Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 82.2% → 90.1%

Time: 15.0s

Alternatives: 14

Speedup: 2.2×

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: 82.2% 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: 90.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \left(\frac{0.5}{d} \cdot M\_m\right) \cdot D\_m\\ \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{d \cdot 2}\right)}^{2} \leq 4 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{D\_m \cdot M\_m}{\left(4 \cdot d\right) \cdot \left(\frac{d}{D\_m \cdot M\_m} \cdot \ell\right)}, -h, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(t\_0 \cdot \frac{-h}{\ell}, t\_0, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* (* (/ 0.5 d) M_m) D_m)))
   (if (<= (pow (/ (* D_m M_m) (* d 2.0)) 2.0) 4e-88)
     (*
      (sqrt
       (fma (/ (* D_m M_m) (* (* 4.0 d) (* (/ d (* D_m M_m)) l))) (- h) 1.0))
      w0)
     (* (sqrt (fma (* t_0 (/ (- h) l)) t_0 1.0)) w0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = ((0.5 / d) * M_m) * D_m;
	double tmp;
	if (pow(((D_m * M_m) / (d * 2.0)), 2.0) <= 4e-88) {
		tmp = sqrt(fma(((D_m * M_m) / ((4.0 * d) * ((d / (D_m * M_m)) * l))), -h, 1.0)) * w0;
	} else {
		tmp = sqrt(fma((t_0 * (-h / l)), t_0, 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(Float64(Float64(0.5 / d) * M_m) * D_m)
	tmp = 0.0
	if ((Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0) <= 4e-88)
		tmp = Float64(sqrt(fma(Float64(Float64(D_m * M_m) / Float64(Float64(4.0 * d) * Float64(Float64(d / Float64(D_m * M_m)) * l))), Float64(-h), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(t_0 * Float64(Float64(-h) / l)), t_0, 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 4e-88], N[(N[Sqrt[N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(N[(4.0 * d), $MachinePrecision] * N[(N[(d / N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-h) + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(t$95$0 * N[((-h) / l), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 3.99999999999999974e-88

   1. Initial program 91.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. frac-2neg N/A

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

      10. associate-/r/ N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

   4. Applied rewrites 92.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 98.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.6

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

   6. Applied rewrites 98.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 99.3

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 99.3

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.3

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

   8. Applied rewrites 99.3%

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

   if 3.99999999999999974e-88 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 69.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 75.8%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq 4 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{D \cdot M}{\left(4 \cdot d\right) \cdot \left(\frac{d}{D \cdot M} \cdot \ell\right)}, -h, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right) \cdot \frac{-h}{\ell}, \left(\frac{0.5}{d} \cdot M\right) \cdot D, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{d \cdot 2}\right)}^{2} \leq -50000:\\ \;\;\;\;\sqrt{\left(-0.25 \cdot h\right) \cdot \left(\left(\frac{M\_m}{d} \cdot D\_m\right) \cdot \frac{D\_m \cdot M\_m}{\ell \cdot d}\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* d 2.0)) 2.0)) -50000.0)
   (* (sqrt (* (* -0.25 h) (* (* (/ M_m d) D_m) (/ (* D_m M_m) (* l d))))) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -50000.0) {
		tmp = sqrt(((-0.25 * h) * (((M_m / d) * D_m) * ((D_m * M_m) / (l * d))))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((h / l) * (((d_m * m_m) / (d * 2.0d0)) ** 2.0d0)) <= (-50000.0d0)) then
        tmp = sqrt((((-0.25d0) * h) * (((m_m / d) * d_m) * ((d_m * m_m) / (l * d))))) * w0
    else
        tmp = 1.0d0 * w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * Math.pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -50000.0) {
		tmp = Math.sqrt(((-0.25 * h) * (((M_m / d) * D_m) * ((D_m * M_m) / (l * d))))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if ((h / l) * math.pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -50000.0:
		tmp = math.sqrt(((-0.25 * h) * (((M_m / d) * D_m) * ((D_m * M_m) / (l * d))))) * w0
	else:
		tmp = 1.0 * w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0)) <= -50000.0)
		tmp = Float64(sqrt(Float64(Float64(-0.25 * h) * Float64(Float64(Float64(M_m / d) * D_m) * Float64(Float64(D_m * M_m) / Float64(l * d))))) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((h / l) * (((D_m * M_m) / (d * 2.0)) ^ 2.0)) <= -50000.0)
		tmp = sqrt(((-0.25 * h) * (((M_m / d) * D_m) * ((D_m * M_m) / (l * d))))) * w0;
	else
		tmp = 1.0 * w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -50000.0], N[(N[Sqrt[N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5e4

   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. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 41.4%

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

   7. Applied rewrites 45.8%

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

   9. Applied rewrites 59.9%

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

   if -5e4 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.1%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 96.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq -50000:\\ \;\;\;\;\sqrt{\left(-0.25 \cdot h\right) \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{D \cdot M}{\ell \cdot d}\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{d \cdot 2}\right)}^{2} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(-h\right) \cdot \left(D\_m \cdot M\_m\right), \frac{0.25 \cdot \left(D\_m \cdot M\_m\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* d 2.0)) 2.0)) -5e-8)
   (*
    (sqrt
     (fma (* (- h) (* D_m M_m)) (/ (* 0.25 (* D_m M_m)) (* (* d d) l)) 1.0))
    w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -5e-8) {
		tmp = sqrt(fma((-h * (D_m * M_m)), ((0.25 * (D_m * M_m)) / ((d * d) * l)), 1.0)) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0)) <= -5e-8)
		tmp = Float64(sqrt(fma(Float64(Float64(-h) * Float64(D_m * M_m)), Float64(Float64(0.25 * Float64(D_m * M_m)) / Float64(Float64(d * d) * l)), 1.0)) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -5e-8], N[(N[Sqrt[N[(N[((-h) * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.9999999999999998e-8

   1. Initial program 67.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. frac-2neg N/A

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

      10. associate-/r/ N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

   4. Applied rewrites 56.6%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 53.2%

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

   if -4.9999999999999998e-8 < (*.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 h around 0

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

   5. Applied rewrites 97.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(-h\right) \cdot \left(D \cdot M\right), \frac{0.25 \cdot \left(D \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{d \cdot 2}\right)}^{2} \leq 5 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{D\_m \cdot M\_m}{\frac{d}{D\_m \cdot M\_m} \cdot \left(4 \cdot d\right)}}{\ell}, -h, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{-0.5}{d} \cdot D\_m\right) \cdot M\_m, \left(\frac{h}{\ell} \cdot D\_m\right) \cdot \left(\frac{0.5}{d} \cdot M\_m\right), 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (pow (/ (* D_m M_m) (* d 2.0)) 2.0) 5e+85)
   (*
    (sqrt
     (fma (/ (/ (* D_m M_m) (* (/ d (* D_m M_m)) (* 4.0 d))) l) (- h) 1.0))
    w0)
   (*
    (sqrt
     (fma
      (* (* (/ -0.5 d) D_m) M_m)
      (* (* (/ h l) D_m) (* (/ 0.5 d) M_m))
      1.0))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (pow(((D_m * M_m) / (d * 2.0)), 2.0) <= 5e+85) {
		tmp = sqrt(fma((((D_m * M_m) / ((d / (D_m * M_m)) * (4.0 * d))) / l), -h, 1.0)) * w0;
	} else {
		tmp = sqrt(fma((((-0.5 / d) * D_m) * M_m), (((h / l) * D_m) * ((0.5 / d) * M_m)), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if ((Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0) <= 5e+85)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(D_m * M_m) / Float64(Float64(d / Float64(D_m * M_m)) * Float64(4.0 * d))) / l), Float64(-h), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(-0.5 / d) * D_m) * M_m), Float64(Float64(Float64(h / l) * D_m) * Float64(Float64(0.5 / d) * M_m)), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 5e+85], N[(N[Sqrt[N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(N[(d / N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(4.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * (-h) + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(-0.5 / d), $MachinePrecision] * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 5.0000000000000001e85

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. frac-2neg N/A

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

      10. associate-/r/ N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

   4. Applied rewrites 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 97.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 97.7

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

   6. Applied rewrites 97.7%

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

   if 5.0000000000000001e85 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 62.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 65.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 70.5

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

   6. Applied rewrites 70.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq 5 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{D \cdot M}{\frac{d}{D \cdot M} \cdot \left(4 \cdot d\right)}}{\ell}, -h, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{-0.5}{d} \cdot D\right) \cdot M, \left(\frac{h}{\ell} \cdot D\right) \cdot \left(\frac{0.5}{d} \cdot M\right), 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.3% accurate, 0.8× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* d 2.0)) 2.0)) -1e+54)
   (* (sqrt (* (* (/ (* D_m M_m) (* (* d d) l)) (* D_m M_m)) (* -0.25 h))) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -1e+54) {
		tmp = sqrt(((((D_m * M_m) / ((d * d) * l)) * (D_m * M_m)) * (-0.25 * h))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((h / l) * (((d_m * m_m) / (d * 2.0d0)) ** 2.0d0)) <= (-1d+54)) then
        tmp = sqrt(((((d_m * m_m) / ((d * d) * l)) * (d_m * m_m)) * ((-0.25d0) * h))) * w0
    else
        tmp = 1.0d0 * w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * Math.pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -1e+54) {
		tmp = Math.sqrt(((((D_m * M_m) / ((d * d) * l)) * (D_m * M_m)) * (-0.25 * h))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if ((h / l) * math.pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -1e+54:
		tmp = math.sqrt(((((D_m * M_m) / ((d * d) * l)) * (D_m * M_m)) * (-0.25 * h))) * w0
	else:
		tmp = 1.0 * w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0)) <= -1e+54)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(D_m * M_m) / Float64(Float64(d * d) * l)) * Float64(D_m * M_m)) * Float64(-0.25 * h))) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((h / l) * (((D_m * M_m) / (d * 2.0)) ^ 2.0)) <= -1e+54)
		tmp = sqrt(((((D_m * M_m) / ((d * d) * l)) * (D_m * M_m)) * (-0.25 * h))) * w0;
	else
		tmp = 1.0 * w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e+54], N[(N[Sqrt[N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.0000000000000001e54

   1. Initial program 64.5%

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

   3. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 43.3%

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

   7. Applied rewrites 47.8%

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

   9. Applied rewrites 53.9%

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

   if -1.0000000000000001e54 < (*.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 h around 0

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

   5. Applied rewrites 94.9%

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

4. Final simplification 81.1%

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

Alternative 6: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{d \cdot 2}\right)}^{2} \leq -5 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\left(\left(\frac{D\_m}{\left(d \cdot d\right) \cdot \ell} \cdot \left(D\_m \cdot M\_m\right)\right) \cdot M\_m\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* d 2.0)) 2.0)) -5e+111)
   (* (sqrt (* (* (* (/ D_m (* (* d d) l)) (* D_m M_m)) M_m) (* -0.25 h))) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -5e+111) {
		tmp = sqrt(((((D_m / ((d * d) * l)) * (D_m * M_m)) * M_m) * (-0.25 * h))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((h / l) * (((d_m * m_m) / (d * 2.0d0)) ** 2.0d0)) <= (-5d+111)) then
        tmp = sqrt(((((d_m / ((d * d) * l)) * (d_m * m_m)) * m_m) * ((-0.25d0) * h))) * w0
    else
        tmp = 1.0d0 * w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * Math.pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -5e+111) {
		tmp = Math.sqrt(((((D_m / ((d * d) * l)) * (D_m * M_m)) * M_m) * (-0.25 * h))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if ((h / l) * math.pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -5e+111:
		tmp = math.sqrt(((((D_m / ((d * d) * l)) * (D_m * M_m)) * M_m) * (-0.25 * h))) * w0
	else:
		tmp = 1.0 * w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0)) <= -5e+111)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(D_m / Float64(Float64(d * d) * l)) * Float64(D_m * M_m)) * M_m) * Float64(-0.25 * h))) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((h / l) * (((D_m * M_m) / (d * 2.0)) ^ 2.0)) <= -5e+111)
		tmp = sqrt(((((D_m / ((d * d) * l)) * (D_m * M_m)) * M_m) * (-0.25 * h))) * w0;
	else
		tmp = 1.0 * w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -5e+111], N[(N[Sqrt[N[(N[(N[(N[(D$95$m / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(-0.25 * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.9999999999999997e111

   1. Initial program 64.1%

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

   3. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 43.7%

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

   7. Applied rewrites 50.1%

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

   if -4.9999999999999997e111 < (*.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 h around 0

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

   5. Applied rewrites 94.4%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq -5 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\left(\left(\frac{D}{\left(d \cdot d\right) \cdot \ell} \cdot \left(D \cdot M\right)\right) \cdot M\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{d \cdot 2}\right)}^{2} \leq -4 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D\_m \cdot D\_m\right), \frac{\left(w0 \cdot M\_m\right) \cdot \left(\frac{h}{\ell \cdot d} \cdot M\_m\right)}{d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* d 2.0)) 2.0)) -4e+188)
   (fma (* -0.125 (* D_m D_m)) (/ (* (* w0 M_m) (* (/ h (* l d)) M_m)) d) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -4e+188) {
		tmp = fma((-0.125 * (D_m * D_m)), (((w0 * M_m) * ((h / (l * d)) * M_m)) / d), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0)) <= -4e+188)
		tmp = fma(Float64(-0.125 * Float64(D_m * D_m)), Float64(Float64(Float64(w0 * M_m) * Float64(Float64(h / Float64(l * d)) * M_m)) / d), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -4e+188], N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w0 * M$95$m), $MachinePrecision] * N[(N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.0000000000000001e188

   1. Initial program 61.0%

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

   3. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 42.5%

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

   7. Applied rewrites 49.5%

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

   9. Applied rewrites 50.6%

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

   if -4.0000000000000001e188 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.8%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 91.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq -4 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{\left(w0 \cdot M\right) \cdot \left(\frac{h}{\ell \cdot d} \cdot M\right)}{d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{d \cdot 2}\right)}^{2} \leq -4 \cdot 10^{+188}:\\ \;\;\;\;\left(\left(\frac{w0}{\left(d \cdot d\right) \cdot \ell} \cdot M\_m\right) \cdot \left(h \cdot M\_m\right)\right) \cdot \left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* d 2.0)) 2.0)) -4e+188)
   (* (* (* (/ w0 (* (* d d) l)) M_m) (* h M_m)) (* -0.125 (* D_m D_m)))
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -4e+188) {
		tmp = (((w0 / ((d * d) * l)) * M_m) * (h * M_m)) * (-0.125 * (D_m * D_m));
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((h / l) * (((d_m * m_m) / (d * 2.0d0)) ** 2.0d0)) <= (-4d+188)) then
        tmp = (((w0 / ((d * d) * l)) * m_m) * (h * m_m)) * ((-0.125d0) * (d_m * d_m))
    else
        tmp = 1.0d0 * w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * Math.pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -4e+188) {
		tmp = (((w0 / ((d * d) * l)) * M_m) * (h * M_m)) * (-0.125 * (D_m * D_m));
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if ((h / l) * math.pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -4e+188:
		tmp = (((w0 / ((d * d) * l)) * M_m) * (h * M_m)) * (-0.125 * (D_m * D_m))
	else:
		tmp = 1.0 * w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0)) <= -4e+188)
		tmp = Float64(Float64(Float64(Float64(w0 / Float64(Float64(d * d) * l)) * M_m) * Float64(h * M_m)) * Float64(-0.125 * Float64(D_m * D_m)));
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((h / l) * (((D_m * M_m) / (d * 2.0)) ^ 2.0)) <= -4e+188)
		tmp = (((w0 / ((d * d) * l)) * M_m) * (h * M_m)) * (-0.125 * (D_m * D_m));
	else
		tmp = 1.0 * w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -4e+188], N[(N[(N[(N[(w0 / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.0000000000000001e188

   1. Initial program 61.0%

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

   3. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 42.5%

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

   6. Taylor expanded in h around inf

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

   8. Applied rewrites 42.5%

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

   10. Applied rewrites 46.4%

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

   if -4.0000000000000001e188 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.8%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 91.0%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq -4 \cdot 10^{+188}:\\ \;\;\;\;\left(\left(\frac{w0}{\left(d \cdot d\right) \cdot \ell} \cdot M\right) \cdot \left(h \cdot M\right)\right) \cdot \left(-0.125 \cdot \left(D \cdot D\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -1.2 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\frac{M\_m}{4 \cdot d} \cdot D\_m\right) \cdot \left(\frac{D\_m}{d} \cdot M\_m\right)}{\ell}, -h, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5 \cdot D\_m, \frac{\left(-M\_m\right) \cdot h}{\ell \cdot d} \cdot \left(\left(\frac{0.5}{d} \cdot M\_m\right) \cdot D\_m\right), 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= h -1.2e-280)
   (*
    (sqrt
     (fma (/ (* (* (/ M_m (* 4.0 d)) D_m) (* (/ D_m d) M_m)) l) (- h) 1.0))
    w0)
   (*
    (sqrt
     (fma
      (* 0.5 D_m)
      (* (/ (* (- M_m) h) (* l d)) (* (* (/ 0.5 d) M_m) D_m))
      1.0))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (h <= -1.2e-280) {
		tmp = sqrt(fma(((((M_m / (4.0 * d)) * D_m) * ((D_m / d) * M_m)) / l), -h, 1.0)) * w0;
	} else {
		tmp = sqrt(fma((0.5 * D_m), (((-M_m * h) / (l * d)) * (((0.5 / d) * M_m) * D_m)), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (h <= -1.2e-280)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(M_m / Float64(4.0 * d)) * D_m) * Float64(Float64(D_m / d) * M_m)) / l), Float64(-h), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(0.5 * D_m), Float64(Float64(Float64(Float64(-M_m) * h) / Float64(l * d)) * Float64(Float64(Float64(0.5 / d) * M_m) * D_m)), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[h, -1.2e-280], N[(N[Sqrt[N[(N[(N[(N[(N[(M$95$m / N[(4.0 * d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * (-h) + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(0.5 * D$95$m), $MachinePrecision] * N[(N[(N[((-M$95$m) * h), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -1.1999999999999999e-280

   1. Initial program 82.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. frac-2neg N/A

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

      10. associate-/r/ N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

   4. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 84.8

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 84.8

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

   6. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow-prod-down N/A

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

      8. inv-pow N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. inv-pow N/A

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

      12. clear-num N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-/l* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 87.1

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 87.1

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

   8. Applied rewrites 87.1%

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

   if -1.1999999999999999e-280 < h

   1. Initial program 79.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 84.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 84.1%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.2 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\frac{M}{4 \cdot d} \cdot D\right) \cdot \left(\frac{D}{d} \cdot M\right)}{\ell}, -h, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5 \cdot D, \frac{\left(-M\right) \cdot h}{\ell \cdot d} \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right), 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 8 \cdot 10^{-84}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{D\_m \cdot M\_m}{\left(4 \cdot d\right) \cdot \left(\frac{d}{D\_m \cdot M\_m} \cdot \ell\right)}, -h, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(0.25 \cdot \left(M\_m \cdot M\_m\right)\right) \cdot \frac{D\_m}{d}\right) \cdot \frac{-h}{\ell}, \frac{D\_m}{d}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 8e-84)
   (*
    (sqrt
     (fma (/ (* D_m M_m) (* (* 4.0 d) (* (/ d (* D_m M_m)) l))) (- h) 1.0))
    w0)
   (*
    (sqrt
     (fma (* (* (* 0.25 (* M_m M_m)) (/ D_m d)) (/ (- h) l)) (/ D_m d) 1.0))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 8e-84) {
		tmp = sqrt(fma(((D_m * M_m) / ((4.0 * d) * ((d / (D_m * M_m)) * l))), -h, 1.0)) * w0;
	} else {
		tmp = sqrt(fma((((0.25 * (M_m * M_m)) * (D_m / d)) * (-h / l)), (D_m / d), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 8e-84)
		tmp = Float64(sqrt(fma(Float64(Float64(D_m * M_m) / Float64(Float64(4.0 * d) * Float64(Float64(d / Float64(D_m * M_m)) * l))), Float64(-h), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(0.25 * Float64(M_m * M_m)) * Float64(D_m / d)) * Float64(Float64(-h) / l)), Float64(D_m / d), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 8e-84], N[(N[Sqrt[N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(N[(4.0 * d), $MachinePrecision] * N[(N[(d / N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-h) + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(0.25 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[((-h) / l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 8.0000000000000003e-84

   1. Initial program 83.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. frac-2neg N/A

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

      10. associate-/r/ N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

   4. Applied rewrites 78.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 86.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 86.6

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

   6. Applied rewrites 86.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 89.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 89.1

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 89.1

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

   8. Applied rewrites 89.1%

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

   if 8.0000000000000003e-84 < M

   1. Initial program 75.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   4. Applied rewrites 67.2%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 8 \cdot 10^{-84}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{D \cdot M}{\left(4 \cdot d\right) \cdot \left(\frac{d}{D \cdot M} \cdot \ell\right)}, -h, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(0.25 \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{d}\right) \cdot \frac{-h}{\ell}, \frac{D}{d}, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 8 \cdot 10^{-84}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{D\_m \cdot M\_m}{\left(4 \cdot d\right) \cdot \left(\frac{d}{D\_m \cdot M\_m} \cdot \ell\right)}, -h, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(0.25 \cdot \left(M\_m \cdot M\_m\right)\right) \cdot \frac{D\_m}{d}, \frac{-D\_m}{d} \cdot \frac{h}{\ell}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 8e-84)
   (*
    (sqrt
     (fma (/ (* D_m M_m) (* (* 4.0 d) (* (/ d (* D_m M_m)) l))) (- h) 1.0))
    w0)
   (*
    (sqrt
     (fma (* (* 0.25 (* M_m M_m)) (/ D_m d)) (* (/ (- D_m) d) (/ h l)) 1.0))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 8e-84) {
		tmp = sqrt(fma(((D_m * M_m) / ((4.0 * d) * ((d / (D_m * M_m)) * l))), -h, 1.0)) * w0;
	} else {
		tmp = sqrt(fma(((0.25 * (M_m * M_m)) * (D_m / d)), ((-D_m / d) * (h / l)), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 8e-84)
		tmp = Float64(sqrt(fma(Float64(Float64(D_m * M_m) / Float64(Float64(4.0 * d) * Float64(Float64(d / Float64(D_m * M_m)) * l))), Float64(-h), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(0.25 * Float64(M_m * M_m)) * Float64(D_m / d)), Float64(Float64(Float64(-D_m) / d) * Float64(h / l)), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 8e-84], N[(N[Sqrt[N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(N[(4.0 * d), $MachinePrecision] * N[(N[(d / N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-h) + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(0.25 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[((-D$95$m) / d), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 8.0000000000000003e-84

   1. Initial program 83.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. frac-2neg N/A

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

      10. associate-/r/ N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

   4. Applied rewrites 78.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 86.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 86.6

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

   6. Applied rewrites 86.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 89.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 89.1

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 89.1

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

   8. Applied rewrites 89.1%

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

   if 8.0000000000000003e-84 < M

   1. Initial program 75.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. times-frac N/A

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

      15. associate-*r* N/A

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

      16. associate-*l* N/A

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

   4. Applied rewrites 67.1%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 8 \cdot 10^{-84}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{D \cdot M}{\left(4 \cdot d\right) \cdot \left(\frac{d}{D \cdot M} \cdot \ell\right)}, -h, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(0.25 \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{d}, \frac{-D}{d} \cdot \frac{h}{\ell}, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 84.9%

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

5. Final simplification 84.9%

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

Alternative 13: 87.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \sqrt{\mathsf{fma}\left(\frac{D\_m \cdot M\_m}{\left(4 \cdot d\right) \cdot \left(\frac{d}{D\_m \cdot M\_m} \cdot \ell\right)}, -h, 1\right)} \cdot w0 \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (*
  (sqrt (fma (/ (* D_m M_m) (* (* 4.0 d) (* (/ d (* D_m M_m)) l))) (- h) 1.0))
  w0))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return sqrt(fma(((D_m * M_m) / ((4.0 * d) * ((d / (D_m * M_m)) * l))), -h, 1.0)) * w0;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	return Float64(sqrt(fma(Float64(Float64(D_m * M_m) / Float64(Float64(4.0 * d) * Float64(Float64(d / Float64(D_m * M_m)) * l))), Float64(-h), 1.0)) * w0)
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := N[(N[Sqrt[N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(N[(4.0 * d), $MachinePrecision] * N[(N[(d / N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-h) + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. frac-2neg N/A

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

   10. associate-/r/ N/A

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

   11. neg-mul-1 N/A

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

   12. associate-*l/ N/A

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

   13. metadata-eval N/A

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

   14. frac-2neg N/A

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

4. Applied rewrites 76.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 83.6

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 83.6

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

6. Applied rewrites 83.6%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 85.5

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 85.5

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 85.5

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

8. Applied rewrites 85.5%

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

9. Final simplification 85.5%

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

Alternative 14: 69.1% accurate, 26.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return 1.0 * w0;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in h around 0

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

5. Applied rewrites 64.6%

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

6. Final simplification 64.6%

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

Reproduce

herbie shell --seed 2024235 
(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))))))