrt(((D_m * (M_m * (M_m * (h * (D_m * -0.25))))) / (l * (d * d))))
	else:
		tmp = w0 * 1.0
	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(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+63)
		tmp = Float64(w0 * sqrt(Float64(Float64(D_m * Float64(M_m * Float64(M_m * Float64(h * Float64(D_m * -0.25))))) / Float64(l * Float64(d * d)))));
	else
		tmp = Float64(w0 * 1.0);
	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 (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+63)
		tmp = w0 * sqrt(((D_m * (M_m * (M_m * (h * (D_m * -0.25))))) / (l * (d * d))));
	else
		tmp = w0 * 1.0;
	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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+63], N[(w0 * N[Sqrt[N[(N[(D$95$m * N[(M$95$m * N[(M$95$m * N[(h * N[(D$95$m * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 62.1%

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

   3. Taylor expanded in M around 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. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 39.7

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

   5. Applied rewrites 39.7%

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

   7. Applied rewrites 48.2%

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

   9. Applied rewrites 50.4%

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

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

   1. Initial program 89.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.8%

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

4. Final simplification 82.5%

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

Alternative 10: 80.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.0005:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, \frac{D\_m \cdot -0.125}{d} \cdot \left(M\_m \cdot \left(M\_m \cdot \frac{h}{d \cdot \ell}\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -0.0005)
   (* w0 (fma D_m (* (/ (* D_m -0.125) d) (* M_m (* M_m (/ h (* d l))))) 1.0))
   (* w0 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) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.0005) {
		tmp = w0 * fma(D_m, (((D_m * -0.125) / d) * (M_m * (M_m * (h / (d * l))))), 1.0);
	} else {
		tmp = w0 * 1.0;
	}
	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(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.0005)
		tmp = Float64(w0 * fma(D_m, Float64(Float64(Float64(D_m * -0.125) / d) * Float64(M_m * Float64(M_m * Float64(h / Float64(d * l))))), 1.0));
	else
		tmp = Float64(w0 * 1.0);
	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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.0005], N[(w0 * N[(D$95$m * N[(N[(N[(D$95$m * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[(M$95$m * N[(M$95$m * N[(h / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e-4

   1. Initial program 65.7%

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

   3. Taylor expanded in M around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.5

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

   7. Applied rewrites 37.2%

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

   9. Applied rewrites 41.5%

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

   11. Applied rewrites 44.5%

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

   if -5.0000000000000001e-4 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.4%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.0005:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D, \frac{D \cdot -0.125}{d} \cdot \left(M \cdot \left(M \cdot \frac{h}{d \cdot \ell}\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.8% accurate, 0.8× speedup

Math

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

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) (- INFINITY))
   (* w0 (* (* -0.125 (* D_m D_m)) (* M_m (/ (* M_m (/ h (* d l))) d))))
   (* w0 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) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -((double) INFINITY)) {
		tmp = w0 * ((-0.125 * (D_m * D_m)) * (M_m * ((M_m * (h / (d * l))) / d)));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

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 ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * ((-0.125 * (D_m * D_m)) * (M_m * ((M_m * (h / (d * l))) / d)));
	} else {
		tmp = w0 * 1.0;
	}
	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 (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -math.inf:
		tmp = w0 * ((-0.125 * (D_m * D_m)) * (M_m * ((M_m * (h / (d * l))) / d)))
	else:
		tmp = w0 * 1.0
	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(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= Float64(-Inf))
		tmp = Float64(w0 * Float64(Float64(-0.125 * Float64(D_m * D_m)) * Float64(M_m * Float64(Float64(M_m * Float64(h / Float64(d * l))) / d))));
	else
		tmp = Float64(w0 * 1.0);
	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 (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -Inf)
		tmp = w0 * ((-0.125 * (D_m * D_m)) * (M_m * ((M_m * (h / (d * l))) / d)));
	else
		tmp = w0 * 1.0;
	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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(w0 * N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * N[(N[(M$95$m * N[(h / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -inf.0

   1. Initial program 55.6%

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

   3. Taylor expanded in M around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 40.2

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

   5. Applied rewrites 40.2%

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

   6. Taylor expanded in D around inf

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

   8. Applied rewrites 40.2%

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

   10. Applied rewrites 45.4%

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

   12. Applied rewrites 49.0%

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

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 90.7%

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

4. Final simplification 80.2%

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

Alternative 12: 79.0% accurate, 0.8× speedup

Math

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

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) (- INFINITY))
   (* w0 (* (* -0.125 (* D_m D_m)) (* M_m (* M_m (/ h (* l (* d d)))))))
   (* w0 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) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -((double) INFINITY)) {
		tmp = w0 * ((-0.125 * (D_m * D_m)) * (M_m * (M_m * (h / (l * (d * d))))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

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 ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * ((-0.125 * (D_m * D_m)) * (M_m * (M_m * (h / (l * (d * d))))));
	} else {
		tmp = w0 * 1.0;
	}
	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 (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -math.inf:
		tmp = w0 * ((-0.125 * (D_m * D_m)) * (M_m * (M_m * (h / (l * (d * d))))))
	else:
		tmp = w0 * 1.0
	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(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= Float64(-Inf))
		tmp = Float64(w0 * Float64(Float64(-0.125 * Float64(D_m * D_m)) * Float64(M_m * Float64(M_m * Float64(h / Float64(l * Float64(d * d)))))));
	else
		tmp = Float64(w0 * 1.0);
	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 (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -Inf)
		tmp = w0 * ((-0.125 * (D_m * D_m)) * (M_m * (M_m * (h / (l * (d * d))))));
	else
		tmp = w0 * 1.0;
	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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(w0 * N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * N[(M$95$m * N[(h / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -inf.0

   1. Initial program 55.6%

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

   3. Taylor expanded in M around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 40.2

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

   5. Applied rewrites 40.2%

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

   6. Taylor expanded in D around inf

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

   8. Applied rewrites 40.2%

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

   10. Applied rewrites 45.4%

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

   12. Applied rewrites 45.4%

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

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 90.7%

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

4. Final simplification 79.4%

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

Alternative 13: 67.9% accurate, 26.2× speedup

Math

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

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 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 * 1.0;
}

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 = w0 * 1.0d0
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 w0 * 1.0;
}

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 w0 * 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 * 1.0)
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 = w0 * 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 * 1.0), $MachinePrecision]

Derivation

1. Initial program 81.4%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 69.1%

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

Reproduce

herbie shell --seed 2024221 
(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))))))