Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.1% → 79.5%

Time: 20.4s

Alternatives: 25

Speedup: 3.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 1 - \frac{\frac{M\_m}{d} \cdot \left(\left(D\_m \cdot 0.5\right) \cdot 0.5\right)}{{h}^{-1}} \cdot \frac{M\_m \cdot \left(D\_m \cdot \frac{0.5}{d}\right)}{\ell}\\ t_1 := {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ t_2 := \sqrt{-d}\\ t_3 := \left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m\\ \mathbf{if}\;d \leq -1.25 \cdot 10^{-176}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \frac{t\_2}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -1.22 \cdot 10^{-300}:\\ \;\;\;\;\frac{t\_2}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(t\_3 \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), t\_3, 1\right)\right)\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{-260}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(t\_3, \frac{M\_m \cdot D\_m}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (-
          1.0
          (*
           (/ (* (/ M_m d) (* (* D_m 0.5) 0.5)) (pow h -1.0))
           (/ (* M_m (* D_m (/ 0.5 d))) l))))
        (t_1 (pow (/ d l) (/ 1.0 2.0)))
        (t_2 (sqrt (- d)))
        (t_3 (* (* M_m (/ 0.5 d)) D_m)))
   (if (<= d -1.25e-176)
     (* t_0 (* t_1 (/ t_2 (sqrt (- h)))))
     (if (<= d -1.22e-300)
       (*
        (/ t_2 (sqrt (- l)))
        (* (sqrt (/ d h)) (fma (* t_3 (* -0.5 (/ h l))) t_3 1.0)))
       (if (<= d 5.2e-260)
         (*
          (/ d (sqrt (* l h)))
          (fma t_3 (* (/ (* M_m D_m) l) (* -0.25 (/ h d))) 1.0))
         (* (* (/ (sqrt d) (sqrt h)) t_1) t_0))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 - ((((M_m / d) * ((D_m * 0.5) * 0.5)) / pow(h, -1.0)) * ((M_m * (D_m * (0.5 / d))) / l));
	double t_1 = pow((d / l), (1.0 / 2.0));
	double t_2 = sqrt(-d);
	double t_3 = (M_m * (0.5 / d)) * D_m;
	double tmp;
	if (d <= -1.25e-176) {
		tmp = t_0 * (t_1 * (t_2 / sqrt(-h)));
	} else if (d <= -1.22e-300) {
		tmp = (t_2 / sqrt(-l)) * (sqrt((d / h)) * fma((t_3 * (-0.5 * (h / l))), t_3, 1.0));
	} else if (d <= 5.2e-260) {
		tmp = (d / sqrt((l * h))) * fma(t_3, (((M_m * D_m) / l) * (-0.25 * (h / d))), 1.0);
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_1) * t_0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(1.0 - Float64(Float64(Float64(Float64(M_m / d) * Float64(Float64(D_m * 0.5) * 0.5)) / (h ^ -1.0)) * Float64(Float64(M_m * Float64(D_m * Float64(0.5 / d))) / l)))
	t_1 = Float64(d / l) ^ Float64(1.0 / 2.0)
	t_2 = sqrt(Float64(-d))
	t_3 = Float64(Float64(M_m * Float64(0.5 / d)) * D_m)
	tmp = 0.0
	if (d <= -1.25e-176)
		tmp = Float64(t_0 * Float64(t_1 * Float64(t_2 / sqrt(Float64(-h)))));
	elseif (d <= -1.22e-300)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * fma(Float64(t_3 * Float64(-0.5 * Float64(h / l))), t_3, 1.0)));
	elseif (d <= 5.2e-260)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * fma(t_3, Float64(Float64(Float64(M_m * D_m) / l) * Float64(-0.25 * Float64(h / d))), 1.0));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_1) * t_0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(D$95$m * 0.5), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$3 = N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[d, -1.25e-176], N[(t$95$0 * N[(t$95$1 * N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.22e-300], N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$3 * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.2e-260], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(-0.25 * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.25e-176

   1. Initial program 80.2%

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

   4. Applied rewrites 84.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 84.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{{h}^{-1}}\right) \]

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lower-/.f64 92.4

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

   6. Applied rewrites 92.4%

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

   if -1.25e-176 < d < -1.22e-300

   1. Initial program 37.9%

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

   4. Applied rewrites 36.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 37.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 60.5

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

   8. Applied rewrites 60.5%

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

   if -1.22e-300 < d < 5.19999999999999987e-260

   1. Initial program 21.4%

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

   4. Applied rewrites 21.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 30.9%

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

   7. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 31.1

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 31.1%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 60.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]

   if 5.19999999999999987e-260 < d

   1. Initial program 64.6%

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

   4. Applied rewrites 67.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 67.0

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{{h}^{-1}}\right) \]

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 81.9

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

   6. Applied rewrites 81.9%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{{h}^{-1}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{-176}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{{h}^{-1}} \cdot \frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -1.22 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right)\right)\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{-260}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{{h}^{-1}} \cdot \frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m\\ t_1 := \mathsf{fma}\left(t\_0 \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), t\_0, 1\right)\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -3.05 \cdot 10^{+137}:\\ \;\;\;\;\frac{t\_2}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot t\_1\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\sqrt{\frac{d}{\ell}} \cdot t\_1\right) \cdot t\_2}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M\_m}{d} \cdot \left(\left(D\_m \cdot 0.5\right) \cdot 0.5\right)}{{h}^{-1}} \cdot \frac{M\_m \cdot \left(D\_m \cdot \frac{0.5}{d}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (* M_m (/ 0.5 d)) D_m))
        (t_1 (fma (* t_0 (* -0.5 (/ h l))) t_0 1.0))
        (t_2 (sqrt (- d))))
   (if (<= l -3.05e+137)
     (* (/ t_2 (sqrt (- l))) (* (sqrt (/ d h)) t_1))
     (if (<= l -2e-310)
       (/ (* (* (sqrt (/ d l)) t_1) t_2) (sqrt (- h)))
       (*
        (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (/ 1.0 2.0)))
        (-
         1.0
         (*
          (/ (* (/ M_m d) (* (* D_m 0.5) 0.5)) (pow h -1.0))
          (/ (* M_m (* D_m (/ 0.5 d))) l))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m * (0.5 / d)) * D_m;
	double t_1 = fma((t_0 * (-0.5 * (h / l))), t_0, 1.0);
	double t_2 = sqrt(-d);
	double tmp;
	if (l <= -3.05e+137) {
		tmp = (t_2 / sqrt(-l)) * (sqrt((d / h)) * t_1);
	} else if (l <= -2e-310) {
		tmp = ((sqrt((d / l)) * t_1) * t_2) / sqrt(-h);
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), (1.0 / 2.0))) * (1.0 - ((((M_m / d) * ((D_m * 0.5) * 0.5)) / pow(h, -1.0)) * ((M_m * (D_m * (0.5 / d))) / l)));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m * Float64(0.5 / d)) * D_m)
	t_1 = fma(Float64(t_0 * Float64(-0.5 * Float64(h / l))), t_0, 1.0)
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -3.05e+137)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * t_1));
	elseif (l <= -2e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(d / l)) * t_1) * t_2) / sqrt(Float64(-h)));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(Float64(M_m / d) * Float64(Float64(D_m * 0.5) * 0.5)) / (h ^ -1.0)) * Float64(Float64(M_m * Float64(D_m * Float64(0.5 / d))) / l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -3.05e+137], N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(D$95$m * 0.5), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.05000000000000002e137

   1. Initial program 70.1%

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

   4. Applied rewrites 70.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 69.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 80.4

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

   8. Applied rewrites 80.4%

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

   if -3.05000000000000002e137 < l < -1.999999999999994e-310

   1. Initial program 73.3%

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

   4. Applied rewrites 84.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 85.9%

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

   if -1.999999999999994e-310 < l

   1. Initial program 61.7%

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

   4. Applied rewrites 63.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 63.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{{h}^{-1}}\right) \]

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 78.5

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

   6. Applied rewrites 78.5%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.05 \cdot 10^{+137}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right)\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{{h}^{-1}} \cdot \frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m\\ t_2 := \mathsf{fma}\left(t\_1 \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), t\_1, 1\right)\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -3.05 \cdot 10^{+137}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot t\_2\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(t\_3 \cdot t\_2\right) \cdot t\_0}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_2\right) \cdot t\_3\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (- d)))
        (t_1 (* (* M_m (/ 0.5 d)) D_m))
        (t_2 (fma (* t_1 (* -0.5 (/ h l))) t_1 1.0))
        (t_3 (sqrt (/ d l))))
   (if (<= l -3.05e+137)
     (* (/ t_0 (sqrt (- l))) (* (sqrt (/ d h)) t_2))
     (if (<= l -2e-310)
       (/ (* (* t_3 t_2) t_0) (sqrt (- h)))
       (* (* (/ (sqrt d) (sqrt h)) t_2) t_3)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(-d);
	double t_1 = (M_m * (0.5 / d)) * D_m;
	double t_2 = fma((t_1 * (-0.5 * (h / l))), t_1, 1.0);
	double t_3 = sqrt((d / l));
	double tmp;
	if (l <= -3.05e+137) {
		tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * t_2);
	} else if (l <= -2e-310) {
		tmp = ((t_3 * t_2) * t_0) / sqrt(-h);
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_2) * t_3;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(Float64(M_m * Float64(0.5 / d)) * D_m)
	t_2 = fma(Float64(t_1 * Float64(-0.5 * Float64(h / l))), t_1, 1.0)
	t_3 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -3.05e+137)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * t_2));
	elseif (l <= -2e-310)
		tmp = Float64(Float64(Float64(t_3 * t_2) * t_0) / sqrt(Float64(-h)));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_2) * t_3);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.05e+137], N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[(N[(t$95$3 * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.05000000000000002e137

   1. Initial program 70.1%

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

   4. Applied rewrites 70.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 69.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 80.4

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

   8. Applied rewrites 80.4%

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

   if -3.05000000000000002e137 < l < -1.999999999999994e-310

   1. Initial program 73.3%

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

   4. Applied rewrites 84.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 85.9%

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

   if -1.999999999999994e-310 < l

   1. Initial program 61.7%

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

   4. Applied rewrites 60.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 62.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow1/2 N/A

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

      8. lower-sqrt.f64 77.0

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

   8. Applied rewrites 77.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.05 \cdot 10^{+137}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right)\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \mathsf{fma}\left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m\\ t_1 := \mathsf{fma}\left(t\_0 \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), t\_0, 1\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+137}:\\ \;\;\;\;\frac{t\_3}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot t\_1\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_1 \cdot \frac{t\_3}{\sqrt{-h}}\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_1\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (* M_m (/ 0.5 d)) D_m))
        (t_1 (fma (* t_0 (* -0.5 (/ h l))) t_0 1.0))
        (t_2 (sqrt (/ d l)))
        (t_3 (sqrt (- d))))
   (if (<= l -3.4e+137)
     (* (/ t_3 (sqrt (- l))) (* (sqrt (/ d h)) t_1))
     (if (<= l -2e-310)
       (* (* t_1 (/ t_3 (sqrt (- h)))) t_2)
       (* (* (/ (sqrt d) (sqrt h)) t_1) t_2)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m * (0.5 / d)) * D_m;
	double t_1 = fma((t_0 * (-0.5 * (h / l))), t_0, 1.0);
	double t_2 = sqrt((d / l));
	double t_3 = sqrt(-d);
	double tmp;
	if (l <= -3.4e+137) {
		tmp = (t_3 / sqrt(-l)) * (sqrt((d / h)) * t_1);
	} else if (l <= -2e-310) {
		tmp = (t_1 * (t_3 / sqrt(-h))) * t_2;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_1) * t_2;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m * Float64(0.5 / d)) * D_m)
	t_1 = fma(Float64(t_0 * Float64(-0.5 * Float64(h / l))), t_0, 1.0)
	t_2 = sqrt(Float64(d / l))
	t_3 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -3.4e+137)
		tmp = Float64(Float64(t_3 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * t_1));
	elseif (l <= -2e-310)
		tmp = Float64(Float64(t_1 * Float64(t_3 / sqrt(Float64(-h)))) * t_2);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_1) * t_2);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -3.4e+137], N[(N[(t$95$3 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[(t$95$1 * N[(t$95$3 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.39999999999999986e137

   1. Initial program 70.1%

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

   4. Applied rewrites 70.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 69.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 80.4

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

   8. Applied rewrites 80.4%

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

   if -3.39999999999999986e137 < l < -1.999999999999994e-310

   1. Initial program 73.3%

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

   4. Applied rewrites 72.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 74.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 86.8

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

   8. Applied rewrites 86.8%

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

   if -1.999999999999994e-310 < l

   1. Initial program 61.7%

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

   4. Applied rewrites 60.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 62.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow1/2 N/A

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

      8. lower-sqrt.f64 77.0

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

   8. Applied rewrites 77.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+137}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \mathsf{fma}\left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m\\ t_3 := \mathsf{fma}\left(t\_2, \frac{M\_m \cdot D\_m}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_0 \cdot t\_3\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 8.6 \cdot 10^{+132}:\\ \;\;\;\;\frac{\sqrt{d} \cdot t\_3}{\sqrt{h}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(t\_1 \cdot \mathsf{fma}\left(t\_2 \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), t\_2, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (sqrt (/ d h)))
        (t_2 (* (* M_m (/ 0.5 d)) D_m))
        (t_3 (fma t_2 (* (/ (* M_m D_m) l) (* -0.25 (/ h d))) 1.0)))
   (if (<= l -2e-310)
     (* (* t_0 t_3) t_1)
     (if (<= l 8.6e+132)
       (* (/ (* (sqrt d) t_3) (sqrt h)) t_0)
       (*
        (/ (sqrt d) (sqrt l))
        (* t_1 (fma (* t_2 (* -0.5 (/ h l))) t_2 1.0)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = sqrt((d / h));
	double t_2 = (M_m * (0.5 / d)) * D_m;
	double t_3 = fma(t_2, (((M_m * D_m) / l) * (-0.25 * (h / d))), 1.0);
	double tmp;
	if (l <= -2e-310) {
		tmp = (t_0 * t_3) * t_1;
	} else if (l <= 8.6e+132) {
		tmp = ((sqrt(d) * t_3) / sqrt(h)) * t_0;
	} else {
		tmp = (sqrt(d) / sqrt(l)) * (t_1 * fma((t_2 * (-0.5 * (h / l))), t_2, 1.0));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = Float64(Float64(M_m * Float64(0.5 / d)) * D_m)
	t_3 = fma(t_2, Float64(Float64(Float64(M_m * D_m) / l) * Float64(-0.25 * Float64(h / d))), 1.0)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(t_0 * t_3) * t_1);
	elseif (l <= 8.6e+132)
		tmp = Float64(Float64(Float64(sqrt(d) * t_3) / sqrt(h)) * t_0);
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(t_1 * fma(Float64(t_2 * Float64(-0.5 * Float64(h / l))), t_2, 1.0)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(-0.25 * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(N[(t$95$0 * t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 8.6e+132], N[(N[(N[(N[Sqrt[d], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[(t$95$2 * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 72.2%

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

   4. Applied rewrites 71.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 72.8%

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

   7. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 72.2

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 72.2%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 72.2%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   if -1.999999999999994e-310 < l < 8.59999999999999964e132

   1. Initial program 63.1%

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

   4. Applied rewrites 61.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 63.1%

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

   7. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 65.0

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 65.0%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 78.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

   if 8.59999999999999964e132 < l

   1. Initial program 58.3%

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

   4. Applied rewrites 58.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 61.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 75.3

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

   8. Applied rewrites 75.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 8.6 \cdot 10^{+132}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m\\ t_1 := \mathsf{fma}\left(t\_0 \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), t\_0, 1\right)\\ \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_1\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (* M_m (/ 0.5 d)) D_m))
        (t_1 (fma (* t_0 (* -0.5 (/ h l))) t_0 1.0)))
   (if (<= h -2e-310)
     (* (/ (sqrt (- d)) (sqrt (- l))) (* (sqrt (/ d h)) t_1))
     (* (* (/ (sqrt d) (sqrt h)) t_1) (sqrt (/ d l))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m * (0.5 / d)) * D_m;
	double t_1 = fma((t_0 * (-0.5 * (h / l))), t_0, 1.0);
	double tmp;
	if (h <= -2e-310) {
		tmp = (sqrt(-d) / sqrt(-l)) * (sqrt((d / h)) * t_1);
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_1) * sqrt((d / l));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m * Float64(0.5 / d)) * D_m)
	t_1 = fma(Float64(t_0 * Float64(-0.5 * Float64(h / l))), t_0, 1.0)
	tmp = 0.0
	if (h <= -2e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * t_1));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_1) * sqrt(Float64(d / l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]}, If[LessEqual[h, -2e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 72.2%

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

   4. Applied rewrites 71.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 72.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 78.2

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

   8. Applied rewrites 78.2%

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

   if -1.999999999999994e-310 < h

   1. Initial program 61.7%

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

   4. Applied rewrites 60.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 62.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow1/2 N/A

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

      8. lower-sqrt.f64 77.0

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

   8. Applied rewrites 77.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \mathsf{fma}\left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \mathsf{fma}\left(\left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m, \frac{M\_m \cdot D\_m}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_0 \cdot t\_1\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+194}:\\ \;\;\;\;\frac{\sqrt{d} \cdot t\_1}{\sqrt{h}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (fma
          (* (* M_m (/ 0.5 d)) D_m)
          (* (/ (* M_m D_m) l) (* -0.25 (/ h d)))
          1.0)))
   (if (<= l -2e-310)
     (* (* t_0 t_1) (sqrt (/ d h)))
     (if (<= l 6.2e+194)
       (* (/ (* (sqrt d) t_1) (sqrt h)) t_0)
       (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = fma(((M_m * (0.5 / d)) * D_m), (((M_m * D_m) / l) * (-0.25 * (h / d))), 1.0);
	double tmp;
	if (l <= -2e-310) {
		tmp = (t_0 * t_1) * sqrt((d / h));
	} else if (l <= 6.2e+194) {
		tmp = ((sqrt(d) * t_1) / sqrt(h)) * t_0;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = fma(Float64(Float64(M_m * Float64(0.5 / d)) * D_m), Float64(Float64(Float64(M_m * D_m) / l) * Float64(-0.25 * Float64(h / d))), 1.0)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(t_0 * t_1) * sqrt(Float64(d / h)));
	elseif (l <= 6.2e+194)
		tmp = Float64(Float64(Float64(sqrt(d) * t_1) / sqrt(h)) * t_0);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(-0.25 * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.2e+194], N[(N[(N[(N[Sqrt[d], $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 72.2%

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

   4. Applied rewrites 71.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 72.8%

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

   7. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 72.2

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 72.2%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 72.2%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   if -1.999999999999994e-310 < l < 6.1999999999999999e194

   1. Initial program 63.5%

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

   4. Applied rewrites 62.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 63.5%

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

   7. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 65.2

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 65.2%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 77.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

   if 6.1999999999999999e194 < l

   1. Initial program 54.1%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 70.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 79.6%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+194}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m\\ \mathbf{if}\;\ell \leq 1.55 \cdot 10^{-230}:\\ \;\;\;\;\left(t\_0 \cdot \mathsf{fma}\left(t\_1, \frac{M\_m \cdot D\_m}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \mathsf{fma}\left(t\_1 \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), t\_1, 1\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (* (* M_m (/ 0.5 d)) D_m)))
   (if (<= l 1.55e-230)
     (*
      (* t_0 (fma t_1 (* (/ (* M_m D_m) l) (* -0.25 (/ h d))) 1.0))
      (sqrt (/ d h)))
     (*
      (* (/ (sqrt d) (sqrt h)) (fma (* t_1 (* -0.5 (/ h l))) t_1 1.0))
      t_0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = (M_m * (0.5 / d)) * D_m;
	double tmp;
	if (l <= 1.55e-230) {
		tmp = (t_0 * fma(t_1, (((M_m * D_m) / l) * (-0.25 * (h / d))), 1.0)) * sqrt((d / h));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * fma((t_1 * (-0.5 * (h / l))), t_1, 1.0)) * t_0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64(M_m * Float64(0.5 / d)) * D_m)
	tmp = 0.0
	if (l <= 1.55e-230)
		tmp = Float64(Float64(t_0 * fma(t_1, Float64(Float64(Float64(M_m * D_m) / l) * Float64(-0.25 * Float64(h / d))), 1.0)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * fma(Float64(t_1 * Float64(-0.5 * Float64(h / l))), t_1, 1.0)) * t_0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[l, 1.55e-230], N[(N[(t$95$0 * N[(t$95$1 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(-0.25 * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.55e-230

   1. Initial program 73.5%

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

   4. Applied rewrites 72.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 74.1%

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

   7. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 74.1

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 74.1%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 74.1%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   if 1.55e-230 < l

   1. Initial program 58.3%

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

   4. Applied rewrites 58.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 59.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow1/2 N/A

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

      8. lower-sqrt.f64 75.9

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

   8. Applied rewrites 75.9%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{-230}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \mathsf{fma}\left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{0.5}{d}\\ \mathbf{if}\;\ell \leq -2.35 \cdot 10^{+104}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \cdot \mathsf{fma}\left(t\_0, \left(\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot D\_m\right) \cdot t\_0\right) \cdot D\_m, 1\right)\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(t\_0 \cdot D\_m, \frac{M\_m \cdot D\_m}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* M_m (/ 0.5 d))))
   (if (<= l -2.35e+104)
     (/ (* (sqrt (/ d l)) (sqrt (- d))) (sqrt (- h)))
     (if (<= l 4.4e-281)
       (*
        (sqrt (* (/ (/ d l) h) d))
        (fma t_0 (* (* (* (* -0.5 (/ h l)) D_m) t_0) D_m) 1.0))
       (if (<= l 5.8e+205)
         (*
          (/ d (sqrt (* l h)))
          (fma (* t_0 D_m) (* (/ (* M_m D_m) l) (* -0.25 (/ h d))) 1.0))
         (/ d (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * (0.5 / d);
	double tmp;
	if (l <= -2.35e+104) {
		tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h);
	} else if (l <= 4.4e-281) {
		tmp = sqrt((((d / l) / h) * d)) * fma(t_0, ((((-0.5 * (h / l)) * D_m) * t_0) * D_m), 1.0);
	} else if (l <= 5.8e+205) {
		tmp = (d / sqrt((l * h))) * fma((t_0 * D_m), (((M_m * D_m) / l) * (-0.25 * (h / d))), 1.0);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m * Float64(0.5 / d))
	tmp = 0.0
	if (l <= -2.35e+104)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (l <= 4.4e-281)
		tmp = Float64(sqrt(Float64(Float64(Float64(d / l) / h) * d)) * fma(t_0, Float64(Float64(Float64(Float64(-0.5 * Float64(h / l)) * D_m) * t_0) * D_m), 1.0));
	elseif (l <= 5.8e+205)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * fma(Float64(t_0 * D_m), Float64(Float64(Float64(M_m * D_m) / l) * Float64(-0.25 * Float64(h / d))), 1.0));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.35e+104], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.4e-281], N[(N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.8e+205], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(-0.25 * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.35000000000000008e104

   1. Initial program 69.2%

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in h around 0

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

      2. lower-/.f64 67.2

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   7. Applied rewrites 67.2%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   if -2.35000000000000008e104 < l < 4.40000000000000008e-281

   1. Initial program 74.9%

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

   4. Applied rewrites 65.8%

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

   6. Applied rewrites 66.9%

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

   if 4.40000000000000008e-281 < l < 5.8000000000000003e205

   1. Initial program 61.6%

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

   4. Applied rewrites 60.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 61.7%

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

   7. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 63.4

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 63.4%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 74.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]

   if 5.8000000000000003e205 < l

   1. Initial program 54.4%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 75.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 82.2%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.35 \cdot 10^{+104}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \cdot \mathsf{fma}\left(M \cdot \frac{0.5}{d}, \left(\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot D\right) \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot D, 1\right)\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{0.5}{d}\\ \mathbf{if}\;\ell \leq -2.35 \cdot 10^{+104}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -1.4 \cdot 10^{-295}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot D\_m\right) \cdot t\_0\right) \cdot \frac{0.5}{d}\right) \cdot M\_m, D\_m, 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(t\_0 \cdot D\_m, \frac{M\_m \cdot D\_m}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* M_m (/ 0.5 d))))
   (if (<= l -2.35e+104)
     (/ (* (sqrt (/ d l)) (sqrt (- d))) (sqrt (- h)))
     (if (<= l -1.4e-295)
       (*
        (fma (* (* (* (* (* -0.5 (/ h l)) D_m) t_0) (/ 0.5 d)) M_m) D_m 1.0)
        (sqrt (* (/ (/ d l) h) d)))
       (if (<= l 5.8e+205)
         (*
          (/ d (sqrt (* l h)))
          (fma (* t_0 D_m) (* (/ (* M_m D_m) l) (* -0.25 (/ h d))) 1.0))
         (/ d (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * (0.5 / d);
	double tmp;
	if (l <= -2.35e+104) {
		tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h);
	} else if (l <= -1.4e-295) {
		tmp = fma((((((-0.5 * (h / l)) * D_m) * t_0) * (0.5 / d)) * M_m), D_m, 1.0) * sqrt((((d / l) / h) * d));
	} else if (l <= 5.8e+205) {
		tmp = (d / sqrt((l * h))) * fma((t_0 * D_m), (((M_m * D_m) / l) * (-0.25 * (h / d))), 1.0);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m * Float64(0.5 / d))
	tmp = 0.0
	if (l <= -2.35e+104)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (l <= -1.4e-295)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(-0.5 * Float64(h / l)) * D_m) * t_0) * Float64(0.5 / d)) * M_m), D_m, 1.0) * sqrt(Float64(Float64(Float64(d / l) / h) * d)));
	elseif (l <= 5.8e+205)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * fma(Float64(t_0 * D_m), Float64(Float64(Float64(M_m * D_m) / l) * Float64(-0.25 * Float64(h / d))), 1.0));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.35e+104], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.4e-295], N[(N[(N[(N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.8e+205], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(-0.25 * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.35000000000000008e104

   1. Initial program 69.2%

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in h around 0

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

      2. lower-/.f64 67.2

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   7. Applied rewrites 67.2%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   if -2.35000000000000008e104 < l < -1.4e-295

   1. Initial program 74.9%

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

   4. Applied rewrites 64.9%

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

   6. Applied rewrites 64.9%

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

   if -1.4e-295 < l < 5.8000000000000003e205

   1. Initial program 62.6%

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

   4. Applied rewrites 61.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 62.7%

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

   7. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 65.2

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 65.2%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 73.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]

   if 5.8000000000000003e205 < l

   1. Initial program 54.4%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 75.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 82.2%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.35 \cdot 10^{+104}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -1.4 \cdot 10^{-295}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot D\right) \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot \frac{0.5}{d}\right) \cdot M, D, 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{0.5}{d}\\ \mathbf{if}\;\ell \leq -1.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -1.4 \cdot 10^{-295}:\\ \;\;\;\;\mathsf{fma}\left(M\_m, \left(\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot D\_m\right) \cdot t\_0\right) \cdot \left(D\_m \cdot \frac{0.5}{d}\right), 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(t\_0 \cdot D\_m, \frac{M\_m \cdot D\_m}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* M_m (/ 0.5 d))))
   (if (<= l -1.5e+102)
     (/ (* (sqrt (/ d l)) (sqrt (- d))) (sqrt (- h)))
     (if (<= l -1.4e-295)
       (*
        (fma M_m (* (* (* (* -0.5 (/ h l)) D_m) t_0) (* D_m (/ 0.5 d))) 1.0)
        (sqrt (* (/ (/ d l) h) d)))
       (if (<= l 5.8e+205)
         (*
          (/ d (sqrt (* l h)))
          (fma (* t_0 D_m) (* (/ (* M_m D_m) l) (* -0.25 (/ h d))) 1.0))
         (/ d (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * (0.5 / d);
	double tmp;
	if (l <= -1.5e+102) {
		tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h);
	} else if (l <= -1.4e-295) {
		tmp = fma(M_m, ((((-0.5 * (h / l)) * D_m) * t_0) * (D_m * (0.5 / d))), 1.0) * sqrt((((d / l) / h) * d));
	} else if (l <= 5.8e+205) {
		tmp = (d / sqrt((l * h))) * fma((t_0 * D_m), (((M_m * D_m) / l) * (-0.25 * (h / d))), 1.0);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m * Float64(0.5 / d))
	tmp = 0.0
	if (l <= -1.5e+102)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (l <= -1.4e-295)
		tmp = Float64(fma(M_m, Float64(Float64(Float64(Float64(-0.5 * Float64(h / l)) * D_m) * t_0) * Float64(D_m * Float64(0.5 / d))), 1.0) * sqrt(Float64(Float64(Float64(d / l) / h) * d)));
	elseif (l <= 5.8e+205)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * fma(Float64(t_0 * D_m), Float64(Float64(Float64(M_m * D_m) / l) * Float64(-0.25 * Float64(h / d))), 1.0));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.5e+102], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.4e-295], N[(N[(M$95$m * N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.8e+205], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(-0.25 * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.4999999999999999e102

   1. Initial program 69.2%

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in h around 0

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

      2. lower-/.f64 67.2

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   7. Applied rewrites 67.2%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   if -1.4999999999999999e102 < l < -1.4e-295

   1. Initial program 74.9%

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

   4. Applied rewrites 64.9%

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

   6. Applied rewrites 63.6%

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

   if -1.4e-295 < l < 5.8000000000000003e205

   1. Initial program 62.6%

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

   4. Applied rewrites 61.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 62.7%

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

   7. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 65.2

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 65.2%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 73.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]

   if 5.8000000000000003e205 < l

   1. Initial program 54.4%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 75.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 82.2%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -1.4 \cdot 10^{-295}:\\ \;\;\;\;\mathsf{fma}\left(M, \left(\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot D\right) \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1.6 \cdot 10^{-139}:\\ \;\;\;\;\frac{t\_0 \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \left(\left(M\_m \cdot M\_m\right) \cdot h\right)}{d}, \frac{D\_m \cdot D\_m}{d}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_0\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m, \frac{M\_m \cdot D\_m}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= l -1.6e-139)
     (/ (* t_0 (sqrt (- d))) (sqrt (- h)))
     (if (<= l -2e-310)
       (*
        (*
         (/ (fma (/ (* -0.125 (* (* M_m M_m) h)) d) (/ (* D_m D_m) d) l) l)
         (sqrt (/ d h)))
        t_0)
       (if (<= l 5.8e+205)
         (*
          (/ d (sqrt (* l h)))
          (fma
           (* (* M_m (/ 0.5 d)) D_m)
           (* (/ (* M_m D_m) l) (* -0.25 (/ h d)))
           1.0))
         (/ d (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (l <= -1.6e-139) {
		tmp = (t_0 * sqrt(-d)) / sqrt(-h);
	} else if (l <= -2e-310) {
		tmp = ((fma(((-0.125 * ((M_m * M_m) * h)) / d), ((D_m * D_m) / d), l) / l) * sqrt((d / h))) * t_0;
	} else if (l <= 5.8e+205) {
		tmp = (d / sqrt((l * h))) * fma(((M_m * (0.5 / d)) * D_m), (((M_m * D_m) / l) * (-0.25 * (h / d))), 1.0);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -1.6e-139)
		tmp = Float64(Float64(t_0 * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (l <= -2e-310)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(-0.125 * Float64(Float64(M_m * M_m) * h)) / d), Float64(Float64(D_m * D_m) / d), l) / l) * sqrt(Float64(d / h))) * t_0);
	elseif (l <= 5.8e+205)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * fma(Float64(Float64(M_m * Float64(0.5 / d)) * D_m), Float64(Float64(Float64(M_m * D_m) / l) * Float64(-0.25 * Float64(h / d))), 1.0));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.6e-139], N[(N[(t$95$0 * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[(N[(N[(N[(N[(-0.125 * N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] + l), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[l, 5.8e+205], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(-0.25 * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.6e-139

   1. Initial program 73.2%

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

   4. Applied rewrites 80.0%

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

   5. Taylor expanded in h around 0

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

      2. lower-/.f64 59.2

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   7. Applied rewrites 59.2%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   if -1.6e-139 < l < -1.999999999999994e-310

   1. Initial program 68.7%

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

   4. Applied rewrites 70.7%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 64.2%

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

   if -1.999999999999994e-310 < l < 5.8000000000000003e205

   1. Initial program 63.3%

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

   4. Applied rewrites 62.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 63.3%

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

   7. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 64.9

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 64.9%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 73.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]

   if 5.8000000000000003e205 < l

   1. Initial program 54.4%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 75.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 82.2%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.6 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}, \frac{D \cdot D}{d}, \ell\right)}{\ell} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m, \frac{M\_m \cdot D\_m}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{if}\;\ell \leq 8.5 \cdot 10^{-259}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot t\_0\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (fma
          (* (* M_m (/ 0.5 d)) D_m)
          (* (/ (* M_m D_m) l) (* -0.25 (/ h d)))
          1.0)))
   (if (<= l 8.5e-259)
     (* (* (sqrt (/ d l)) t_0) (sqrt (/ d h)))
     (if (<= l 5.8e+205)
       (* (/ d (sqrt (* l h))) t_0)
       (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = fma(((M_m * (0.5 / d)) * D_m), (((M_m * D_m) / l) * (-0.25 * (h / d))), 1.0);
	double tmp;
	if (l <= 8.5e-259) {
		tmp = (sqrt((d / l)) * t_0) * sqrt((d / h));
	} else if (l <= 5.8e+205) {
		tmp = (d / sqrt((l * h))) * t_0;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = fma(Float64(Float64(M_m * Float64(0.5 / d)) * D_m), Float64(Float64(Float64(M_m * D_m) / l) * Float64(-0.25 * Float64(h / d))), 1.0)
	tmp = 0.0
	if (l <= 8.5e-259)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * t_0) * sqrt(Float64(d / h)));
	elseif (l <= 5.8e+205)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * t_0);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(-0.25 * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[l, 8.5e-259], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.8e+205], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 8.4999999999999994e-259

   1. Initial program 73.1%

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

   4. Applied rewrites 72.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 73.7%

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

   7. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 73.8

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 73.8%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 73.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   if 8.4999999999999994e-259 < l < 5.8000000000000003e205

   1. Initial program 60.6%

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

   4. Applied rewrites 60.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 60.6%

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

   7. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 61.4

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 61.4%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 73.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]

   if 5.8000000000000003e205 < l

   1. Initial program 54.4%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 75.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 82.2%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{-259}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m\\ \mathbf{if}\;\ell \leq 8.5 \cdot 10^{-259}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(M\_m \cdot D\_m\right) \cdot -0.25}{\ell} \cdot \frac{h}{d}, t\_0, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(t\_0, \frac{M\_m \cdot D\_m}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (* M_m (/ 0.5 d)) D_m)))
   (if (<= l 8.5e-259)
     (*
      (* (fma (* (/ (* (* M_m D_m) -0.25) l) (/ h d)) t_0 1.0) (sqrt (/ d h)))
      (sqrt (/ d l)))
     (if (<= l 5.8e+205)
       (*
        (/ d (sqrt (* l h)))
        (fma t_0 (* (/ (* M_m D_m) l) (* -0.25 (/ h d))) 1.0))
       (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m * (0.5 / d)) * D_m;
	double tmp;
	if (l <= 8.5e-259) {
		tmp = (fma(((((M_m * D_m) * -0.25) / l) * (h / d)), t_0, 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (l <= 5.8e+205) {
		tmp = (d / sqrt((l * h))) * fma(t_0, (((M_m * D_m) / l) * (-0.25 * (h / d))), 1.0);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m * Float64(0.5 / d)) * D_m)
	tmp = 0.0
	if (l <= 8.5e-259)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(M_m * D_m) * -0.25) / l) * Float64(h / d)), t_0, 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (l <= 5.8e+205)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * fma(t_0, Float64(Float64(Float64(M_m * D_m) / l) * Float64(-0.25 * Float64(h / d))), 1.0));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[l, 8.5e-259], N[(N[(N[(N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * -0.25), $MachinePrecision] / l), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.8e+205], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(-0.25 * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 8.4999999999999994e-259

   1. Initial program 73.1%

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

   4. Applied rewrites 72.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 73.7%

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

   7. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 73.8

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 73.8%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if 8.4999999999999994e-259 < l < 5.8000000000000003e205

   1. Initial program 60.6%

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

   4. Applied rewrites 60.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 60.6%

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

   7. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 61.4

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 61.4%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 73.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]

   if 5.8000000000000003e205 < l

   1. Initial program 54.4%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 75.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 82.2%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{-259}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot -0.25}{\ell} \cdot \frac{h}{d}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(-h, \frac{\frac{D\_m \cdot D\_m}{d}}{d} \cdot \frac{0.125 \cdot \left(M\_m \cdot M\_m\right)}{\ell}, 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m, \frac{M\_m \cdot D\_m}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -7.8e+101)
   (/ (* (sqrt (/ d l)) (sqrt (- d))) (sqrt (- h)))
   (if (<= l -2e-310)
     (*
      (fma (- h) (* (/ (/ (* D_m D_m) d) d) (/ (* 0.125 (* M_m M_m)) l)) 1.0)
      (sqrt (* (/ (/ d l) h) d)))
     (if (<= l 5.8e+205)
       (*
        (/ d (sqrt (* l h)))
        (fma
         (* (* M_m (/ 0.5 d)) D_m)
         (* (/ (* M_m D_m) l) (* -0.25 (/ h d)))
         1.0))
       (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -7.8e+101) {
		tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h);
	} else if (l <= -2e-310) {
		tmp = fma(-h, ((((D_m * D_m) / d) / d) * ((0.125 * (M_m * M_m)) / l)), 1.0) * sqrt((((d / l) / h) * d));
	} else if (l <= 5.8e+205) {
		tmp = (d / sqrt((l * h))) * fma(((M_m * (0.5 / d)) * D_m), (((M_m * D_m) / l) * (-0.25 * (h / d))), 1.0);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -7.8e+101)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (l <= -2e-310)
		tmp = Float64(fma(Float64(-h), Float64(Float64(Float64(Float64(D_m * D_m) / d) / d) * Float64(Float64(0.125 * Float64(M_m * M_m)) / l)), 1.0) * sqrt(Float64(Float64(Float64(d / l) / h) * d)));
	elseif (l <= 5.8e+205)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * fma(Float64(Float64(M_m * Float64(0.5 / d)) * D_m), Float64(Float64(Float64(M_m * D_m) / l) * Float64(-0.25 * Float64(h / d))), 1.0));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -7.8e+101], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[((-h) * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * N[(N[(0.125 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.8e+205], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(-0.25 * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -7.8e101

   1. Initial program 69.2%

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in h around 0

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

      2. lower-/.f64 67.2

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   7. Applied rewrites 67.2%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   if -7.8e101 < l < -1.999999999999994e-310

   1. Initial program 74.0%

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. distribute-neg-frac2 N/A

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

      5. mul-1-neg N/A

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

      6. rgt-mult-inverse N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 52.8%

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

   if -1.999999999999994e-310 < l < 5.8000000000000003e205

   1. Initial program 63.3%

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

   4. Applied rewrites 62.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\color{blue}{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. clear-num N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. unpow2 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. associate-*r* N/A

          \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d}, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 63.3%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   7. Taylor expanded in h around 0

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot \left(D \cdot \left(M \cdot h\right)\right)}{d \cdot \ell}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{4} \cdot \color{blue}{\left(\left(D \cdot M\right) \cdot h\right)}}{d \cdot \ell}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\color{blue}{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right) \cdot h}}{d \cdot \ell}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right) \cdot h}{\color{blue}{\ell \cdot d}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. times-frac N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot \left(D \cdot M\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot \left(D \cdot M\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot \left(D \cdot M\right)}{\ell}} \cdot \frac{h}{d}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{4} \cdot \left(D \cdot M\right)}}{\ell} \cdot \frac{h}{d}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{4} \cdot \color{blue}{\left(M \cdot D\right)}}{\ell} \cdot \frac{h}{d}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{4} \cdot \color{blue}{\left(M \cdot D\right)}}{\ell} \cdot \frac{h}{d}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lower-/.f64 64.9

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 64.9%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 73.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]

   if 5.8000000000000003e205 < l

   1. Initial program 54.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 75.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 82.2%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(-h, \frac{\frac{D \cdot D}{d}}{d} \cdot \frac{0.125 \cdot \left(M \cdot M\right)}{\ell}, 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \left(\left(M\_m \cdot M\_m\right) \cdot h\right)}{d}, \frac{D\_m \cdot D\_m}{d}, \ell\right)}{\ell} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m, \frac{M\_m \cdot D\_m}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.35e-139)
   (/ (* (sqrt (/ d l)) (sqrt (- d))) (sqrt (- h)))
   (if (<= l -2e-310)
     (*
      (/ (fma (/ (* -0.125 (* (* M_m M_m) h)) d) (/ (* D_m D_m) d) l) l)
      (sqrt (* (/ (/ d l) h) d)))
     (if (<= l 5.8e+205)
       (*
        (/ d (sqrt (* l h)))
        (fma
         (* (* M_m (/ 0.5 d)) D_m)
         (* (/ (* M_m D_m) l) (* -0.25 (/ h d)))
         1.0))
       (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.35e-139) {
		tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h);
	} else if (l <= -2e-310) {
		tmp = (fma(((-0.125 * ((M_m * M_m) * h)) / d), ((D_m * D_m) / d), l) / l) * sqrt((((d / l) / h) * d));
	} else if (l <= 5.8e+205) {
		tmp = (d / sqrt((l * h))) * fma(((M_m * (0.5 / d)) * D_m), (((M_m * D_m) / l) * (-0.25 * (h / d))), 1.0);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.35e-139)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (l <= -2e-310)
		tmp = Float64(Float64(fma(Float64(Float64(-0.125 * Float64(Float64(M_m * M_m) * h)) / d), Float64(Float64(D_m * D_m) / d), l) / l) * sqrt(Float64(Float64(Float64(d / l) / h) * d)));
	elseif (l <= 5.8e+205)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * fma(Float64(Float64(M_m * Float64(0.5 / d)) * D_m), Float64(Float64(Float64(M_m * D_m) / l) * Float64(-0.25 * Float64(h / d))), 1.0));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.35e-139], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[(N[(N[(N[(-0.125 * N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] + l), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.8e+205], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(-0.25 * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.3499999999999999e-139

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}} \]

   5. Taylor expanded in h around 0

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

      2. lower-/.f64 59.2

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   7. Applied rewrites 59.2%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   if -1.3499999999999999e-139 < l < -1.999999999999994e-310

   1. Initial program 68.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 64.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\frac{\ell + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

   7. Applied rewrites 61.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot -0.125}{d}, \frac{D \cdot D}{d}, \ell\right)}{\ell}} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

   if -1.999999999999994e-310 < l < 5.8000000000000003e205

   1. Initial program 63.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2} + 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{{\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{\color{blue}{\left(-1 \cdot 2\right)}} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. pow-pow N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{{\left({\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-1}\right)}^{2}} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. inv-pow N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\color{blue}{\left(\frac{1}{\frac{2}{M} \cdot \frac{d}{D}}\right)}}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\color{blue}{\frac{2}{M} \cdot \frac{d}{D}}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\color{blue}{\frac{2}{M}} \cdot \frac{d}{D}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\frac{2}{M} \cdot \color{blue}{\frac{d}{D}}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. frac-times N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\color{blue}{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. clear-num N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. unpow2 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. associate-*r* N/A

          \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d}, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 63.3%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   7. Taylor expanded in h around 0

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot \left(D \cdot \left(M \cdot h\right)\right)}{d \cdot \ell}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{4} \cdot \color{blue}{\left(\left(D \cdot M\right) \cdot h\right)}}{d \cdot \ell}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\color{blue}{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right) \cdot h}}{d \cdot \ell}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right) \cdot h}{\color{blue}{\ell \cdot d}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. times-frac N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot \left(D \cdot M\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot \left(D \cdot M\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot \left(D \cdot M\right)}{\ell}} \cdot \frac{h}{d}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{4} \cdot \left(D \cdot M\right)}}{\ell} \cdot \frac{h}{d}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{4} \cdot \color{blue}{\left(M \cdot D\right)}}{\ell} \cdot \frac{h}{d}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{4} \cdot \color{blue}{\left(M \cdot D\right)}}{\ell} \cdot \frac{h}{d}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lower-/.f64 64.9

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 64.9%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 73.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]

   if 5.8000000000000003e205 < l

   1. Initial program 54.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 75.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 82.2%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}, \frac{D \cdot D}{d}, \ell\right)}{\ell} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 60.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\frac{M\_m \cdot M\_m}{d}}{d} \cdot \left(\left(\frac{D\_m \cdot D\_m}{\ell} \cdot h\right) \cdot -0.125\right)\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m, \frac{M\_m \cdot D\_m}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.7e-220)
   (/ (* (sqrt (/ d l)) (sqrt (- d))) (sqrt (- h)))
   (if (<= l -2e-310)
     (*
      (* (/ (/ (* M_m M_m) d) d) (* (* (/ (* D_m D_m) l) h) -0.125))
      (sqrt (* (/ (/ d l) h) d)))
     (if (<= l 5.8e+205)
       (*
        (/ d (sqrt (* l h)))
        (fma
         (* (* M_m (/ 0.5 d)) D_m)
         (* (/ (* M_m D_m) l) (* -0.25 (/ h d)))
         1.0))
       (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.7e-220) {
		tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h);
	} else if (l <= -2e-310) {
		tmp = ((((M_m * M_m) / d) / d) * ((((D_m * D_m) / l) * h) * -0.125)) * sqrt((((d / l) / h) * d));
	} else if (l <= 5.8e+205) {
		tmp = (d / sqrt((l * h))) * fma(((M_m * (0.5 / d)) * D_m), (((M_m * D_m) / l) * (-0.25 * (h / d))), 1.0);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.7e-220)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (l <= -2e-310)
		tmp = Float64(Float64(Float64(Float64(Float64(M_m * M_m) / d) / d) * Float64(Float64(Float64(Float64(D_m * D_m) / l) * h) * -0.125)) * sqrt(Float64(Float64(Float64(d / l) / h) * d)));
	elseif (l <= 5.8e+205)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * fma(Float64(Float64(M_m * Float64(0.5 / d)) * D_m), Float64(Float64(Float64(M_m * D_m) / l) * Float64(-0.25 * Float64(h / d))), 1.0));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.7e-220], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * h), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.8e+205], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(-0.25 * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.69999999999999997e-220

   1. Initial program 72.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}} \]

   5. Taylor expanded in h around 0

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

      2. lower-/.f64 57.4

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   7. Applied rewrites 57.4%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   if -1.69999999999999997e-220 < l < -1.999999999999994e-310

   1. Initial program 69.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{\color{blue}{\left({D}^{2} \cdot h\right) \cdot {M}^{2}}}{{d}^{2} \cdot \ell}\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      3. associate-*l/ N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot {M}^{2}\right)}\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot {M}^{2}\right)} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}} \cdot {M}^{2}\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot h\right)\right) \cdot {M}^{2}}{{d}^{2} \cdot \ell}} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot h\right)\right) \cdot {M}^{2}}{\color{blue}{\ell \cdot {d}^{2}}} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot h\right)}{\ell} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot h\right)}{\ell} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      10. associate-/l* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)} \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)} \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      12. *-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\color{blue}{h \cdot {D}^{2}}}{\ell}\right) \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      13. associate-/l* N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \color{blue}{\left(h \cdot \frac{{D}^{2}}{\ell}\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \color{blue}{\left(h \cdot \frac{{D}^{2}}{\ell}\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      15. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(h \cdot \color{blue}{\frac{{D}^{2}}{\ell}}\right)\right) \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      16. unpow2 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(h \cdot \frac{\color{blue}{D \cdot D}}{\ell}\right)\right) \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(h \cdot \frac{\color{blue}{D \cdot D}}{\ell}\right)\right) \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

   7. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\left(\left(-0.125 \cdot \left(h \cdot \frac{D \cdot D}{\ell}\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}\right)} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]

   if -1.999999999999994e-310 < l < 5.8000000000000003e205

   1. Initial program 63.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2} + 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{{\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{\color{blue}{\left(-1 \cdot 2\right)}} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. pow-pow N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{{\left({\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-1}\right)}^{2}} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. inv-pow N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\color{blue}{\left(\frac{1}{\frac{2}{M} \cdot \frac{d}{D}}\right)}}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\color{blue}{\frac{2}{M} \cdot \frac{d}{D}}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\color{blue}{\frac{2}{M}} \cdot \frac{d}{D}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\frac{2}{M} \cdot \color{blue}{\frac{d}{D}}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. frac-times N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\color{blue}{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. clear-num N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. unpow2 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. associate-*r* N/A

          \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d}, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 63.3%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   7. Taylor expanded in h around 0

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot \left(D \cdot \left(M \cdot h\right)\right)}{d \cdot \ell}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{4} \cdot \color{blue}{\left(\left(D \cdot M\right) \cdot h\right)}}{d \cdot \ell}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\color{blue}{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right) \cdot h}}{d \cdot \ell}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right) \cdot h}{\color{blue}{\ell \cdot d}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. times-frac N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot \left(D \cdot M\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot \left(D \cdot M\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot \left(D \cdot M\right)}{\ell}} \cdot \frac{h}{d}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{4} \cdot \left(D \cdot M\right)}}{\ell} \cdot \frac{h}{d}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{4} \cdot \color{blue}{\left(M \cdot D\right)}}{\ell} \cdot \frac{h}{d}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{4} \cdot \color{blue}{\left(M \cdot D\right)}}{\ell} \cdot \frac{h}{d}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lower-/.f64 64.9

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 64.9%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 73.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]

   if 5.8000000000000003e205 < l

   1. Initial program 54.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 75.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 82.2%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\frac{M \cdot M}{d}}{d} \cdot \left(\left(\frac{D \cdot D}{\ell} \cdot h\right) \cdot -0.125\right)\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 59.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m, \frac{M\_m \cdot D\_m}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -2e-310)
   (/ (* (sqrt (/ d l)) (sqrt (- d))) (sqrt (- h)))
   (if (<= l 5.8e+205)
     (*
      (/ d (sqrt (* l h)))
      (fma
       (* (* M_m (/ 0.5 d)) D_m)
       (* (/ (* M_m D_m) l) (* -0.25 (/ h d)))
       1.0))
     (/ d (* (sqrt l) (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2e-310) {
		tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h);
	} else if (l <= 5.8e+205) {
		tmp = (d / sqrt((l * h))) * fma(((M_m * (0.5 / d)) * D_m), (((M_m * D_m) / l) * (-0.25 * (h / d))), 1.0);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (l <= 5.8e+205)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * fma(Float64(Float64(M_m * Float64(0.5 / d)) * D_m), Float64(Float64(Float64(M_m * D_m) / l) * Float64(-0.25 * Float64(h / d))), 1.0));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -2e-310], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.8e+205], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(-0.25 * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 72.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}} \]

   5. Taylor expanded in h around 0

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

      2. lower-/.f64 51.9

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   7. Applied rewrites 51.9%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   if -1.999999999999994e-310 < l < 5.8000000000000003e205

   1. Initial program 63.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2} + 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{{\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{\color{blue}{\left(-1 \cdot 2\right)}} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. pow-pow N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{{\left({\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-1}\right)}^{2}} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. inv-pow N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\color{blue}{\left(\frac{1}{\frac{2}{M} \cdot \frac{d}{D}}\right)}}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\color{blue}{\frac{2}{M} \cdot \frac{d}{D}}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\color{blue}{\frac{2}{M}} \cdot \frac{d}{D}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\frac{2}{M} \cdot \color{blue}{\frac{d}{D}}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. frac-times N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{1}{\color{blue}{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. clear-num N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. unpow2 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. associate-*r* N/A

          \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \frac{h}{\ell}\right) \cdot \frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d}, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 63.3%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right), \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   7. Taylor expanded in h around 0

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot \left(D \cdot \left(M \cdot h\right)\right)}{d \cdot \ell}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{4} \cdot \color{blue}{\left(\left(D \cdot M\right) \cdot h\right)}}{d \cdot \ell}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\color{blue}{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right) \cdot h}}{d \cdot \ell}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right) \cdot h}{\color{blue}{\ell \cdot d}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. times-frac N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot \left(D \cdot M\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot \left(D \cdot M\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot \left(D \cdot M\right)}{\ell}} \cdot \frac{h}{d}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{4} \cdot \left(D \cdot M\right)}}{\ell} \cdot \frac{h}{d}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{4} \cdot \color{blue}{\left(M \cdot D\right)}}{\ell} \cdot \frac{h}{d}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{4} \cdot \color{blue}{\left(M \cdot D\right)}}{\ell} \cdot \frac{h}{d}, \left(M \cdot \frac{\frac{1}{2}}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lower-/.f64 64.9

          \[\leadsto \left(\mathsf{fma}\left(\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \color{blue}{\frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 64.9%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d}}, \left(M \cdot \frac{0.5}{d}\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 73.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(\frac{h}{d} \cdot -0.25\right) \cdot \frac{D \cdot M}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]

   if 5.8000000000000003e205 < l

   1. Initial program 54.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 75.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 82.2%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D, \frac{M \cdot D}{\ell} \cdot \left(-0.25 \cdot \frac{h}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 46.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{-199}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -6e-199)
   (/ (* (sqrt (/ d l)) (sqrt (- d))) (sqrt (- h)))
   (if (<= d 4.3e-128)
     (* (sqrt (/ 1.0 (* l h))) (- d))
     (/ d (* (sqrt l) (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -6e-199) {
		tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 4.3e-128) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-6d-199)) then
        tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h)
    else if (d <= 4.3d-128) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -6e-199) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt(-d)) / Math.sqrt(-h);
	} else if (d <= 4.3e-128) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -6e-199:
		tmp = (math.sqrt((d / l)) * math.sqrt(-d)) / math.sqrt(-h)
	elif d <= 4.3e-128:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -6e-199)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 4.3e-128)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -6e-199)
		tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h);
	elseif (d <= 4.3e-128)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -6e-199], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.3e-128], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.99999999999999966e-199

   1. Initial program 80.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}} \]

   5. Taylor expanded in h around 0

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

      2. lower-/.f64 57.2

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   7. Applied rewrites 57.2%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   if -5.99999999999999966e-199 < d < 4.29999999999999994e-128

   1. Initial program 40.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 34.4

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 4.29999999999999994e-128 < d

   1. Initial program 69.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 53.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 58.9%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{-199}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 44.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5.1 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5.1e-54)
   (/ (sqrt (/ d l)) (sqrt (/ h d)))
   (if (<= d 4.3e-128)
     (* (sqrt (/ 1.0 (* l h))) (- d))
     (/ d (* (sqrt l) (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5.1e-54) {
		tmp = sqrt((d / l)) / sqrt((h / d));
	} else if (d <= 4.3e-128) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-5.1d-54)) then
        tmp = sqrt((d / l)) / sqrt((h / d))
    else if (d <= 4.3d-128) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5.1e-54) {
		tmp = Math.sqrt((d / l)) / Math.sqrt((h / d));
	} else if (d <= 4.3e-128) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -5.1e-54:
		tmp = math.sqrt((d / l)) / math.sqrt((h / d))
	elif d <= 4.3e-128:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5.1e-54)
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	elseif (d <= 4.3e-128)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -5.1e-54)
		tmp = sqrt((d / l)) / sqrt((h / d));
	elseif (d <= 4.3e-128)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5.1e-54], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.3e-128], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.1000000000000001e-54

   1. Initial program 87.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 7.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 7.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.9%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

   if -5.1000000000000001e-54 < d < 4.29999999999999994e-128

   1. Initial program 46.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 32.0

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 4.29999999999999994e-128 < d

   1. Initial program 69.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 53.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 58.9%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.1 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 44.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5.1 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5.1e-54)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= d 4.3e-128)
     (* (sqrt (/ 1.0 (* l h))) (- d))
     (/ d (* (sqrt l) (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5.1e-54) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= 4.3e-128) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-5.1d-54)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (d <= 4.3d-128) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5.1e-54) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= 4.3e-128) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -5.1e-54:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= 4.3e-128:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5.1e-54)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= 4.3e-128)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -5.1e-54)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (d <= 4.3e-128)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5.1e-54], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.3e-128], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.1000000000000001e-54

   1. Initial program 87.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lower-/.f64 61.8

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

   7. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

   if -5.1000000000000001e-54 < d < 4.29999999999999994e-128

   1. Initial program 46.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 32.0

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 4.29999999999999994e-128 < d

   1. Initial program 69.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 53.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 58.9%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.1 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 44.2% accurate, 8.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 6.3 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -3e+150)
   (sqrt (* (/ d h) (/ d l)))
   (if (<= l 6.3e-197)
     (* (sqrt (/ 1.0 (* l h))) (- d))
     (/ d (* (sqrt l) (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -3e+150) {
		tmp = sqrt(((d / h) * (d / l)));
	} else if (l <= 6.3e-197) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-3d+150)) then
        tmp = sqrt(((d / h) * (d / l)))
    else if (l <= 6.3d-197) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -3e+150) {
		tmp = Math.sqrt(((d / h) * (d / l)));
	} else if (l <= 6.3e-197) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -3e+150:
		tmp = math.sqrt(((d / h) * (d / l)))
	elif l <= 6.3e-197:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -3e+150)
		tmp = sqrt(Float64(Float64(d / h) * Float64(d / l)));
	elseif (l <= 6.3e-197)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -3e+150)
		tmp = sqrt(((d / h) * (d / l)));
	elseif (l <= 6.3e-197)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -3e+150], N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6.3e-197], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.00000000000000012e150

   1. Initial program 75.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 8.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 8.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.0%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   if -3.00000000000000012e150 < l < 6.2999999999999994e-197

   1. Initial program 72.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 38.8

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 6.2999999999999994e-197 < l

   1. Initial program 57.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 49.9

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 53.5%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 6.3 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 45.0% accurate, 9.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.3 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 4.3e-128)
   (* (sqrt (/ 1.0 (* l h))) (- d))
   (/ d (* (sqrt l) (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 4.3e-128) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 4.3d-128) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 4.3e-128) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 4.3e-128:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 4.3e-128)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 4.3e-128)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 4.3e-128], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.29999999999999994e-128

   1. Initial program 66.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 37.0

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 37.0%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 4.29999999999999994e-128 < d

   1. Initial program 69.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 53.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 58.9%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.3 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 41.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.3 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 6.3e-197) (* (sqrt (/ 1.0 (* l h))) (- d)) (/ d (sqrt (* l h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 6.3e-197) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 6.3d-197) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 6.3e-197) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 6.3e-197:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 6.3e-197)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 6.3e-197)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 6.3e-197], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.2999999999999994e-197

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 37.9

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 37.9%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 6.2999999999999994e-197 < l

   1. Initial program 57.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 49.9

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.3 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 26.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ d (sqrt (* l h))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d / sqrt((l * h));
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d / sqrt((l * h))
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d / Math.sqrt((l * h));
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((l * h))

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((l * h));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.1%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in h around 0

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   6. lower-*.f64 25.7

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 25.7%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Step-by-step derivation

7. Applied rewrites 25.7%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))