Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.8% → 82.7%

Time: 17.0s

Alternatives: 18

Speedup: 3.7×

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: 65.8% 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: 82.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-\ell}\\ t_1 := M\_m \cdot \frac{D\_m}{d}\\ t_2 := \sqrt{-d}\\ t_3 := \frac{t\_2}{\sqrt{-h}}\\ t_4 := 1 - \frac{\frac{M\_m}{d} \cdot \left(0.25 \cdot D\_m\right)}{\ell \cdot \left({h}^{-1} \cdot \frac{2}{t\_1}\right)}\\ \mathbf{if}\;h \leq -2.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{\left(D\_m \cdot 0.25\right) \cdot M\_m}{\left(-\ell\right) \cdot d} \cdot \left(t\_1 \cdot 0.5\right), h, 1\right) \cdot t\_3\right) \cdot t\_2}{t\_0}\\ \mathbf{elif}\;h \leq -1.25 \cdot 10^{-141}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{\frac{M\_m \cdot \left(0.25 \cdot D\_m\right)}{d}}{-\ell} \cdot \left(\left(D\_m \cdot \frac{M\_m}{d}\right) \cdot 0.5\right), h, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_2}{t\_0}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_3 \cdot {\left(\sqrt{\frac{\ell}{d}}\right)}^{-1}\right) \cdot t\_4\\ \mathbf{elif}\;h \leq 6.4 \cdot 10^{+140}:\\ \;\;\;\;\left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.5 \cdot \left(\frac{D\_m}{d} \cdot M\_m\right)\right) \cdot h, \frac{M\_m}{\ell} \cdot \left(-0.25 \cdot \frac{D\_m}{d}\right), 1\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

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 (sqrt (- l)))
        (t_1 (* M_m (/ D_m d)))
        (t_2 (sqrt (- d)))
        (t_3 (/ t_2 (sqrt (- h))))
        (t_4
         (-
          1.0
          (/ (* (/ M_m d) (* 0.25 D_m)) (* l (* (pow h -1.0) (/ 2.0 t_1)))))))
   (if (<= h -2.1e+154)
     (/
      (*
       (* (fma (* (/ (* (* D_m 0.25) M_m) (* (- l) d)) (* t_1 0.5)) h 1.0) t_3)
       t_2)
      t_0)
     (if (<= h -1.25e-141)
       (/
        (*
         (*
          (fma
           (* (/ (/ (* M_m (* 0.25 D_m)) d) (- l)) (* (* D_m (/ M_m d)) 0.5))
           h
           1.0)
          (sqrt (/ d h)))
         t_2)
        t_0)
       (if (<= h -2e-310)
         (* (* t_3 (pow (sqrt (/ l d)) -1.0)) t_4)
         (if (<= h 6.4e+140)
           (* (* (sqrt (pow (* l h) -1.0)) d) t_4)
           (*
            (*
             (fma
              (* (* 0.5 (* (/ D_m d) M_m)) h)
              (* (/ M_m l) (* -0.25 (/ D_m d)))
              1.0)
             (/ (sqrt d) (sqrt h)))
            (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 = sqrt(-l);
	double t_1 = M_m * (D_m / d);
	double t_2 = sqrt(-d);
	double t_3 = t_2 / sqrt(-h);
	double t_4 = 1.0 - (((M_m / d) * (0.25 * D_m)) / (l * (pow(h, -1.0) * (2.0 / t_1))));
	double tmp;
	if (h <= -2.1e+154) {
		tmp = ((fma(((((D_m * 0.25) * M_m) / (-l * d)) * (t_1 * 0.5)), h, 1.0) * t_3) * t_2) / t_0;
	} else if (h <= -1.25e-141) {
		tmp = ((fma(((((M_m * (0.25 * D_m)) / d) / -l) * ((D_m * (M_m / d)) * 0.5)), h, 1.0) * sqrt((d / h))) * t_2) / t_0;
	} else if (h <= -2e-310) {
		tmp = (t_3 * pow(sqrt((l / d)), -1.0)) * t_4;
	} else if (h <= 6.4e+140) {
		tmp = (sqrt(pow((l * h), -1.0)) * d) * t_4;
	} else {
		tmp = (fma(((0.5 * ((D_m / d) * M_m)) * h), ((M_m / l) * (-0.25 * (D_m / d))), 1.0) * (sqrt(d) / sqrt(h))) * 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 = sqrt(Float64(-l))
	t_1 = Float64(M_m * Float64(D_m / d))
	t_2 = sqrt(Float64(-d))
	t_3 = Float64(t_2 / sqrt(Float64(-h)))
	t_4 = Float64(1.0 - Float64(Float64(Float64(M_m / d) * Float64(0.25 * D_m)) / Float64(l * Float64((h ^ -1.0) * Float64(2.0 / t_1)))))
	tmp = 0.0
	if (h <= -2.1e+154)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(Float64(D_m * 0.25) * M_m) / Float64(Float64(-l) * d)) * Float64(t_1 * 0.5)), h, 1.0) * t_3) * t_2) / t_0);
	elseif (h <= -1.25e-141)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(Float64(M_m * Float64(0.25 * D_m)) / d) / Float64(-l)) * Float64(Float64(D_m * Float64(M_m / d)) * 0.5)), h, 1.0) * sqrt(Float64(d / h))) * t_2) / t_0);
	elseif (h <= -2e-310)
		tmp = Float64(Float64(t_3 * (sqrt(Float64(l / d)) ^ -1.0)) * t_4);
	elseif (h <= 6.4e+140)
		tmp = Float64(Float64(sqrt((Float64(l * h) ^ -1.0)) * d) * t_4);
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.5 * Float64(Float64(D_m / d) * M_m)) * h), Float64(Float64(M_m / l) * Float64(-0.25 * Float64(D_m / d))), 1.0) * Float64(sqrt(d) / sqrt(h))) * 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[Sqrt[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * N[(N[Power[h, -1.0], $MachinePrecision] * N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2.1e+154], N[(N[(N[(N[(N[(N[(N[(N[(D$95$m * 0.25), $MachinePrecision] * M$95$m), $MachinePrecision] / N[((-l) * d), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[h, -1.25e-141], N[(N[(N[(N[(N[(N[(N[(N[(M$95$m * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / (-l)), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[(t$95$3 * N[Power[N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[h, 6.4e+140], N[(N[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * t$95$4), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(-0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if h < -2.09999999999999994e154

   1. Initial program 49.2%

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

   4. Applied rewrites 49.1%

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

   6. Applied rewrites 47.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. neg-mul-1 N/A

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

      16. lift-neg.f64 N/A

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

      17. lower-*.f64 46.8

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

   8. Applied rewrites 46.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 80.0

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

   10. Applied rewrites 80.0%

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

   if -2.09999999999999994e154 < h < -1.25e-141

   1. Initial program 69.9%

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

   4. Applied rewrites 79.6%

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

   6. Applied rewrites 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 89.7

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

   8. Applied rewrites 89.7%

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

   if -1.25e-141 < h < -1.999999999999994e-310

   1. Initial program 61.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 61.4

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

   4. Applied rewrites 61.4%

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

   6. Applied rewrites 65.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{M}{d} \cdot \left(0.25 \cdot D\right)\right) \cdot 1}{\ell \cdot \left({h}^{-1} \cdot \frac{2}{M \cdot \frac{D}{d}}\right)}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lower-/.f64 83.6

          \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\left(\frac{M}{d} \cdot \left(0.25 \cdot D\right)\right) \cdot 1}{\ell \cdot \left({h}^{-1} \cdot \frac{2}{M \cdot \frac{D}{d}}\right)}\right) \]

   8. Applied rewrites 83.6%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\left(\frac{M}{d} \cdot \left(0.25 \cdot D\right)\right) \cdot 1}{\ell \cdot \left({h}^{-1} \cdot \frac{2}{M \cdot \frac{D}{d}}\right)}\right) \]

   if -1.999999999999994e-310 < h < 6.40000000000000021e140

   1. Initial program 71.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 70.6

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

   4. Applied rewrites 70.6%

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

   6. Applied rewrites 73.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{M}{d} \cdot \left(0.25 \cdot D\right)\right) \cdot 1}{\ell \cdot \left({h}^{-1} \cdot \frac{2}{M \cdot \frac{D}{d}}\right)}}\right) \]

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.4

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{M}{d} \cdot \left(0.25 \cdot D\right)\right) \cdot 1}{\ell \cdot \left({h}^{-1} \cdot \frac{2}{M \cdot \frac{D}{d}}\right)}\right) \]

   9. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{M}{d} \cdot \left(0.25 \cdot D\right)\right) \cdot 1}{\ell \cdot \left({h}^{-1} \cdot \frac{2}{M \cdot \frac{D}{d}}\right)}\right) \]

   if 6.40000000000000021e140 < h

   1. Initial program 67.5%

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.0%

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

   8. Applied rewrites 78.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 93.1

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

   10. Applied rewrites 93.1%

       \[\leadsto \left(\mathsf{fma}\left(\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot h, \frac{M}{\ell} \cdot \left(-0.25 \cdot \frac{D}{d}\right), 1\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{\left(D \cdot 0.25\right) \cdot M}{\left(-\ell\right) \cdot d} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5\right), h, 1\right) \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq -1.25 \cdot 10^{-141}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{\frac{M \cdot \left(0.25 \cdot D\right)}{d}}{-\ell} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot 0.5\right), h, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\sqrt{\frac{\ell}{d}}\right)}^{-1}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(0.25 \cdot D\right)}{\ell \cdot \left({h}^{-1} \cdot \frac{2}{M \cdot \frac{D}{d}}\right)}\right)\\ \mathbf{elif}\;h \leq 6.4 \cdot 10^{+140}:\\ \;\;\;\;\left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(0.25 \cdot D\right)}{\ell \cdot \left({h}^{-1} \cdot \frac{2}{M \cdot \frac{D}{d}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot h, \frac{M}{\ell} \cdot \left(-0.25 \cdot \frac{D}{d}\right), 1\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+278}:\\ \;\;\;\;\left(\frac{\left(\left(D\_m \cdot D\_m\right) \cdot h\right) \cdot -0.125}{d} \cdot \frac{\frac{M\_m \cdot M\_m}{\ell}}{d}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(0.5 \cdot M\_m\right) \cdot \frac{D\_m}{d}\right) \cdot \left(h \cdot \frac{M\_m}{\ell}\right)\right) \cdot \frac{D\_m}{d}, -0.25, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot \frac{D\_m}{\ell}, \frac{M\_m}{d} \cdot \left(\left(h \cdot \left(\frac{D\_m}{d} \cdot 0.5\right)\right) \cdot M\_m\right), 1\right) \cdot t\_1\\ \end{array} \end{array} \]

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
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_1 (/ d (sqrt (* l h)))))
   (if (<= t_0 -1e+278)
     (*
      (* (/ (* (* (* D_m D_m) h) -0.125) d) (/ (/ (* M_m M_m) l) d))
      (sqrt (* (/ d l) (/ d h))))
     (if (<= t_0 0.0)
       (*
        (fma
         (* (* (* (* 0.5 M_m) (/ D_m d)) (* h (/ M_m l))) (/ D_m d))
         -0.25
         1.0)
        t_1)
       (if (<= t_0 5e+174)
         (* (sqrt (/ d h)) (sqrt (/ d l)))
         (if (<= t_0 INFINITY)
           (/ (* (sqrt (/ h l)) d) h)
           (*
            (fma
             (* -0.25 (/ D_m l))
             (* (/ M_m d) (* (* h (* (/ D_m d) 0.5)) M_m))
             1.0)
            t_1)))))))

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 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = d / sqrt((l * h));
	double tmp;
	if (t_0 <= -1e+278) {
		tmp = (((((D_m * D_m) * h) * -0.125) / d) * (((M_m * M_m) / l) / d)) * sqrt(((d / l) * (d / h)));
	} else if (t_0 <= 0.0) {
		tmp = fma(((((0.5 * M_m) * (D_m / d)) * (h * (M_m / l))) * (D_m / d)), -0.25, 1.0) * t_1;
	} else if (t_0 <= 5e+174) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = fma((-0.25 * (D_m / l)), ((M_m / d) * ((h * ((D_m / d) * 0.5)) * M_m)), 1.0) * t_1;
	}
	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((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_0 <= -1e+278)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(D_m * D_m) * h) * -0.125) / d) * Float64(Float64(Float64(M_m * M_m) / l) / d)) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(0.5 * M_m) * Float64(D_m / d)) * Float64(h * Float64(M_m / l))) * Float64(D_m / d)), -0.25, 1.0) * t_1);
	elseif (t_0 <= 5e+174)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(fma(Float64(-0.25 * Float64(D_m / l)), Float64(Float64(M_m / d) * Float64(Float64(h * Float64(Float64(D_m / d) * 0.5)) * M_m)), 1.0) * t_1);
	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[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+278], N[(N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(N[(N[(0.5 * M$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 5e+174], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(-0.25 * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(h * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.99999999999999964e277

   1. Initial program 84.1%

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

   4. Applied rewrites 39.6%

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

   6. Applied rewrites 40.8%

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

   8. Applied rewrites 72.1%

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

   9. Taylor expanded in d around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l/ N/A

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

       4. associate-*l* N/A

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

       5. associate-*r/ N/A

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

       6. associate-*l/ N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. times-frac N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. *-commutative N/A

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

   11. Applied rewrites 60.7%

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

   if -9.99999999999999964e277 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   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

   4. Applied rewrites 39.9%

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

   6. Applied rewrites 31.7%

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

   8. Applied rewrites 60.8%

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

   10. Applied rewrites 35.5%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.9999999999999997e174

   1. Initial program 99.0%

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

   4. Applied rewrites 42.1%

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

   6. Applied rewrites 40.9%

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

   8. Applied rewrites 97.8%

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

   9. Taylor expanded in d around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. rem-square-sqrt N/A

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

       5. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{d}{h}}\right)\right)}\right)\right) \cdot \sqrt{\frac{d}{\ell}} \]

       6. remove-double-neg N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 96.9

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

   11. Applied rewrites 96.9%

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

   if 4.9999999999999997e174 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 57.2%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 79.2%

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   4. Applied rewrites 2.6%

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

   6. Applied rewrites 16.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. neg-mul-1 N/A

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

      16. lift-neg.f64 N/A

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

      17. lower-*.f64 16.2

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

   8. Applied rewrites 16.2%

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

   10. Applied rewrites 27.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25 \cdot \frac{D}{\ell}, \frac{M}{d} \cdot \left(\left(h \cdot \left(\frac{D}{d} \cdot 0.5\right)\right) \cdot M\right), 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{+278}:\\ \;\;\;\;\left(\frac{\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.125}{d} \cdot \frac{\frac{M \cdot M}{\ell}}{d}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\ell}\right)\right) \cdot \frac{D}{d}, -0.25, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot \frac{D}{\ell}, \frac{M}{d} \cdot \left(\left(h \cdot \left(\frac{D}{d} \cdot 0.5\right)\right) \cdot M\right), 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \frac{t\_0 \cdot d}{h}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \frac{\left(-d\right) \cdot t\_0}{h}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-190}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+147}:\\ \;\;\;\;1 \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 (/ h l)))
        (t_1 (/ (* t_0 d) h))
        (t_2
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_3 (/ (* (- d) t_0) h)))
   (if (<= t_2 -2e-190)
     t_3
     (if (<= t_2 5e-145)
       t_1
       (if (<= t_2 5e+147)
         (* 1.0 (sqrt (* (/ d l) (/ d h))))
         (if (<= t_2 INFINITY) t_1 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((h / l));
	double t_1 = (t_0 * d) / h;
	double t_2 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = (-d * t_0) / h;
	double tmp;
	if (t_2 <= -2e-190) {
		tmp = t_3;
	} else if (t_2 <= 5e-145) {
		tmp = t_1;
	} else if (t_2 <= 5e+147) {
		tmp = 1.0 * sqrt(((d / l) * (d / h)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

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 t_0 = Math.sqrt((h / l));
	double t_1 = (t_0 * d) / h;
	double t_2 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = (-d * t_0) / h;
	double tmp;
	if (t_2 <= -2e-190) {
		tmp = t_3;
	} else if (t_2 <= 5e-145) {
		tmp = t_1;
	} else if (t_2 <= 5e+147) {
		tmp = 1.0 * Math.sqrt(((d / l) * (d / h)));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	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):
	t_0 = math.sqrt((h / l))
	t_1 = (t_0 * d) / h
	t_2 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_3 = (-d * t_0) / h
	tmp = 0
	if t_2 <= -2e-190:
		tmp = t_3
	elif t_2 <= 5e-145:
		tmp = t_1
	elif t_2 <= 5e+147:
		tmp = 1.0 * math.sqrt(((d / l) * (d / h)))
	elif t_2 <= math.inf:
		tmp = t_1
	else:
		tmp = 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(h / l))
	t_1 = Float64(Float64(t_0 * d) / h)
	t_2 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_3 = Float64(Float64(Float64(-d) * t_0) / h)
	tmp = 0.0
	if (t_2 <= -2e-190)
		tmp = t_3;
	elseif (t_2 <= 5e-145)
		tmp = t_1;
	elseif (t_2 <= 5e+147)
		tmp = Float64(1.0 * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	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)
	t_0 = sqrt((h / l));
	t_1 = (t_0 * d) / h;
	t_2 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_3 = (-d * t_0) / h;
	tmp = 0.0;
	if (t_2 <= -2e-190)
		tmp = t_3;
	elseif (t_2 <= 5e-145)
		tmp = t_1;
	elseif (t_2 <= 5e+147)
		tmp = 1.0 * sqrt(((d / l) * (d / h)));
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	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_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[((-d) * t$95$0), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-190], t$95$3, If[LessEqual[t$95$2, 5e-145], t$95$1, If[LessEqual[t$95$2, 5e+147], N[(1.0 * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e-190 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.1%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \frac{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   7. Step-by-step derivation

   8. Applied rewrites 22.3%

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

   if -2e-190 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.9999999999999998e-145 or 5.0000000000000002e147 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 62.8%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 45.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 77.2%

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

   if 4.9999999999999998e-145 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e147

   1. Initial program 98.9%

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

   4. Applied rewrites 41.7%

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

   6. Applied rewrites 39.9%

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in d around inf

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

   11. Applied rewrites 96.9%

       \[\leadsto \color{blue}{1} \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+147}:\\ \;\;\;\;1 \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \frac{t\_0 \cdot d}{h}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \frac{\left(-d\right) \cdot t\_0}{h}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-190}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 (/ h l)))
        (t_1 (/ (* t_0 d) h))
        (t_2
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_3 (/ (* (- d) t_0) h)))
   (if (<= t_2 -2e-190)
     t_3
     (if (<= t_2 2e-98)
       t_1
       (if (<= t_2 5e+147)
         (sqrt (* (/ (/ d l) h) d))
         (if (<= t_2 INFINITY) t_1 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((h / l));
	double t_1 = (t_0 * d) / h;
	double t_2 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = (-d * t_0) / h;
	double tmp;
	if (t_2 <= -2e-190) {
		tmp = t_3;
	} else if (t_2 <= 2e-98) {
		tmp = t_1;
	} else if (t_2 <= 5e+147) {
		tmp = sqrt((((d / l) / h) * d));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

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 t_0 = Math.sqrt((h / l));
	double t_1 = (t_0 * d) / h;
	double t_2 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = (-d * t_0) / h;
	double tmp;
	if (t_2 <= -2e-190) {
		tmp = t_3;
	} else if (t_2 <= 2e-98) {
		tmp = t_1;
	} else if (t_2 <= 5e+147) {
		tmp = Math.sqrt((((d / l) / h) * d));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	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):
	t_0 = math.sqrt((h / l))
	t_1 = (t_0 * d) / h
	t_2 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_3 = (-d * t_0) / h
	tmp = 0
	if t_2 <= -2e-190:
		tmp = t_3
	elif t_2 <= 2e-98:
		tmp = t_1
	elif t_2 <= 5e+147:
		tmp = math.sqrt((((d / l) / h) * d))
	elif t_2 <= math.inf:
		tmp = t_1
	else:
		tmp = 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(h / l))
	t_1 = Float64(Float64(t_0 * d) / h)
	t_2 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_3 = Float64(Float64(Float64(-d) * t_0) / h)
	tmp = 0.0
	if (t_2 <= -2e-190)
		tmp = t_3;
	elseif (t_2 <= 2e-98)
		tmp = t_1;
	elseif (t_2 <= 5e+147)
		tmp = sqrt(Float64(Float64(Float64(d / l) / h) * d));
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	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)
	t_0 = sqrt((h / l));
	t_1 = (t_0 * d) / h;
	t_2 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_3 = (-d * t_0) / h;
	tmp = 0.0;
	if (t_2 <= -2e-190)
		tmp = t_3;
	elseif (t_2 <= 2e-98)
		tmp = t_1;
	elseif (t_2 <= 5e+147)
		tmp = sqrt((((d / l) / h) * d));
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	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_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[((-d) * t$95$0), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-190], t$95$3, If[LessEqual[t$95$2, 2e-98], t$95$1, If[LessEqual[t$95$2, 5e+147], N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e-190 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.1%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \frac{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   7. Step-by-step derivation

   8. Applied rewrites 22.3%

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

   if -2e-190 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.99999999999999988e-98 or 5.0000000000000002e147 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 65.1%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 47.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 77.7%

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

   if 1.99999999999999988e-98 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e147

   1. Initial program 99.6%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 44.0

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

   5. Applied rewrites 44.0%

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

   7. Applied rewrites 43.9%

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

   9. Applied rewrites 94.6%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot \frac{D\_m}{d}\right) \cdot M\_m, \left(0.5 \cdot \left(\frac{D\_m}{d} \cdot M\_m\right)\right) \cdot \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot \frac{D\_m}{\ell}, \frac{M\_m}{d} \cdot \left(\left(h \cdot \left(\frac{D\_m}{d} \cdot 0.5\right)\right) \cdot M\_m\right), 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l))))))
   (if (<= t_0 -2e-190)
     (*
      (fma
       (* (* -0.25 (/ D_m d)) M_m)
       (* (* 0.5 (* (/ D_m d) M_m)) (/ h l))
       1.0)
      (sqrt (* (/ d l) (/ d h))))
     (if (<= t_0 5e+174)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (if (<= t_0 INFINITY)
         (/ (* (sqrt (/ h l)) d) h)
         (*
          (fma
           (* -0.25 (/ D_m l))
           (* (/ M_m d) (* (* h (* (/ D_m d) 0.5)) M_m))
           1.0)
          (/ 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 t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -2e-190) {
		tmp = fma(((-0.25 * (D_m / d)) * M_m), ((0.5 * ((D_m / d) * M_m)) * (h / l)), 1.0) * sqrt(((d / l) * (d / h)));
	} else if (t_0 <= 5e+174) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = fma((-0.25 * (D_m / l)), ((M_m / d) * ((h * ((D_m / d) * 0.5)) * M_m)), 1.0) * (d / 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)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -2e-190)
		tmp = Float64(fma(Float64(Float64(-0.25 * Float64(D_m / d)) * M_m), Float64(Float64(0.5 * Float64(Float64(D_m / d) * M_m)) * Float64(h / l)), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (t_0 <= 5e+174)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(fma(Float64(-0.25 * Float64(D_m / l)), Float64(Float64(M_m / d) * Float64(Float64(h * Float64(Float64(D_m / d) * 0.5)) * M_m)), 1.0) * Float64(d / sqrt(Float64(l * 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[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-190], N[(N[(N[(N[(-0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(0.5 * N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+174], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(-0.25 * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(h * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e-190

   1. Initial program 86.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 86.3

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

   4. Applied rewrites 86.3%

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

   6. Applied rewrites 89.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{M}{d} \cdot \left(0.25 \cdot D\right)\right) \cdot 1}{\ell \cdot \left({h}^{-1} \cdot \frac{2}{M \cdot \frac{D}{d}}\right)}}\right) \]

   8. Applied rewrites 74.9%

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

   if -2e-190 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.9999999999999997e174

   1. Initial program 87.5%

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

   4. Applied rewrites 39.1%

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

   6. Applied rewrites 36.8%

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

   8. Applied rewrites 86.3%

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

   9. Taylor expanded in d around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. rem-square-sqrt N/A

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

       5. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{d}{h}}\right)\right)}\right)\right) \cdot \sqrt{\frac{d}{\ell}} \]

       6. remove-double-neg N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 85.8

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

   11. Applied rewrites 85.8%

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

   if 4.9999999999999997e174 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 57.2%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 79.2%

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   4. Applied rewrites 2.6%

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

   6. Applied rewrites 16.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. neg-mul-1 N/A

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

      16. lift-neg.f64 N/A

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

      17. lower-*.f64 16.2

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

   8. Applied rewrites 16.2%

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

   10. Applied rewrites 27.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25 \cdot \frac{D}{\ell}, \frac{M}{d} \cdot \left(\left(h \cdot \left(\frac{D}{d} \cdot 0.5\right)\right) \cdot M\right), 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot \frac{D}{d}\right) \cdot M, \left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot \frac{D}{\ell}, \frac{M}{d} \cdot \left(\left(h \cdot \left(\frac{D}{d} \cdot 0.5\right)\right) \cdot M\right), 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 \cdot \left(\frac{D\_m}{d} \cdot M\_m\right)\right) \cdot h, \frac{\left(-0.25 \cdot D\_m\right) \cdot M\_m}{\ell \cdot d}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot \frac{D\_m}{\ell}, \frac{M\_m}{d} \cdot \left(\left(h \cdot \left(\frac{D\_m}{d} \cdot 0.5\right)\right) \cdot M\_m\right), 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l))))))
   (if (<= t_0 -5e-45)
     (*
      (fma
       (* (* 0.5 (* (/ D_m d) M_m)) h)
       (/ (* (* -0.25 D_m) M_m) (* l d))
       1.0)
      (sqrt (* (/ d l) (/ d h))))
     (if (<= t_0 5e+174)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (if (<= t_0 INFINITY)
         (/ (* (sqrt (/ h l)) d) h)
         (*
          (fma
           (* -0.25 (/ D_m l))
           (* (/ M_m d) (* (* h (* (/ D_m d) 0.5)) M_m))
           1.0)
          (/ 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 t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -5e-45) {
		tmp = fma(((0.5 * ((D_m / d) * M_m)) * h), (((-0.25 * D_m) * M_m) / (l * d)), 1.0) * sqrt(((d / l) * (d / h)));
	} else if (t_0 <= 5e+174) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = fma((-0.25 * (D_m / l)), ((M_m / d) * ((h * ((D_m / d) * 0.5)) * M_m)), 1.0) * (d / 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)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -5e-45)
		tmp = Float64(fma(Float64(Float64(0.5 * Float64(Float64(D_m / d) * M_m)) * h), Float64(Float64(Float64(-0.25 * D_m) * M_m) / Float64(l * d)), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (t_0 <= 5e+174)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(fma(Float64(-0.25 * Float64(D_m / l)), Float64(Float64(M_m / d) * Float64(Float64(h * Float64(Float64(D_m / d) * 0.5)) * M_m)), 1.0) * Float64(d / sqrt(Float64(l * 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[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-45], N[(N[(N[(N[(0.5 * N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(N[(N[(-0.25 * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+174], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(-0.25 * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(h * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.99999999999999976e-45

   1. Initial program 86.8%

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

   4. Applied rewrites 41.4%

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

   6. Applied rewrites 40.2%

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

   8. Applied rewrites 71.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. remove-double-neg N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. distribute-lft-neg-out N/A

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

      19. remove-double-neg N/A

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

      20. lower-*.f64 69.1

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

   10. Applied rewrites 69.1%

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

   if -4.99999999999999976e-45 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.9999999999999997e174

   1. Initial program 87.1%

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

   4. Applied rewrites 39.6%

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

   6. Applied rewrites 37.4%

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in d around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. rem-square-sqrt N/A

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

       5. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{d}{h}}\right)\right)}\right)\right) \cdot \sqrt{\frac{d}{\ell}} \]

       6. remove-double-neg N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 81.6

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

   11. Applied rewrites 81.6%

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

   if 4.9999999999999997e174 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 57.2%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 79.2%

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   4. Applied rewrites 2.6%

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

   6. Applied rewrites 16.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. neg-mul-1 N/A

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

      16. lift-neg.f64 N/A

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

      17. lower-*.f64 16.2

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

   8. Applied rewrites 16.2%

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

   10. Applied rewrites 27.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25 \cdot \frac{D}{\ell}, \frac{M}{d} \cdot \left(\left(h \cdot \left(\frac{D}{d} \cdot 0.5\right)\right) \cdot M\right), 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot h, \frac{\left(-0.25 \cdot D\right) \cdot M}{\ell \cdot d}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot \frac{D}{\ell}, \frac{M}{d} \cdot \left(\left(h \cdot \left(\frac{D}{d} \cdot 0.5\right)\right) \cdot M\right), 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{\left(-d\right) \cdot t\_0}{h}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-190}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (sqrt (/ h l)))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_2 (/ (* (- d) t_0) h)))
   (if (<= t_1 -2e-190)
     t_2
     (if (<= t_1 5e+174)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (if (<= t_1 INFINITY) (/ (* t_0 d) h) 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 = sqrt((h / l));
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (-d * t_0) / h;
	double tmp;
	if (t_1 <= -2e-190) {
		tmp = t_2;
	} else if (t_1 <= 5e+174) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = t_2;
	}
	return tmp;
}

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 t_0 = Math.sqrt((h / l));
	double t_1 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (-d * t_0) / h;
	double tmp;
	if (t_1 <= -2e-190) {
		tmp = t_2;
	} else if (t_1 <= 5e+174) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = t_2;
	}
	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):
	t_0 = math.sqrt((h / l))
	t_1 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = (-d * t_0) / h
	tmp = 0
	if t_1 <= -2e-190:
		tmp = t_2
	elif t_1 <= 5e+174:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif t_1 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = 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 = sqrt(Float64(h / l))
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(Float64(Float64(-d) * t_0) / h)
	tmp = 0.0
	if (t_1 <= -2e-190)
		tmp = t_2;
	elseif (t_1 <= 5e+174)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = t_2;
	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)
	t_0 = sqrt((h / l));
	t_1 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = (-d * t_0) / h;
	tmp = 0.0;
	if (t_1 <= -2e-190)
		tmp = t_2;
	elseif (t_1 <= 5e+174)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (t_1 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = t_2;
	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_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-d) * t$95$0), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-190], t$95$2, If[LessEqual[t$95$1, 5e+174], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e-190 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.1%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \frac{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   7. Step-by-step derivation

   8. Applied rewrites 22.3%

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

   if -2e-190 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.9999999999999997e174

   1. Initial program 87.5%

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

   4. Applied rewrites 39.1%

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

   6. Applied rewrites 36.8%

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

   8. Applied rewrites 86.3%

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

   9. Taylor expanded in d around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. rem-square-sqrt N/A

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

       5. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{d}{h}}\right)\right)}\right)\right) \cdot \sqrt{\frac{d}{\ell}} \]

       6. remove-double-neg N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 85.8

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

   11. Applied rewrites 85.8%

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

   if 4.9999999999999997e174 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 57.2%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 79.2%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+174}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-0.25 \cdot \frac{D\_m}{d}\right) \cdot M\_m, \left(0.5 \cdot \left(\frac{D\_m}{d} \cdot M\_m\right)\right) \cdot \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot \frac{D\_m}{\ell}, \frac{M\_m}{d} \cdot \left(\left(h \cdot \left(\frac{D\_m}{d} \cdot 0.5\right)\right) \cdot M\_m\right), 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l))))))
   (if (<= t_0 5e+174)
     (*
      (*
       (fma
        (* (* -0.25 (/ D_m d)) M_m)
        (* (* 0.5 (* (/ D_m d) M_m)) (/ h l))
        1.0)
       (sqrt (/ d h)))
      (sqrt (/ d l)))
     (if (<= t_0 INFINITY)
       (/ (* (sqrt (/ h l)) d) h)
       (*
        (fma
         (* -0.25 (/ D_m l))
         (* (/ M_m d) (* (* h (* (/ D_m d) 0.5)) M_m))
         1.0)
        (/ 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 t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 5e+174) {
		tmp = (fma(((-0.25 * (D_m / d)) * M_m), ((0.5 * ((D_m / d) * M_m)) * (h / l)), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = fma((-0.25 * (D_m / l)), ((M_m / d) * ((h * ((D_m / d) * 0.5)) * M_m)), 1.0) * (d / 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)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 5e+174)
		tmp = Float64(Float64(fma(Float64(Float64(-0.25 * Float64(D_m / d)) * M_m), Float64(Float64(0.5 * Float64(Float64(D_m / d) * M_m)) * Float64(h / l)), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(fma(Float64(-0.25 * Float64(D_m / l)), Float64(Float64(M_m / d) * Float64(Float64(h * Float64(Float64(D_m / d) * 0.5)) * M_m)), 1.0) * Float64(d / sqrt(Float64(l * 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[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+174], N[(N[(N[(N[(N[(-0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(0.5 * N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(-0.25 * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(h * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.9999999999999997e174

   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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 86.4

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

   4. Applied rewrites 86.4%

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

   6. Applied rewrites 87.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{M}{d} \cdot \left(0.25 \cdot D\right)\right) \cdot 1}{\ell \cdot \left({h}^{-1} \cdot \frac{2}{M \cdot \frac{D}{d}}\right)}}\right) \]

   8. Applied rewrites 88.7%

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

   if 4.9999999999999997e174 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 57.2%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 79.2%

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   4. Applied rewrites 2.6%

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

   6. Applied rewrites 16.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. neg-mul-1 N/A

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

      16. lift-neg.f64 N/A

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

      17. lower-*.f64 16.2

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

   8. Applied rewrites 16.2%

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

   10. Applied rewrites 27.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25 \cdot \frac{D}{\ell}, \frac{M}{d} \cdot \left(\left(h \cdot \left(\frac{D}{d} \cdot 0.5\right)\right) \cdot M\right), 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+174}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-0.25 \cdot \frac{D}{d}\right) \cdot M, \left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot \frac{D}{\ell}, \frac{M}{d} \cdot \left(\left(h \cdot \left(\frac{D}{d} \cdot 0.5\right)\right) \cdot M\right), 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(0.5 \cdot \left(\frac{D\_m}{d} \cdot M\_m\right)\right) \cdot h, \frac{M\_m}{\ell} \cdot \left(-0.25 \cdot \frac{D\_m}{d}\right), 1\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot t\_1\\ \mathbf{elif}\;h \leq 6.4 \cdot 10^{+140}:\\ \;\;\;\;\left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \frac{\frac{M\_m}{d} \cdot \left(0.25 \cdot D\_m\right)}{\ell \cdot \left({h}^{-1} \cdot \frac{2}{M\_m \cdot \frac{D\_m}{d}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot t\_1\\ \end{array} \end{array} \]

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
          (* (* 0.5 (* (/ D_m d) M_m)) h)
          (* (/ M_m l) (* -0.25 (/ D_m d)))
          1.0))
        (t_1 (sqrt (/ d l))))
   (if (<= h -2e-310)
     (* (* t_0 (/ (sqrt (- d)) (sqrt (- h)))) t_1)
     (if (<= h 6.4e+140)
       (*
        (* (sqrt (pow (* l h) -1.0)) d)
        (-
         1.0
         (/
          (* (/ M_m d) (* 0.25 D_m))
          (* l (* (pow h -1.0) (/ 2.0 (* M_m (/ D_m d))))))))
       (* (* t_0 (/ (sqrt d) (sqrt h))) t_1)))))

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(((0.5 * ((D_m / d) * M_m)) * h), ((M_m / l) * (-0.25 * (D_m / d))), 1.0);
	double t_1 = sqrt((d / l));
	double tmp;
	if (h <= -2e-310) {
		tmp = (t_0 * (sqrt(-d) / sqrt(-h))) * t_1;
	} else if (h <= 6.4e+140) {
		tmp = (sqrt(pow((l * h), -1.0)) * d) * (1.0 - (((M_m / d) * (0.25 * D_m)) / (l * (pow(h, -1.0) * (2.0 / (M_m * (D_m / d)))))));
	} else {
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * t_1;
	}
	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(0.5 * Float64(Float64(D_m / d) * M_m)) * h), Float64(Float64(M_m / l) * Float64(-0.25 * Float64(D_m / d))), 1.0)
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -2e-310)
		tmp = Float64(Float64(t_0 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))) * t_1);
	elseif (h <= 6.4e+140)
		tmp = Float64(Float64(sqrt((Float64(l * h) ^ -1.0)) * d) * Float64(1.0 - Float64(Float64(Float64(M_m / d) * Float64(0.25 * D_m)) / Float64(l * Float64((h ^ -1.0) * Float64(2.0 / Float64(M_m * Float64(D_m / d))))))));
	else
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(h))) * t_1);
	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[(0.5 * N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(-0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -2e-310], N[(N[(t$95$0 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[h, 6.4e+140], N[(N[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * N[(N[Power[h, -1.0], $MachinePrecision] * N[(2.0 / N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 62.7%

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

   4. Applied rewrites 67.8%

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

   6. Applied rewrites 70.3%

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

   8. Applied rewrites 63.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 76.9

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

   10. Applied rewrites 76.9%

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

   if -1.999999999999994e-310 < h < 6.40000000000000021e140

   1. Initial program 71.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 70.6

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

   4. Applied rewrites 70.6%

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

   6. Applied rewrites 73.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{M}{d} \cdot \left(0.25 \cdot D\right)\right) \cdot 1}{\ell \cdot \left({h}^{-1} \cdot \frac{2}{M \cdot \frac{D}{d}}\right)}}\right) \]

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.4

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{M}{d} \cdot \left(0.25 \cdot D\right)\right) \cdot 1}{\ell \cdot \left({h}^{-1} \cdot \frac{2}{M \cdot \frac{D}{d}}\right)}\right) \]

   9. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{M}{d} \cdot \left(0.25 \cdot D\right)\right) \cdot 1}{\ell \cdot \left({h}^{-1} \cdot \frac{2}{M \cdot \frac{D}{d}}\right)}\right) \]

   if 6.40000000000000021e140 < h

   1. Initial program 67.5%

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.0%

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

   8. Applied rewrites 78.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 93.1

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

   10. Applied rewrites 93.1%

       \[\leadsto \left(\mathsf{fma}\left(\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot h, \frac{M}{\ell} \cdot \left(-0.25 \cdot \frac{D}{d}\right), 1\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot h, \frac{M}{\ell} \cdot \left(-0.25 \cdot \frac{D}{d}\right), 1\right) \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq 6.4 \cdot 10^{+140}:\\ \;\;\;\;\left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(0.25 \cdot D\right)}{\ell \cdot \left({h}^{-1} \cdot \frac{2}{M \cdot \frac{D}{d}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot h, \frac{M}{\ell} \cdot \left(-0.25 \cdot \frac{D}{d}\right), 1\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(0.5 \cdot \left(\frac{D\_m}{d} \cdot M\_m\right)\right) \cdot h, \frac{M\_m}{\ell} \cdot \left(-0.25 \cdot \frac{D\_m}{d}\right), 1\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot t\_1\\ \mathbf{elif}\;h \leq 6.4 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(0.5 \cdot M\_m\right) \cdot \frac{D\_m}{d}\right) \cdot \left(h \cdot \frac{M\_m}{\ell}\right)\right) \cdot \frac{D\_m}{d}, -0.25, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot t\_1\\ \end{array} \end{array} \]

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
          (* (* 0.5 (* (/ D_m d) M_m)) h)
          (* (/ M_m l) (* -0.25 (/ D_m d)))
          1.0))
        (t_1 (sqrt (/ d l))))
   (if (<= h -2e-310)
     (* (* t_0 (/ (sqrt (- d)) (sqrt (- h)))) t_1)
     (if (<= h 6.4e+140)
       (*
        (fma
         (* (* (* (* 0.5 M_m) (/ D_m d)) (* h (/ M_m l))) (/ D_m d))
         -0.25
         1.0)
        (/ d (sqrt (* l h))))
       (* (* t_0 (/ (sqrt d) (sqrt h))) t_1)))))

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(((0.5 * ((D_m / d) * M_m)) * h), ((M_m / l) * (-0.25 * (D_m / d))), 1.0);
	double t_1 = sqrt((d / l));
	double tmp;
	if (h <= -2e-310) {
		tmp = (t_0 * (sqrt(-d) / sqrt(-h))) * t_1;
	} else if (h <= 6.4e+140) {
		tmp = fma(((((0.5 * M_m) * (D_m / d)) * (h * (M_m / l))) * (D_m / d)), -0.25, 1.0) * (d / sqrt((l * h)));
	} else {
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * t_1;
	}
	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(0.5 * Float64(Float64(D_m / d) * M_m)) * h), Float64(Float64(M_m / l) * Float64(-0.25 * Float64(D_m / d))), 1.0)
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -2e-310)
		tmp = Float64(Float64(t_0 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))) * t_1);
	elseif (h <= 6.4e+140)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(0.5 * M_m) * Float64(D_m / d)) * Float64(h * Float64(M_m / l))) * Float64(D_m / d)), -0.25, 1.0) * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(h))) * t_1);
	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[(0.5 * N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(-0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -2e-310], N[(N[(t$95$0 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[h, 6.4e+140], N[(N[(N[(N[(N[(N[(0.5 * M$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 62.7%

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

   4. Applied rewrites 67.8%

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

   6. Applied rewrites 70.3%

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

   8. Applied rewrites 63.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 76.9

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

   10. Applied rewrites 76.9%

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

   if -1.999999999999994e-310 < h < 6.40000000000000021e140

   1. Initial program 71.3%

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.0%

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

   8. Applied rewrites 72.4%

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

   10. Applied rewrites 84.8%

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

   if 6.40000000000000021e140 < h

   1. Initial program 67.5%

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.0%

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

   8. Applied rewrites 78.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 93.1

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

   10. Applied rewrites 93.1%

       \[\leadsto \left(\mathsf{fma}\left(\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot h, \frac{M}{\ell} \cdot \left(-0.25 \cdot \frac{D}{d}\right), 1\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 74.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq 3 \cdot 10^{-309}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{D\_m}{d} \cdot 0.5, M\_m \cdot \left(\left(h \cdot \left(-0.25 \cdot \frac{M\_m}{\ell}\right)\right) \cdot \frac{D\_m}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_0\\ \mathbf{elif}\;h \leq 6.4 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(0.5 \cdot M\_m\right) \cdot \frac{D\_m}{d}\right) \cdot \left(h \cdot \frac{M\_m}{\ell}\right)\right) \cdot \frac{D\_m}{d}, -0.25, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.5 \cdot \left(\frac{D\_m}{d} \cdot M\_m\right)\right) \cdot h, \frac{M\_m}{\ell} \cdot \left(-0.25 \cdot \frac{D\_m}{d}\right), 1\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot t\_0\\ \end{array} \end{array} \]

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))))
   (if (<= h 3e-309)
     (*
      (*
       (fma
        (* (/ D_m d) 0.5)
        (* M_m (* (* h (* -0.25 (/ M_m l))) (/ D_m d)))
        1.0)
       (sqrt (/ d h)))
      t_0)
     (if (<= h 6.4e+140)
       (*
        (fma
         (* (* (* (* 0.5 M_m) (/ D_m d)) (* h (/ M_m l))) (/ D_m d))
         -0.25
         1.0)
        (/ d (sqrt (* l h))))
       (*
        (*
         (fma
          (* (* 0.5 (* (/ D_m d) M_m)) h)
          (* (/ M_m l) (* -0.25 (/ D_m d)))
          1.0)
         (/ (sqrt d) (sqrt h)))
        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 tmp;
	if (h <= 3e-309) {
		tmp = (fma(((D_m / d) * 0.5), (M_m * ((h * (-0.25 * (M_m / l))) * (D_m / d))), 1.0) * sqrt((d / h))) * t_0;
	} else if (h <= 6.4e+140) {
		tmp = fma(((((0.5 * M_m) * (D_m / d)) * (h * (M_m / l))) * (D_m / d)), -0.25, 1.0) * (d / sqrt((l * h)));
	} else {
		tmp = (fma(((0.5 * ((D_m / d) * M_m)) * h), ((M_m / l) * (-0.25 * (D_m / d))), 1.0) * (sqrt(d) / sqrt(h))) * 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))
	tmp = 0.0
	if (h <= 3e-309)
		tmp = Float64(Float64(fma(Float64(Float64(D_m / d) * 0.5), Float64(M_m * Float64(Float64(h * Float64(-0.25 * Float64(M_m / l))) * Float64(D_m / d))), 1.0) * sqrt(Float64(d / h))) * t_0);
	elseif (h <= 6.4e+140)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(0.5 * M_m) * Float64(D_m / d)) * Float64(h * Float64(M_m / l))) * Float64(D_m / d)), -0.25, 1.0) * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.5 * Float64(Float64(D_m / d) * M_m)) * h), Float64(Float64(M_m / l) * Float64(-0.25 * Float64(D_m / d))), 1.0) * Float64(sqrt(d) / sqrt(h))) * 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]}, If[LessEqual[h, 3e-309], N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision] * N[(M$95$m * N[(N[(h * N[(-0.25 * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[h, 6.4e+140], N[(N[(N[(N[(N[(N[(0.5 * M$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(-0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < 3.000000000000001e-309

   1. Initial program 62.7%

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

   4. Applied rewrites 67.8%

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

   6. Applied rewrites 70.3%

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

   8. Applied rewrites 63.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   10. Applied rewrites 62.0%

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

   if 3.000000000000001e-309 < h < 6.40000000000000021e140

   1. Initial program 71.3%

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.0%

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

   8. Applied rewrites 72.4%

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

   10. Applied rewrites 84.8%

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

   if 6.40000000000000021e140 < h

   1. Initial program 67.5%

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.0%

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

   8. Applied rewrites 78.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 93.1

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

   10. Applied rewrites 93.1%

       \[\leadsto \left(\mathsf{fma}\left(\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot h, \frac{M}{\ell} \cdot \left(-0.25 \cdot \frac{D}{d}\right), 1\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 46.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-271}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 3.6e-271)
   (* (- d) (sqrt (pow (* l h) -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 <= 3.6e-271) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} 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 <= 3.6d-271) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    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 <= 3.6e-271) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} 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 <= 3.6e-271:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	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 <= 3.6e-271)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	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 <= 3.6e-271)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	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, 3.6e-271], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.5999999999999998e-271

   1. Initial program 64.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 37.7

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

   5. Applied rewrites 37.7%

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

   if 3.5999999999999998e-271 < l

   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 d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 45.7

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

   5. Applied rewrites 45.7%

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

   7. Applied rewrites 46.1%

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

   9. Applied rewrites 55.4%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-271}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{-269}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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 3.4e-269) (* (- d) (sqrt (pow (* l h) -1.0))) (/ 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 <= 3.4e-269) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} 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 <= 3.4d-269) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    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 <= 3.4e-269) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} 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 <= 3.4e-269:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	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 <= 3.4e-269)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	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 <= 3.4e-269)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	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, 3.4e-269], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.3999999999999997e-269

   1. Initial program 64.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 37.7

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

   5. Applied rewrites 37.7%

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

   if 3.3999999999999997e-269 < l

   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 d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 45.7

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

   5. Applied rewrites 45.7%

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

   7. Applied rewrites 46.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{-269}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(0.5 \cdot \left(\frac{D\_m}{d} \cdot M\_m\right)\right) \cdot h\\ t_1 := \left(\mathsf{fma}\left(t\_0, \frac{\left(-0.25 \cdot D\_m\right) \cdot M\_m}{\ell \cdot d}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -1.85 \cdot 10^{-131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;h \leq 9 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{M\_m}{\ell} \cdot \left(-0.25 \cdot \frac{D\_m}{d}\right), 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq 3.95 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(0.5 \cdot M\_m\right) \cdot \frac{D\_m}{d}\right) \cdot \left(h \cdot \frac{M\_m}{\ell}\right)\right) \cdot \frac{D\_m}{d}, -0.25, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (* 0.5 (* (/ D_m d) M_m)) h))
        (t_1
         (*
          (* (fma t_0 (/ (* (* -0.25 D_m) M_m) (* l d)) 1.0) (sqrt (/ d h)))
          (sqrt (/ d l)))))
   (if (<= h -1.85e-131)
     t_1
     (if (<= h 9e-309)
       (*
        (fma t_0 (* (/ M_m l) (* -0.25 (/ D_m d))) 1.0)
        (sqrt (* (/ d l) (/ d h))))
       (if (<= h 3.95e+186)
         (*
          (fma
           (* (* (* (* 0.5 M_m) (/ D_m d)) (* h (/ M_m l))) (/ D_m d))
           -0.25
           1.0)
          (/ d (sqrt (* l h))))
         t_1)))))

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 = (0.5 * ((D_m / d) * M_m)) * h;
	double t_1 = (fma(t_0, (((-0.25 * D_m) * M_m) / (l * d)), 1.0) * sqrt((d / h))) * sqrt((d / l));
	double tmp;
	if (h <= -1.85e-131) {
		tmp = t_1;
	} else if (h <= 9e-309) {
		tmp = fma(t_0, ((M_m / l) * (-0.25 * (D_m / d))), 1.0) * sqrt(((d / l) * (d / h)));
	} else if (h <= 3.95e+186) {
		tmp = fma(((((0.5 * M_m) * (D_m / d)) * (h * (M_m / l))) * (D_m / d)), -0.25, 1.0) * (d / sqrt((l * h)));
	} else {
		tmp = t_1;
	}
	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(0.5 * Float64(Float64(D_m / d) * M_m)) * h)
	t_1 = Float64(Float64(fma(t_0, Float64(Float64(Float64(-0.25 * D_m) * M_m) / Float64(l * d)), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)))
	tmp = 0.0
	if (h <= -1.85e-131)
		tmp = t_1;
	elseif (h <= 9e-309)
		tmp = Float64(fma(t_0, Float64(Float64(M_m / l) * Float64(-0.25 * Float64(D_m / d))), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (h <= 3.95e+186)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(0.5 * M_m) * Float64(D_m / d)) * Float64(h * Float64(M_m / l))) * Float64(D_m / d)), -0.25, 1.0) * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = t_1;
	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[(0.5 * N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(t$95$0 * N[(N[(N[(-0.25 * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1.85e-131], t$95$1, If[LessEqual[h, 9e-309], N[(N[(t$95$0 * N[(N[(M$95$m / l), $MachinePrecision] * N[(-0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 3.95e+186], N[(N[(N[(N[(N[(N[(0.5 * M$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.8500000000000001e-131 or 3.95000000000000001e186 < h

   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

   4. Applied rewrites 55.7%

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

   6. Applied rewrites 59.1%

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

   8. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. remove-double-neg N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. distribute-lft-neg-out N/A

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

      19. remove-double-neg N/A

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

      20. lower-*.f64 66.1

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

   10. Applied rewrites 66.1%

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

   if -1.8500000000000001e-131 < h < 9.0000000000000021e-309

   1. Initial program 59.2%

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

   4. Applied rewrites 62.6%

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

   6. Applied rewrites 60.3%

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

   8. Applied rewrites 53.9%

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

   if 9.0000000000000021e-309 < h < 3.95000000000000001e186

   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 0.0%

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

   6. Applied rewrites 0.0%

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

   8. Applied rewrites 72.0%

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

   10. Applied rewrites 84.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\ell}\right)\right) \cdot \frac{D}{d}, -0.25, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 67.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 \cdot \left(\frac{D\_m}{d} \cdot M\_m\right)\right) \cdot h, \frac{M\_m}{\ell} \cdot \left(-0.25 \cdot \frac{D\_m}{d}\right), 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 1.08 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(0.5 \cdot M\_m\right) \cdot \frac{D\_m}{d}\right) \cdot \left(h \cdot \frac{M\_m}{\ell}\right)\right) \cdot \frac{D\_m}{d}, -0.25, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 -1e-309)
   (*
    (fma (* (* 0.5 (* (/ D_m d) M_m)) h) (* (/ M_m l) (* -0.25 (/ D_m d))) 1.0)
    (sqrt (* (/ d l) (/ d h))))
   (if (<= l 1.08e+104)
     (*
      (fma
       (* (* (* (* 0.5 M_m) (/ D_m d)) (* h (/ M_m l))) (/ D_m d))
       -0.25
       1.0)
      (/ d (sqrt (* l h))))
     (/ 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 <= -1e-309) {
		tmp = fma(((0.5 * ((D_m / d) * M_m)) * h), ((M_m / l) * (-0.25 * (D_m / d))), 1.0) * sqrt(((d / l) * (d / h)));
	} else if (l <= 1.08e+104) {
		tmp = fma(((((0.5 * M_m) * (D_m / d)) * (h * (M_m / l))) * (D_m / d)), -0.25, 1.0) * (d / sqrt((l * h)));
	} 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 <= -1e-309)
		tmp = Float64(fma(Float64(Float64(0.5 * Float64(Float64(D_m / d) * M_m)) * h), Float64(Float64(M_m / l) * Float64(-0.25 * Float64(D_m / d))), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (l <= 1.08e+104)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(0.5 * M_m) * Float64(D_m / d)) * Float64(h * Float64(M_m / l))) * Float64(D_m / d)), -0.25, 1.0) * Float64(d / sqrt(Float64(l * h))));
	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, -1e-309], N[(N[(N[(N[(0.5 * N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(-0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.08e+104], N[(N[(N[(N[(N[(N[(0.5 * M$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $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 < -1.000000000000002e-309

   1. Initial program 62.7%

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

   4. Applied rewrites 67.8%

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

   6. Applied rewrites 70.3%

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

   8. Applied rewrites 52.7%

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

   if -1.000000000000002e-309 < l < 1.07999999999999997e104

   1. Initial program 76.7%

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.0%

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

   8. Applied rewrites 80.6%

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

   10. Applied rewrites 87.3%

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

   if 1.07999999999999997e104 < l

   1. Initial program 59.1%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 51.2

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

   5. Applied rewrites 51.2%

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

   7. Applied rewrites 51.2%

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

   9. Applied rewrites 70.2%

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

Alternative 16: 60.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq 1.08 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(0.5 \cdot M\_m\right) \cdot \frac{D\_m}{d}\right) \cdot \left(h \cdot \frac{M\_m}{\ell}\right)\right) \cdot \frac{D\_m}{d}, -0.25, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 -1e-309)
   (/ (* (sqrt (/ d h)) (sqrt (- d))) (sqrt (- l)))
   (if (<= l 1.08e+104)
     (*
      (fma
       (* (* (* (* 0.5 M_m) (/ D_m d)) (* h (/ M_m l))) (/ D_m d))
       -0.25
       1.0)
      (/ d (sqrt (* l h))))
     (/ 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 <= -1e-309) {
		tmp = (sqrt((d / h)) * sqrt(-d)) / sqrt(-l);
	} else if (l <= 1.08e+104) {
		tmp = fma(((((0.5 * M_m) * (D_m / d)) * (h * (M_m / l))) * (D_m / d)), -0.25, 1.0) * (d / sqrt((l * h)));
	} 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 <= -1e-309)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(-d))) / sqrt(Float64(-l)));
	elseif (l <= 1.08e+104)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(0.5 * M_m) * Float64(D_m / d)) * Float64(h * Float64(M_m / l))) * Float64(D_m / d)), -0.25, 1.0) * Float64(d / sqrt(Float64(l * h))));
	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, -1e-309], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.08e+104], N[(N[(N[(N[(N[(N[(0.5 * M$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $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 < -1.000000000000002e-309

   1. Initial program 62.7%

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

   4. Applied rewrites 67.8%

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in d around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. neg-mul-1 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 40.1

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

   9. Applied rewrites 40.1%

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

   if -1.000000000000002e-309 < l < 1.07999999999999997e104

   1. Initial program 76.7%

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.0%

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

   8. Applied rewrites 80.6%

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

   10. Applied rewrites 87.3%

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

   if 1.07999999999999997e104 < l

   1. Initial program 59.1%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 51.2

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

   5. Applied rewrites 51.2%

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

   7. Applied rewrites 51.2%

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

   9. Applied rewrites 70.2%

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

Alternative 17: 58.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq 4.9 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot \frac{D\_m}{\ell}, \frac{M\_m}{d} \cdot \left(\left(h \cdot \left(\frac{D\_m}{d} \cdot 0.5\right)\right) \cdot M\_m\right), 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 -1e-309)
   (/ (* (sqrt (/ d h)) (sqrt (- d))) (sqrt (- l)))
   (if (<= l 4.9e+110)
     (*
      (fma
       (* -0.25 (/ D_m l))
       (* (/ M_m d) (* (* h (* (/ D_m d) 0.5)) M_m))
       1.0)
      (/ d (sqrt (* l h))))
     (/ 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 <= -1e-309) {
		tmp = (sqrt((d / h)) * sqrt(-d)) / sqrt(-l);
	} else if (l <= 4.9e+110) {
		tmp = fma((-0.25 * (D_m / l)), ((M_m / d) * ((h * ((D_m / d) * 0.5)) * M_m)), 1.0) * (d / sqrt((l * h)));
	} 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 <= -1e-309)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(-d))) / sqrt(Float64(-l)));
	elseif (l <= 4.9e+110)
		tmp = Float64(fma(Float64(-0.25 * Float64(D_m / l)), Float64(Float64(M_m / d) * Float64(Float64(h * Float64(Float64(D_m / d) * 0.5)) * M_m)), 1.0) * Float64(d / sqrt(Float64(l * h))));
	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, -1e-309], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.9e+110], N[(N[(N[(-0.25 * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(h * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $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 < -1.000000000000002e-309

   1. Initial program 62.7%

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

   4. Applied rewrites 67.8%

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in d around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. neg-mul-1 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 40.1

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

   9. Applied rewrites 40.1%

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

   if -1.000000000000002e-309 < l < 4.90000000000000002e110

   1. Initial program 75.8%

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. neg-mul-1 N/A

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

      16. lift-neg.f64 N/A

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

      17. lower-*.f64 0.0

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

   8. Applied rewrites 0.0%

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

   10. Applied rewrites 88.6%

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

   if 4.90000000000000002e110 < l

   1. Initial program 60.4%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 52.3

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

   5. Applied rewrites 52.3%

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

   7. Applied rewrites 52.3%

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

   9. Applied rewrites 71.8%

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

Alternative 18: 26.9% 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 66.4%

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

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
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 26.6

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

5. Applied rewrites 26.6%

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

7. Applied rewrites 26.7%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(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)))))