Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.5% → 83.0%

Time: 23.5s

Alternatives: 21

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.5% 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: 83.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{t\_0}{\sqrt{-h}}\\ \mathbf{if}\;\ell \leq -7.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-\ell}} \cdot \left(t\_1 \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(t\_1 \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{M\_m}{\frac{d}{D\_m}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (- d))) (t_1 (/ t_0 (sqrt (- h)))))
   (if (<= l -7.6e-70)
     (*
      (/ t_0 (sqrt (- l)))
      (* t_1 (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M_m 2.0) d)) 2.0) -0.5)))))
     (if (<= l -5e-310)
       (*
        (sqrt (/ d l))
        (*
         t_1
         (+ 1.0 (* (/ 1.0 l) (* h (* -0.125 (pow (* D_m (/ M_m d)) 2.0)))))))
       (*
        d
        (/
         (fma (* -0.125 (pow (/ M_m (/ d D_m)) 2.0)) (/ h l) 1.0)
         (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(-d);
	double t_1 = t_0 / sqrt(-h);
	double tmp;
	if (l <= -7.6e-70) {
		tmp = (t_0 / sqrt(-l)) * (t_1 * (1.0 + ((h / l) * (pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else if (l <= -5e-310) {
		tmp = sqrt((d / l)) * (t_1 * (1.0 + ((1.0 / l) * (h * (-0.125 * pow((D_m * (M_m / d)), 2.0))))));
	} else {
		tmp = d * (fma((-0.125 * pow((M_m / (d / D_m)), 2.0)), (h / l), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(t_0 / sqrt(Float64(-h)))
	tmp = 0.0
	if (l <= -7.6e-70)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(t_1 * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))));
	elseif (l <= -5e-310)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(t_1 * Float64(1.0 + Float64(Float64(1.0 / l) * Float64(h * Float64(-0.125 * (Float64(D_m * Float64(M_m / d)) ^ 2.0)))))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.125 * (Float64(M_m / Float64(d / D_m)) ^ 2.0)), Float64(h / l), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7.6e-70], N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(1.0 + N[(N[(1.0 / l), $MachinePrecision] * N[(h * N[(-0.125 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.125 * N[Power[N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.5999999999999995e-70

   1. Initial program 59.2%

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

   3. Simplified 59.1%

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

      1. frac-2neg 59.1%

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

      2. sqrt-div 71.7%

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

   6. Applied egg-rr 71.7%

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

      1. frac-2neg 71.7%

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

      2. sqrt-div 87.2%

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

   8. Applied egg-rr 87.2%

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

   if -7.5999999999999995e-70 < l < -4.999999999999985e-310

   1. Initial program 80.9%

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

   3. Simplified 80.8%

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

      1. associate-*l/ 87.2%

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

      2. clear-num 87.2%

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

      3. *-commutative 87.2%

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

      4. associate-*r/ 87.2%

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

      5. div-inv 87.2%

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

      6. metadata-eval 87.2%

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

   6. Applied egg-rr 87.2%

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

      1. associate-/r/ 87.2%

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

      2. associate-/l* 87.2%

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

   8. Simplified 87.2%

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

   9. Taylor expanded in D around 0 55.5%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative 55.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right)\right)\right)\right) \]

       2. associate-/l* 53.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}\right)\right)\right)\right) \]

       3. unpow2 53.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right)\right)\right)\right)\right) \]

       4. unpow2 53.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right)\right)\right)\right)\right) \]

       5. unpow2 53.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right)\right)\right)\right)\right) \]

       6. times-frac 64.0%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right)\right)\right)\right)\right) \]

       7. swap-sqr 87.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}\right)\right)\right)\right) \]

       8. unpow2 87.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}\right)\right)\right)\right) \]

       9. associate-*r/ 87.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}\right)\right)\right)\right) \]

       10. *-commutative 87.2%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}\right)\right)\right)\right) \]

       11. associate-/l* 87.2%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}\right)\right)\right)\right) \]

   11. Simplified 87.2%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \color{blue}{\left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. frac-2neg 80.8%

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

       2. sqrt-div 87.2%

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

   13. Applied egg-rr 97.6%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\right) \]

   if -4.999999999999985e-310 < l

   1. Initial program 67.3%

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

   3. Simplified 67.4%

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

      1. pow1 67.4%

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

      2. associate-*r* 67.4%

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

      3. sqrt-div 74.3%

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

      4. sqrt-div 80.4%

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

      5. frac-times 80.3%

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

      6. add-sqr-sqrt 80.5%

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

      7. unpow-prod-down 79.8%

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

      8. metadata-eval 79.8%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 79.8%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 79.8%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 79.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 81.7%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 81.7%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 80.2%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 81.7%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. clear-num 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      2. un-div-inv 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. Applied egg-rr 81.7%

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

Alternative 2: 80.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{+218}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{M\_m}{\frac{d}{D\_m}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
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))))
   (if (<= l -2e+218)
     (*
      (/ t_0 (sqrt (- l)))
      (*
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M_m 2.0) d)) 2.0) -0.5)))
       (sqrt (/ d h))))
     (if (<= l -5e-310)
       (*
        (sqrt (/ d l))
        (*
         (/ t_0 (sqrt (- h)))
         (+ 1.0 (* (/ 1.0 l) (* h (* -0.125 (pow (* D_m (/ M_m d)) 2.0)))))))
       (*
        d
        (/
         (fma (* -0.125 (pow (/ M_m (/ d D_m)) 2.0)) (/ h l) 1.0)
         (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -2e+218) {
		tmp = (t_0 / sqrt(-l)) * ((1.0 + ((h / l) * (pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))) * sqrt((d / h)));
	} else if (l <= -5e-310) {
		tmp = sqrt((d / l)) * ((t_0 / sqrt(-h)) * (1.0 + ((1.0 / l) * (h * (-0.125 * pow((D_m * (M_m / d)), 2.0))))));
	} else {
		tmp = d * (fma((-0.125 * pow((M_m / (d / D_m)), 2.0)), (h / l), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
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))
	tmp = 0.0
	if (l <= -2e+218)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt(Float64(d / h))));
	elseif (l <= -5e-310)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(1.0 / l) * Float64(h * Float64(-0.125 * (Float64(D_m * Float64(M_m / d)) ^ 2.0)))))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.125 * (Float64(M_m / Float64(d / D_m)) ^ 2.0)), Float64(h / l), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -2e+218], N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(1.0 / l), $MachinePrecision] * N[(h * N[(-0.125 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.125 * N[Power[N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.00000000000000017e218

   1. Initial program 42.7%

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

   3. Simplified 42.3%

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

      1. frac-2neg 42.3%

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

      2. sqrt-div 75.2%

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

   6. Applied egg-rr 75.2%

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

   if -2.00000000000000017e218 < l < -4.999999999999985e-310

   1. Initial program 72.0%

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

   3. Simplified 71.9%

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

      1. associate-*l/ 75.7%

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

      2. clear-num 75.7%

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

      3. *-commutative 75.7%

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

      4. associate-*r/ 75.7%

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

      5. div-inv 75.7%

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

      6. metadata-eval 75.7%

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

   6. Applied egg-rr 75.7%

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

      1. associate-/r/ 75.7%

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

      2. associate-/l* 75.7%

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

   8. Simplified 75.7%

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

   9. Taylor expanded in D around 0 45.8%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative 45.8%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right)\right)\right)\right) \]

       2. associate-/l* 44.0%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}\right)\right)\right)\right) \]

       3. unpow2 44.0%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right)\right)\right)\right)\right) \]

       4. unpow2 44.0%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right)\right)\right)\right)\right) \]

       5. unpow2 44.0%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right)\right)\right)\right)\right) \]

       6. times-frac 57.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right)\right)\right)\right)\right) \]

       7. swap-sqr 74.7%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}\right)\right)\right)\right) \]

       8. unpow2 74.7%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}\right)\right)\right)\right) \]

       9. associate-*r/ 75.7%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}\right)\right)\right)\right) \]

       10. *-commutative 75.7%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}\right)\right)\right)\right) \]

       11. associate-/l* 75.7%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}\right)\right)\right)\right) \]

   11. Simplified 75.7%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \color{blue}{\left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. frac-2neg 75.1%

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

       2. sqrt-div 86.7%

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

   13. Applied egg-rr 88.2%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\right) \]

   if -4.999999999999985e-310 < l

   1. Initial program 67.3%

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

   3. Simplified 67.4%

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

      1. pow1 67.4%

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

      2. associate-*r* 67.4%

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

      3. sqrt-div 74.3%

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

      4. sqrt-div 80.4%

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

      5. frac-times 80.3%

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

      6. add-sqr-sqrt 80.5%

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

      7. unpow-prod-down 79.8%

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

      8. metadata-eval 79.8%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 79.8%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 79.8%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 79.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 81.7%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 81.7%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 80.2%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 81.7%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. clear-num 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      2. un-div-inv 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. Applied egg-rr 81.7%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{+218}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{M\_m}{\frac{d}{D\_m}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
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 -5e-310)
   (*
    (sqrt (/ d l))
    (*
     (/ (sqrt (- d)) (sqrt (- h)))
     (+ 1.0 (* (/ 1.0 l) (* h (* -0.125 (pow (* D_m (/ M_m d)) 2.0)))))))
   (*
    d
    (/
     (fma (* -0.125 (pow (/ M_m (/ d D_m)) 2.0)) (/ h l) 1.0)
     (* (sqrt h) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
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 <= -5e-310) {
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((1.0 / l) * (h * (-0.125 * pow((D_m * (M_m / d)), 2.0))))));
	} else {
		tmp = d * (fma((-0.125 * pow((M_m / (d / D_m)), 2.0)), (h / l), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
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 <= -5e-310)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(1.0 / l) * Float64(h * Float64(-0.125 * (Float64(D_m * Float64(M_m / d)) ^ 2.0)))))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.125 * (Float64(M_m / Float64(d / D_m)) ^ 2.0)), Float64(h / l), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $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, -5e-310], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(1.0 / l), $MachinePrecision] * N[(h * N[(-0.125 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.125 * N[Power[N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 67.3%

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

   3. Simplified 67.2%

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

      1. associate-*l/ 70.4%

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

      2. clear-num 70.4%

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

      3. *-commutative 70.4%

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

      4. associate-*r/ 70.4%

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

      5. div-inv 70.4%

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

      6. metadata-eval 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. associate-/r/ 70.5%

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

      2. associate-/l* 70.4%

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

   8. Simplified 70.4%

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

   9. Taylor expanded in D around 0 42.6%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative 42.6%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right)\right)\right)\right) \]

       2. associate-/l* 41.1%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}\right)\right)\right)\right) \]

       3. unpow2 41.1%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right)\right)\right)\right)\right) \]

       4. unpow2 41.1%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right)\right)\right)\right)\right) \]

       5. unpow2 41.1%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right)\right)\right)\right)\right) \]

       6. times-frac 53.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right)\right)\right)\right)\right) \]

       7. swap-sqr 69.6%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}\right)\right)\right)\right) \]

       8. unpow2 69.6%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}\right)\right)\right)\right) \]

       9. associate-*r/ 70.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}\right)\right)\right)\right) \]

       10. *-commutative 70.5%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}\right)\right)\right)\right) \]

       11. associate-/l* 70.4%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}\right)\right)\right)\right) \]

   11. Simplified 70.4%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \color{blue}{\left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. frac-2neg 75.1%

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

       2. sqrt-div 87.2%

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

   13. Applied egg-rr 82.5%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\right) \]

   if -4.999999999999985e-310 < l

   1. Initial program 67.3%

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

   3. Simplified 67.4%

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

      1. pow1 67.4%

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

      2. associate-*r* 67.4%

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

      3. sqrt-div 74.3%

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

      4. sqrt-div 80.4%

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

      5. frac-times 80.3%

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

      6. add-sqr-sqrt 80.5%

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

      7. unpow-prod-down 79.8%

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

      8. metadata-eval 79.8%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 79.8%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 79.8%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 79.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 81.7%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 81.7%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 80.2%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 81.7%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. clear-num 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      2. un-div-inv 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. Applied egg-rr 81.7%

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

Alternative 4: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot {\left(\sqrt{h \cdot -0.125} \cdot \frac{D\_m \cdot M\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{M\_m}{\frac{d}{D\_m}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -4.7e-108)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+
      1.0
      (* (/ 1.0 l) (pow (* (sqrt (* h -0.125)) (/ (* D_m M_m) d)) 2.0)))))
   (if (<= d -1e-310)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ -1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
     (*
      d
      (/
       (fma (* -0.125 (pow (/ M_m (/ d D_m)) 2.0)) (/ h l) 1.0)
       (* (sqrt h) (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -4.7e-108) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((1.0 / l) * pow((sqrt((h * -0.125)) * ((D_m * M_m) / d)), 2.0))));
	} else if (d <= -1e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	} else {
		tmp = d * (fma((-0.125 * pow((M_m / (d / D_m)), 2.0)), (h / l), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -4.7e-108)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(1.0 / l) * (Float64(sqrt(Float64(h * -0.125)) * Float64(Float64(D_m * M_m) / d)) ^ 2.0)))));
	elseif (d <= -1e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.125 * (Float64(M_m / Float64(d / D_m)) ^ 2.0)), Float64(h / l), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -4.7e-108], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(1.0 / l), $MachinePrecision] * N[Power[N[(N[Sqrt[N[(h * -0.125), $MachinePrecision]], $MachinePrecision] * N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.125 * N[Power[N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.70000000000000013e-108

   1. Initial program 77.4%

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

   3. Simplified 77.3%

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

      1. associate-*l/ 82.2%

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

      2. clear-num 82.1%

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

      3. *-commutative 82.1%

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

      4. associate-*r/ 82.3%

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

      5. div-inv 82.3%

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

      6. metadata-eval 82.3%

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

   6. Applied egg-rr 82.3%

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

      1. associate-/r/ 82.3%

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

      2. associate-/l* 82.2%

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

   8. Simplified 82.2%

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

   9. Taylor expanded in D around 0 52.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative 52.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right)\right)\right)\right) \]

       2. associate-/l* 52.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}\right)\right)\right)\right) \]

       3. unpow2 52.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right)\right)\right)\right)\right) \]

       4. unpow2 52.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right)\right)\right)\right)\right) \]

       5. unpow2 52.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right)\right)\right)\right)\right) \]

       6. times-frac 64.0%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right)\right)\right)\right)\right) \]

       7. swap-sqr 82.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}\right)\right)\right)\right) \]

       8. unpow2 82.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}\right)\right)\right)\right) \]

       9. associate-*r/ 82.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}\right)\right)\right)\right) \]

       10. *-commutative 82.3%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}\right)\right)\right)\right) \]

       11. associate-/l* 82.2%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}\right)\right)\right)\right) \]

   11. Simplified 82.2%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \color{blue}{\left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 82.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \color{blue}{\left(\sqrt{h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)} \cdot \sqrt{h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right)}\right)\right) \]

       2. pow2 82.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \color{blue}{{\left(\sqrt{h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right)}^{2}}\right)\right) \]

       3. associate-*r* 82.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot {\left(\sqrt{\color{blue}{\left(h \cdot -0.125\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}\right)}^{2}\right)\right) \]

       4. sqrt-prod 82.1%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot {\color{blue}{\left(\sqrt{h \cdot -0.125} \cdot \sqrt{{\left(D \cdot \frac{M}{d}\right)}^{2}}\right)}}^{2}\right)\right) \]

       5. *-commutative 82.1%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot {\left(\sqrt{h \cdot -0.125} \cdot \sqrt{{\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}}\right)}^{2}\right)\right) \]

       6. associate-/r/ 81.0%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot {\left(\sqrt{h \cdot -0.125} \cdot \sqrt{{\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}}\right)}^{2}\right)\right) \]

       7. sqrt-pow1 83.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot {\left(\sqrt{h \cdot -0.125} \cdot \color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{\left(\frac{2}{2}\right)}}\right)}^{2}\right)\right) \]

       8. metadata-eval 83.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot {\left(\sqrt{h \cdot -0.125} \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{\color{blue}{1}}\right)}^{2}\right)\right) \]

       9. pow1 83.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot {\left(\sqrt{h \cdot -0.125} \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right)}^{2}\right)\right) \]

       10. associate-/r/ 83.4%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot {\left(\sqrt{h \cdot -0.125} \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)}^{2}\right)\right) \]

       11. *-commutative 83.4%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot {\left(\sqrt{h \cdot -0.125} \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right)}^{2}\right)\right) \]

       12. associate-*r/ 84.6%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot {\left(\sqrt{h \cdot -0.125} \cdot \color{blue}{\frac{D \cdot M}{d}}\right)}^{2}\right)\right) \]

   13. Applied egg-rr 84.6%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \color{blue}{{\left(\sqrt{h \cdot -0.125} \cdot \frac{D \cdot M}{d}\right)}^{2}}\right)\right) \]

   if -4.70000000000000013e-108 < d < -9.999999999999969e-311

   1. Initial program 49.2%

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

   3. Simplified 46.8%

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

      1. sqrt-unprod 39.3%

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

      2. pow1/2 39.3%

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

      3. frac-times 17.6%

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

      4. pow2 17.6%

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

   6. Applied egg-rr 17.6%

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

      1. unpow1/2 17.6%

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

   8. Simplified 17.6%

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

   9. Taylor expanded in d around -inf 63.1%

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

   if -9.999999999999969e-311 < d

   1. Initial program 67.3%

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

   3. Simplified 67.4%

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

      1. pow1 67.4%

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

      2. associate-*r* 67.4%

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

      3. sqrt-div 74.3%

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

      4. sqrt-div 80.4%

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

      5. frac-times 80.3%

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

      6. add-sqr-sqrt 80.5%

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

      7. unpow-prod-down 79.8%

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

      8. metadata-eval 79.8%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 79.8%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 79.8%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 79.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 81.7%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 81.7%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 80.2%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 81.7%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. clear-num 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      2. un-div-inv 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. Applied egg-rr 81.7%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot {\left(\sqrt{h \cdot -0.125} \cdot \frac{D \cdot M}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -8.6 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.125\right) \cdot {\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{M\_m}{\frac{d}{D\_m}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -8.6e-108)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (/ (* (* h -0.125) (pow (/ (* D_m M_m) d) 2.0)) l))))
   (if (<= d -1e-310)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ -1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
     (*
      d
      (/
       (fma (* -0.125 (pow (/ M_m (/ d D_m)) 2.0)) (/ h l) 1.0)
       (* (sqrt h) (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -8.6e-108) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (((h * -0.125) * pow(((D_m * M_m) / d), 2.0)) / l)));
	} else if (d <= -1e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	} else {
		tmp = d * (fma((-0.125 * pow((M_m / (d / D_m)), 2.0)), (h / l), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -8.6e-108)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(h * -0.125) * (Float64(Float64(D_m * M_m) / d) ^ 2.0)) / l))));
	elseif (d <= -1e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.125 * (Float64(M_m / Float64(d / D_m)) ^ 2.0)), Float64(h / l), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -8.6e-108], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(h * -0.125), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.125 * N[Power[N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.6000000000000001e-108

   1. Initial program 77.4%

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

   3. Simplified 77.3%

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

      1. associate-*l/ 82.2%

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

      2. clear-num 82.1%

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

      3. *-commutative 82.1%

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

      4. associate-*r/ 82.3%

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

      5. div-inv 82.3%

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

      6. metadata-eval 82.3%

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

   6. Applied egg-rr 82.3%

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

      1. associate-/r/ 82.3%

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

      2. associate-/l* 82.2%

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

   8. Simplified 82.2%

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

   9. Taylor expanded in D around 0 52.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative 52.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right)\right)\right)\right) \]

       2. associate-/l* 52.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}\right)\right)\right)\right) \]

       3. unpow2 52.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right)\right)\right)\right)\right) \]

       4. unpow2 52.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right)\right)\right)\right)\right) \]

       5. unpow2 52.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right)\right)\right)\right)\right) \]

       6. times-frac 64.0%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right)\right)\right)\right)\right) \]

       7. swap-sqr 82.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}\right)\right)\right)\right) \]

       8. unpow2 82.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}\right)\right)\right)\right) \]

       9. associate-*r/ 82.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}\right)\right)\right)\right) \]

       10. *-commutative 82.3%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}\right)\right)\right)\right) \]

       11. associate-/l* 82.2%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}\right)\right)\right)\right) \]

   11. Simplified 82.2%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \color{blue}{\left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. associate-*l/ 82.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{1 \cdot \left(h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)}{\ell}}\right)\right) \]

       2. *-un-lft-identity 82.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\color{blue}{h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}}{\ell}\right)\right) \]

       3. associate-*r* 82.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\color{blue}{\left(h \cdot -0.125\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}{\ell}\right)\right) \]

       4. associate-*r/ 82.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.125\right) \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}}{\ell}\right)\right) \]

   13. Applied egg-rr 82.3%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(h \cdot -0.125\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}}\right)\right) \]

   if -8.6000000000000001e-108 < d < -9.999999999999969e-311

   1. Initial program 49.2%

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

   3. Simplified 46.8%

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

      1. sqrt-unprod 39.3%

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

      2. pow1/2 39.3%

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

      3. frac-times 17.6%

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

      4. pow2 17.6%

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

   6. Applied egg-rr 17.6%

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

      1. unpow1/2 17.6%

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

   8. Simplified 17.6%

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

   9. Taylor expanded in d around -inf 63.1%

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

   if -9.999999999999969e-311 < d

   1. Initial program 67.3%

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

   3. Simplified 67.4%

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

      1. pow1 67.4%

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

      2. associate-*r* 67.4%

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

      3. sqrt-div 74.3%

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

      4. sqrt-div 80.4%

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

      5. frac-times 80.3%

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

      6. add-sqr-sqrt 80.5%

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

      7. unpow-prod-down 79.8%

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

      8. metadata-eval 79.8%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 79.8%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 79.8%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 79.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 81.7%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 81.7%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 80.2%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 81.7%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. clear-num 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      2. un-div-inv 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. Applied egg-rr 81.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.6 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.125\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.125\right) \cdot {\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M\_m \cdot \frac{D\_m}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -9.5e-108)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (/ (* (* h -0.125) (pow (/ (* D_m M_m) d) 2.0)) l))))
   (if (<= d -1e-310)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ -1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
     (*
      d
      (/
       (fma (* -0.125 (pow (* M_m (/ D_m d)) 2.0)) (/ h l) 1.0)
       (* (sqrt h) (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -9.5e-108) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (((h * -0.125) * pow(((D_m * M_m) / d), 2.0)) / l)));
	} else if (d <= -1e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	} else {
		tmp = d * (fma((-0.125 * pow((M_m * (D_m / d)), 2.0)), (h / l), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -9.5e-108)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(h * -0.125) * (Float64(Float64(D_m * M_m) / d) ^ 2.0)) / l))));
	elseif (d <= -1e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.125 * (Float64(M_m * Float64(D_m / d)) ^ 2.0)), Float64(h / l), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -9.5e-108], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(h * -0.125), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.125 * N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.5000000000000005e-108

   1. Initial program 77.4%

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

   3. Simplified 77.3%

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

      1. associate-*l/ 82.2%

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

      2. clear-num 82.1%

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

      3. *-commutative 82.1%

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

      4. associate-*r/ 82.3%

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

      5. div-inv 82.3%

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

      6. metadata-eval 82.3%

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

   6. Applied egg-rr 82.3%

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

      1. associate-/r/ 82.3%

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

      2. associate-/l* 82.2%

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

   8. Simplified 82.2%

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

   9. Taylor expanded in D around 0 52.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative 52.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right)\right)\right)\right) \]

       2. associate-/l* 52.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}\right)\right)\right)\right) \]

       3. unpow2 52.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right)\right)\right)\right)\right) \]

       4. unpow2 52.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right)\right)\right)\right)\right) \]

       5. unpow2 52.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right)\right)\right)\right)\right) \]

       6. times-frac 64.0%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right)\right)\right)\right)\right) \]

       7. swap-sqr 82.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}\right)\right)\right)\right) \]

       8. unpow2 82.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}\right)\right)\right)\right) \]

       9. associate-*r/ 82.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}\right)\right)\right)\right) \]

       10. *-commutative 82.3%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}\right)\right)\right)\right) \]

       11. associate-/l* 82.2%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}\right)\right)\right)\right) \]

   11. Simplified 82.2%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \color{blue}{\left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. associate-*l/ 82.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{1 \cdot \left(h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)}{\ell}}\right)\right) \]

       2. *-un-lft-identity 82.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\color{blue}{h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}}{\ell}\right)\right) \]

       3. associate-*r* 82.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\color{blue}{\left(h \cdot -0.125\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}{\ell}\right)\right) \]

       4. associate-*r/ 82.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.125\right) \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}}{\ell}\right)\right) \]

   13. Applied egg-rr 82.3%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(h \cdot -0.125\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}}\right)\right) \]

   if -9.5000000000000005e-108 < d < -9.999999999999969e-311

   1. Initial program 49.2%

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

   3. Simplified 46.8%

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

      1. sqrt-unprod 39.3%

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

      2. pow1/2 39.3%

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

      3. frac-times 17.6%

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

      4. pow2 17.6%

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

   6. Applied egg-rr 17.6%

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

      1. unpow1/2 17.6%

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

   8. Simplified 17.6%

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

   9. Taylor expanded in d around -inf 63.1%

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

   if -9.999999999999969e-311 < d

   1. Initial program 67.3%

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

   3. Simplified 67.4%

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

      1. pow1 67.4%

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

      2. associate-*r* 67.4%

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

      3. sqrt-div 74.3%

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

      4. sqrt-div 80.4%

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

      5. frac-times 80.3%

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

      6. add-sqr-sqrt 80.5%

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

      7. unpow-prod-down 79.8%

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

      8. metadata-eval 79.8%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 79.8%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 79.8%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 79.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 81.7%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 81.7%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 81.7%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 81.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 80.2%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 81.7%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 81.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.125\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\\ \mathbf{if}\;d \leq -8 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.125\right) \cdot {\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
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 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
   (if (<= d -8e-108)
     (*
      (sqrt (/ d l))
      (*
       (sqrt (/ d h))
       (+ 1.0 (/ (* (* h -0.125) (pow (/ (* D_m M_m) d) 2.0)) l))))
     (if (<= d -1e-310)
       (* (* d (sqrt (/ 1.0 (* l h)))) (+ -1.0 t_0))
       (* (- 1.0 t_0) (/ d (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
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 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double tmp;
	if (d <= -8e-108) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (((h * -0.125) * pow(((D_m * M_m) / d), 2.0)) / l)));
	} else if (d <= -1e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else {
		tmp = (1.0 - t_0) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))
    if (d <= (-8d-108)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + (((h * (-0.125d0)) * (((d_m * m_m) / d) ** 2.0d0)) / l)))
    else if (d <= (-1d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + t_0)
    else
        tmp = (1.0d0 - t_0) * (d / (sqrt(h) * sqrt(l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
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 = 0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double tmp;
	if (d <= -8e-108) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + (((h * -0.125) * Math.pow(((D_m * M_m) / d), 2.0)) / l)));
	} else if (d <= -1e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else {
		tmp = (1.0 - t_0) * (d / (Math.sqrt(h) * Math.sqrt(l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[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 = 0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))
	tmp = 0
	if d <= -8e-108:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + (((h * -0.125) * math.pow(((D_m * M_m) / d), 2.0)) / l)))
	elif d <= -1e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + t_0)
	else:
		tmp = (1.0 - t_0) * (d / (math.sqrt(h) * math.sqrt(l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
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(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))
	tmp = 0.0
	if (d <= -8e-108)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(h * -0.125) * (Float64(Float64(D_m * M_m) / d) ^ 2.0)) / l))));
	elseif (d <= -1e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + t_0));
	else
		tmp = Float64(Float64(1.0 - t_0) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
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 = 0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0));
	tmp = 0.0;
	if (d <= -8e-108)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (((h * -0.125) * (((D_m * M_m) / d) ^ 2.0)) / l)));
	elseif (d <= -1e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	else
		tmp = (1.0 - t_0) * (d / (sqrt(h) * sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $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[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8e-108], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(h * -0.125), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.00000000000000032e-108

   1. Initial program 77.4%

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

   3. Simplified 77.3%

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

      1. associate-*l/ 82.2%

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

      2. clear-num 82.1%

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

      3. *-commutative 82.1%

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

      4. associate-*r/ 82.3%

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

      5. div-inv 82.3%

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

      6. metadata-eval 82.3%

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

   6. Applied egg-rr 82.3%

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

      1. associate-/r/ 82.3%

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

      2. associate-/l* 82.2%

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

   8. Simplified 82.2%

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

   9. Taylor expanded in D around 0 52.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative 52.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right)\right)\right)\right) \]

       2. associate-/l* 52.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}\right)\right)\right)\right) \]

       3. unpow2 52.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right)\right)\right)\right)\right) \]

       4. unpow2 52.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right)\right)\right)\right)\right) \]

       5. unpow2 52.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right)\right)\right)\right)\right) \]

       6. times-frac 64.0%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right)\right)\right)\right)\right) \]

       7. swap-sqr 82.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}\right)\right)\right)\right) \]

       8. unpow2 82.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}\right)\right)\right)\right) \]

       9. associate-*r/ 82.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}\right)\right)\right)\right) \]

       10. *-commutative 82.3%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}\right)\right)\right)\right) \]

       11. associate-/l* 82.2%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}\right)\right)\right)\right) \]

   11. Simplified 82.2%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \color{blue}{\left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. associate-*l/ 82.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{1 \cdot \left(h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)}{\ell}}\right)\right) \]

       2. *-un-lft-identity 82.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\color{blue}{h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}}{\ell}\right)\right) \]

       3. associate-*r* 82.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\color{blue}{\left(h \cdot -0.125\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}{\ell}\right)\right) \]

       4. associate-*r/ 82.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.125\right) \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}}{\ell}\right)\right) \]

   13. Applied egg-rr 82.3%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(h \cdot -0.125\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}}\right)\right) \]

   if -8.00000000000000032e-108 < d < -9.999999999999969e-311

   1. Initial program 49.2%

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

   3. Simplified 46.8%

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

      1. sqrt-unprod 39.3%

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

      2. pow1/2 39.3%

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

      3. frac-times 17.6%

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

      4. pow2 17.6%

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

   6. Applied egg-rr 17.6%

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

      1. unpow1/2 17.6%

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

   8. Simplified 17.6%

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

   9. Taylor expanded in d around -inf 63.1%

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

   if -9.999999999999969e-311 < d

   1. Initial program 67.3%

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

   3. Simplified 67.4%

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

      1. *-commutative 67.4%

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

      2. sqrt-div 73.5%

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

      3. sqrt-div 80.4%

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

      4. frac-times 80.3%

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

      5. add-sqr-sqrt 80.5%

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

   6. Applied egg-rr 80.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.125\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{-274}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.125\right) \cdot {\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+134}:\\ \;\;\;\;\frac{d \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
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 2.7e-274)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (/ (* (* h -0.125) (pow (/ (* D_m M_m) d) 2.0)) l))))
   (if (<= l 3.3e+134)
     (/
      (* d (- 1.0 (* h (* 0.125 (/ (pow (* D_m (/ M_m d)) 2.0) l)))))
      (sqrt (* l h)))
     (* d (/ 1.0 (* (sqrt h) (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
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 <= 2.7e-274) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (((h * -0.125) * pow(((D_m * M_m) / d), 2.0)) / l)));
	} else if (l <= 3.3e+134) {
		tmp = (d * (1.0 - (h * (0.125 * (pow((D_m * (M_m / d)), 2.0) / l))))) / sqrt((l * h));
	} else {
		tmp = d * (1.0 / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
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 <= 2.7d-274) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + (((h * (-0.125d0)) * (((d_m * m_m) / d) ** 2.0d0)) / l)))
    else if (l <= 3.3d+134) then
        tmp = (d * (1.0d0 - (h * (0.125d0 * (((d_m * (m_m / d)) ** 2.0d0) / l))))) / sqrt((l * h))
    else
        tmp = d * (1.0d0 / (sqrt(h) * sqrt(l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
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 <= 2.7e-274) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + (((h * -0.125) * Math.pow(((D_m * M_m) / d), 2.0)) / l)));
	} else if (l <= 3.3e+134) {
		tmp = (d * (1.0 - (h * (0.125 * (Math.pow((D_m * (M_m / d)), 2.0) / l))))) / Math.sqrt((l * h));
	} else {
		tmp = d * (1.0 / (Math.sqrt(h) * Math.sqrt(l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[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 <= 2.7e-274:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + (((h * -0.125) * math.pow(((D_m * M_m) / d), 2.0)) / l)))
	elif l <= 3.3e+134:
		tmp = (d * (1.0 - (h * (0.125 * (math.pow((D_m * (M_m / d)), 2.0) / l))))) / math.sqrt((l * h))
	else:
		tmp = d * (1.0 / (math.sqrt(h) * math.sqrt(l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
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 <= 2.7e-274)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(h * -0.125) * (Float64(Float64(D_m * M_m) / d) ^ 2.0)) / l))));
	elseif (l <= 3.3e+134)
		tmp = Float64(Float64(d * Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) / l))))) / sqrt(Float64(l * h)));
	else
		tmp = Float64(d * Float64(1.0 / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
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 <= 2.7e-274)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (((h * -0.125) * (((D_m * M_m) / d) ^ 2.0)) / l)));
	elseif (l <= 3.3e+134)
		tmp = (d * (1.0 - (h * (0.125 * (((D_m * (M_m / d)) ^ 2.0) / l))))) / sqrt((l * h));
	else
		tmp = d * (1.0 / (sqrt(h) * sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $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, 2.7e-274], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(h * -0.125), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.3e+134], N[(N[(d * N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(1.0 / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.7e-274

   1. Initial program 68.3%

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

   3. Simplified 68.2%

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

      1. associate-*l/ 71.2%

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

      2. clear-num 71.2%

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

      3. *-commutative 71.2%

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

      4. associate-*r/ 71.2%

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

      5. div-inv 71.2%

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

      6. metadata-eval 71.2%

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

   6. Applied egg-rr 71.2%

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

      1. associate-/r/ 71.3%

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

      2. associate-/l* 71.2%

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

   8. Simplified 71.2%

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

   9. Taylor expanded in D around 0 43.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative 43.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right)\right)\right)\right) \]

       2. associate-/l* 42.0%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}\right)\right)\right)\right) \]

       3. unpow2 42.0%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right)\right)\right)\right)\right) \]

       4. unpow2 42.0%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right)\right)\right)\right)\right) \]

       5. unpow2 42.0%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right)\right)\right)\right)\right) \]

       6. times-frac 55.0%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right)\right)\right)\right)\right) \]

       7. swap-sqr 70.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}\right)\right)\right)\right) \]

       8. unpow2 70.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}\right)\right)\right)\right) \]

       9. associate-*r/ 71.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}\right)\right)\right)\right) \]

       10. *-commutative 71.3%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}\right)\right)\right)\right) \]

       11. associate-/l* 71.2%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}\right)\right)\right)\right) \]

   11. Simplified 71.2%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{1}{\ell} \cdot \left(h \cdot \color{blue}{\left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. associate-*l/ 71.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{1 \cdot \left(h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)}{\ell}}\right)\right) \]

       2. *-un-lft-identity 71.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\color{blue}{h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}}{\ell}\right)\right) \]

       3. associate-*r* 71.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\color{blue}{\left(h \cdot -0.125\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}{\ell}\right)\right) \]

       4. associate-*r/ 71.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.125\right) \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}}{\ell}\right)\right) \]

   13. Applied egg-rr 71.3%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(h \cdot -0.125\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}}\right)\right) \]

   if 2.7e-274 < l < 3.3e134

   1. Initial program 71.3%

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

   3. Simplified 70.1%

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

      1. pow1 70.1%

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

      2. associate-*r* 70.1%

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

      3. sqrt-div 77.9%

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

      4. sqrt-div 82.5%

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

      5. frac-times 82.5%

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

      6. add-sqr-sqrt 82.6%

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

      7. unpow-prod-down 81.4%

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

      8. metadata-eval 81.4%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 81.4%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 81.4%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 81.4%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 81.4%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 82.7%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 81.4%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 81.4%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 81.4%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 81.4%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 81.4%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 81.4%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 81.4%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 81.4%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 81.5%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 81.4%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 81.4%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 81.4%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 82.7%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. associate-*r/ 83.8%

         \[\leadsto \frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      3. sqrt-unprod 83.8%

         \[\leadsto \frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{M \cdot D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   10. Applied egg-rr 83.8%

       \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{M \cdot D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h \cdot \ell}}} \]

   11. Taylor expanded in h around -inf 59.7%

       \[\leadsto \frac{d \cdot \color{blue}{\left(-1 \cdot \left(h \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} - \frac{1}{h}\right)\right)\right)}}{\sqrt{h \cdot \ell}} \]
   12. Step-by-step derivation

       1. associate-*r* 59.7%

          \[\leadsto \frac{d \cdot \color{blue}{\left(\left(-1 \cdot h\right) \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} - \frac{1}{h}\right)\right)}}{\sqrt{h \cdot \ell}} \]

       2. neg-mul-1 59.7%

          \[\leadsto \frac{d \cdot \left(\color{blue}{\left(-h\right)} \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} - \frac{1}{h}\right)\right)}{\sqrt{h \cdot \ell}} \]

       3. sub-neg 59.7%

          \[\leadsto \frac{d \cdot \left(\left(-h\right) \cdot \color{blue}{\left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} + \left(-\frac{1}{h}\right)\right)}\right)}{\sqrt{h \cdot \ell}} \]

       4. distribute-lft-in 59.7%

          \[\leadsto \frac{d \cdot \color{blue}{\left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) + \left(-h\right) \cdot \left(-\frac{1}{h}\right)\right)}}{\sqrt{h \cdot \ell}} \]

   13. Simplified 87.7%

       \[\leadsto \frac{d \cdot \color{blue}{\left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) + 1\right)}}{\sqrt{h \cdot \ell}} \]

   if 3.3e134 < l

   1. Initial program 57.2%

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

   3. Simplified 59.4%

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

      1. pow1 59.4%

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

      2. associate-*r* 59.4%

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

      3. sqrt-div 63.7%

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

      4. sqrt-div 73.4%

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

      5. frac-times 73.4%

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

      6. add-sqr-sqrt 73.6%

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

      7. unpow-prod-down 73.6%

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

      8. metadata-eval 73.6%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 73.6%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 73.7%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 73.7%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 77.2%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 79.3%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 79.3%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 79.3%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 79.3%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 79.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 79.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 79.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 79.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 79.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 74.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 79.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 79.3%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   9. Taylor expanded in M around 0 68.6%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{-274}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.125\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+134}:\\ \;\;\;\;\frac{d \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}\\ \mathbf{if}\;\ell \leq -4.25 \cdot 10^{+40}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(t\_0 \cdot \left(\frac{h}{\ell} \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+133}:\\ \;\;\;\;\frac{d \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{t\_0}{\ell}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
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_m (/ M_m d)) 2.0)))
   (if (<= l -4.25e+40)
     (* d (- (sqrt (/ (/ 1.0 l) h))))
     (if (<= l -5e-310)
       (* (sqrt (* (/ d l) (/ d h))) (+ 1.0 (* -0.5 (* t_0 (* (/ h l) 0.25)))))
       (if (<= l 7.5e+133)
         (/ (* d (- 1.0 (* h (* 0.125 (/ t_0 l))))) (sqrt (* l h)))
         (* d (/ 1.0 (* (sqrt h) (sqrt l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
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_m * (M_m / d)), 2.0);
	double tmp;
	if (l <= -4.25e+40) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else if (l <= -5e-310) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (t_0 * ((h / l) * 0.25))));
	} else if (l <= 7.5e+133) {
		tmp = (d * (1.0 - (h * (0.125 * (t_0 / l))))) / sqrt((l * h));
	} else {
		tmp = d * (1.0 / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
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) :: t_0
    real(8) :: tmp
    t_0 = (d_m * (m_m / d)) ** 2.0d0
    if (l <= (-4.25d+40)) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    else if (l <= (-5d-310)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + ((-0.5d0) * (t_0 * ((h / l) * 0.25d0))))
    else if (l <= 7.5d+133) then
        tmp = (d * (1.0d0 - (h * (0.125d0 * (t_0 / l))))) / sqrt((l * h))
    else
        tmp = d * (1.0d0 / (sqrt(h) * sqrt(l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
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.pow((D_m * (M_m / d)), 2.0);
	double tmp;
	if (l <= -4.25e+40) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else if (l <= -5e-310) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (t_0 * ((h / l) * 0.25))));
	} else if (l <= 7.5e+133) {
		tmp = (d * (1.0 - (h * (0.125 * (t_0 / l))))) / Math.sqrt((l * h));
	} else {
		tmp = d * (1.0 / (Math.sqrt(h) * Math.sqrt(l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[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.pow((D_m * (M_m / d)), 2.0)
	tmp = 0
	if l <= -4.25e+40:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	elif l <= -5e-310:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (t_0 * ((h / l) * 0.25))))
	elif l <= 7.5e+133:
		tmp = (d * (1.0 - (h * (0.125 * (t_0 / l))))) / math.sqrt((l * h))
	else:
		tmp = d * (1.0 / (math.sqrt(h) * math.sqrt(l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
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(D_m * Float64(M_m / d)) ^ 2.0
	tmp = 0.0
	if (l <= -4.25e+40)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (l <= -5e-310)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64(t_0 * Float64(Float64(h / l) * 0.25)))));
	elseif (l <= 7.5e+133)
		tmp = Float64(Float64(d * Float64(1.0 - Float64(h * Float64(0.125 * Float64(t_0 / l))))) / sqrt(Float64(l * h)));
	else
		tmp = Float64(d * Float64(1.0 / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
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 = (D_m * (M_m / d)) ^ 2.0;
	tmp = 0.0;
	if (l <= -4.25e+40)
		tmp = d * -sqrt(((1.0 / l) / h));
	elseif (l <= -5e-310)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (t_0 * ((h / l) * 0.25))));
	elseif (l <= 7.5e+133)
		tmp = (d * (1.0 - (h * (0.125 * (t_0 / l))))) / sqrt((l * h));
	else
		tmp = d * (1.0 / (sqrt(h) * sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $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[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -4.25e+40], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.5e+133], N[(N[(d * N[(1.0 - N[(h * N[(0.125 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(1.0 / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.24999999999999998e40

   1. Initial program 51.2%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in d around inf 7.0%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 56.7%

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

      4. mul-1-neg 56.7%

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

      5. *-commutative 56.7%

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

      6. associate-/r* 58.6%

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

   8. Simplified 58.6%

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

   if -4.24999999999999998e40 < l < -4.999999999999985e-310

   1. Initial program 77.9%

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

   3. Simplified 77.9%

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

      1. add-sqr-sqrt 77.9%

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

      2. pow2 77.9%

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

      3. sqrt-prod 77.9%

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

      4. sqrt-pow1 77.9%

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

      5. metadata-eval 77.9%

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

      6. associate-*l/ 77.9%

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

      7. div-inv 77.9%

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

      8. metadata-eval 77.9%

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

      9. *-commutative 77.9%

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

      10. pow1 77.9%

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

      11. clear-num 76.5%

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

      12. un-div-inv 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. pow1 76.5%

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

   8. Applied egg-rr 62.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.25\right)\right)\right)\right)}^{1}} \]
   9. Step-by-step derivation

      1. unpow1 62.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.25\right)\right)\right)} \]

      2. *-commutative 62.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.25\right) \cdot \frac{h}{\ell}\right)}\right) \]

      3. associate-*l* 62.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \color{blue}{\left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \left(0.25 \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. associate-/l* 62.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2} \cdot \left(0.25 \cdot \frac{h}{\ell}\right)\right)\right) \]

   10. Simplified 62.9%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.25 \cdot \frac{h}{\ell}\right)\right)\right)} \]

   if -4.999999999999985e-310 < l < 7.49999999999999992e133

   1. Initial program 72.4%

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

   3. Simplified 71.4%

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

      1. pow1 71.4%

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

      2. associate-*r* 71.4%

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

      3. sqrt-div 79.7%

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

      4. sqrt-div 83.9%

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

      5. frac-times 83.9%

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

      6. add-sqr-sqrt 84.0%

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

      7. unpow-prod-down 82.9%

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

      8. metadata-eval 82.9%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 82.9%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 82.9%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 82.9%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 82.9%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 84.1%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 82.9%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 82.9%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 82.9%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 82.9%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 82.9%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 82.9%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 82.9%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 82.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 83.0%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 82.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 82.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 82.9%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 84.1%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. associate-*r/ 85.1%

         \[\leadsto \frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      3. sqrt-unprod 82.9%

         \[\leadsto \frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{M \cdot D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   10. Applied egg-rr 82.9%

       \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{M \cdot D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h \cdot \ell}}} \]

   11. Taylor expanded in h around -inf 59.4%

       \[\leadsto \frac{d \cdot \color{blue}{\left(-1 \cdot \left(h \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} - \frac{1}{h}\right)\right)\right)}}{\sqrt{h \cdot \ell}} \]
   12. Step-by-step derivation

       1. associate-*r* 59.4%

          \[\leadsto \frac{d \cdot \color{blue}{\left(\left(-1 \cdot h\right) \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} - \frac{1}{h}\right)\right)}}{\sqrt{h \cdot \ell}} \]

       2. neg-mul-1 59.4%

          \[\leadsto \frac{d \cdot \left(\color{blue}{\left(-h\right)} \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} - \frac{1}{h}\right)\right)}{\sqrt{h \cdot \ell}} \]

       3. sub-neg 59.4%

          \[\leadsto \frac{d \cdot \left(\left(-h\right) \cdot \color{blue}{\left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} + \left(-\frac{1}{h}\right)\right)}\right)}{\sqrt{h \cdot \ell}} \]

       4. distribute-lft-in 59.4%

          \[\leadsto \frac{d \cdot \color{blue}{\left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) + \left(-h\right) \cdot \left(-\frac{1}{h}\right)\right)}}{\sqrt{h \cdot \ell}} \]

   13. Simplified 86.5%

       \[\leadsto \frac{d \cdot \color{blue}{\left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) + 1\right)}}{\sqrt{h \cdot \ell}} \]

   if 7.49999999999999992e133 < l

   1. Initial program 57.2%

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

   3. Simplified 59.4%

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

      1. pow1 59.4%

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

      2. associate-*r* 59.4%

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

      3. sqrt-div 63.7%

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

      4. sqrt-div 73.4%

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

      5. frac-times 73.4%

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

      6. add-sqr-sqrt 73.6%

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

      7. unpow-prod-down 73.6%

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

      8. metadata-eval 73.6%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 73.6%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 73.7%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 73.7%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 77.2%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 79.3%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 79.3%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 79.3%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 79.3%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 79.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 79.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 79.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 79.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 79.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 74.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 79.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 79.3%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   9. Taylor expanded in M around 0 68.6%

      \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h} \cdot \sqrt{\ell}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.25 \cdot 10^{+40}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+133}:\\ \;\;\;\;\frac{d \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{-177}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+134}:\\ \;\;\;\;\frac{d \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.05e-177)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (if (<= l -5e-310)
     (* d (pow (+ -1.0 (fma h l 1.0)) -0.5))
     (if (<= l 6.5e+134)
       (/
        (* d (- 1.0 (* h (* 0.125 (/ (pow (* D_m (/ M_m d)) 2.0) l)))))
        (sqrt (* l h)))
       (* d (/ 1.0 (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.05e-177) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else if (l <= -5e-310) {
		tmp = d * pow((-1.0 + fma(h, l, 1.0)), -0.5);
	} else if (l <= 6.5e+134) {
		tmp = (d * (1.0 - (h * (0.125 * (pow((D_m * (M_m / d)), 2.0) / l))))) / sqrt((l * h));
	} else {
		tmp = d * (1.0 / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.05e-177)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (l <= -5e-310)
		tmp = Float64(d * (Float64(-1.0 + fma(h, l, 1.0)) ^ -0.5));
	elseif (l <= 6.5e+134)
		tmp = Float64(Float64(d * Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) / l))))) / sqrt(Float64(l * h)));
	else
		tmp = Float64(d * Float64(1.0 / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.05e-177], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -5e-310], N[(d * N[Power[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.5e+134], N[(N[(d * N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(1.0 / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.05e-177

   1. Initial program 63.6%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in d around inf 6.7%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 46.5%

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

      4. mul-1-neg 46.5%

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

      5. *-commutative 46.5%

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

      6. associate-/r* 47.8%

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

   8. Simplified 47.8%

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

   if -1.05e-177 < l < -4.999999999999985e-310

   1. Initial program 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in d around inf 39.9%

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

      1. pow1 39.9%

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

      2. metadata-eval 39.9%

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

      3. sqrt-pow1 14.9%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. sqrt-prod 14.9%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{h \cdot \ell}}} \]

      5. div-inv 14.9%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

      6. sqrt-div 15.0%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}} \]

      7. sqrt-pow1 35.2%

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

      8. metadata-eval 35.2%

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

      9. pow1 35.2%

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

   7. Applied egg-rr 35.2%

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

      1. div-inv 35.2%

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

      2. pow1/2 35.2%

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

      3. pow-flip 35.2%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5\right)}} \]

      4. *-commutative 35.2%

         \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{\left(-0.5\right)} \]

      5. metadata-eval 35.2%

         \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{\color{blue}{-0.5}} \]

   9. Applied egg-rr 35.2%

      \[\leadsto \color{blue}{d \cdot {\left(\ell \cdot h\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 35.2%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot h\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 62.7%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot h\right)} - 1\right)}}^{-0.5} \]

   11. Applied egg-rr 62.7%

       \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot h\right)} - 1\right)}}^{-0.5} \]
   12. Step-by-step derivation

       1. sub-neg 62.7%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot h\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 62.7%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(\ell \cdot h\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 62.7%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(\ell \cdot h\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 62.7%

          \[\leadsto d \cdot {\left(-1 + e^{\color{blue}{\log \left(1 + \ell \cdot h\right)}}\right)}^{-0.5} \]

       5. rem-exp-log 62.7%

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

       6. +-commutative 62.7%

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

       7. *-commutative 62.7%

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

       8. fma-define 62.7%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}\right)}^{-0.5} \]

   13. Simplified 62.7%

       \[\leadsto d \cdot {\color{blue}{\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}}^{-0.5} \]

   if -4.999999999999985e-310 < l < 6.5e134

   1. Initial program 72.4%

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

   3. Simplified 71.4%

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

      1. pow1 71.4%

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

      2. associate-*r* 71.4%

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

      3. sqrt-div 79.7%

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

      4. sqrt-div 83.9%

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

      5. frac-times 83.9%

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

      6. add-sqr-sqrt 84.0%

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

      7. unpow-prod-down 82.9%

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

      8. metadata-eval 82.9%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 82.9%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 82.9%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 82.9%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 82.9%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 84.1%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 82.9%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 82.9%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 82.9%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 82.9%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 82.9%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 82.9%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 82.9%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 82.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 83.0%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 82.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 82.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 82.9%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 84.1%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. associate-*r/ 85.1%

         \[\leadsto \frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      3. sqrt-unprod 82.9%

         \[\leadsto \frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{M \cdot D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   10. Applied egg-rr 82.9%

       \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{M \cdot D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h \cdot \ell}}} \]

   11. Taylor expanded in h around -inf 59.4%

       \[\leadsto \frac{d \cdot \color{blue}{\left(-1 \cdot \left(h \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} - \frac{1}{h}\right)\right)\right)}}{\sqrt{h \cdot \ell}} \]
   12. Step-by-step derivation

       1. associate-*r* 59.4%

          \[\leadsto \frac{d \cdot \color{blue}{\left(\left(-1 \cdot h\right) \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} - \frac{1}{h}\right)\right)}}{\sqrt{h \cdot \ell}} \]

       2. neg-mul-1 59.4%

          \[\leadsto \frac{d \cdot \left(\color{blue}{\left(-h\right)} \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} - \frac{1}{h}\right)\right)}{\sqrt{h \cdot \ell}} \]

       3. sub-neg 59.4%

          \[\leadsto \frac{d \cdot \left(\left(-h\right) \cdot \color{blue}{\left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} + \left(-\frac{1}{h}\right)\right)}\right)}{\sqrt{h \cdot \ell}} \]

       4. distribute-lft-in 59.4%

          \[\leadsto \frac{d \cdot \color{blue}{\left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) + \left(-h\right) \cdot \left(-\frac{1}{h}\right)\right)}}{\sqrt{h \cdot \ell}} \]

   13. Simplified 86.5%

       \[\leadsto \frac{d \cdot \color{blue}{\left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) + 1\right)}}{\sqrt{h \cdot \ell}} \]

   if 6.5e134 < l

   1. Initial program 57.2%

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

   3. Simplified 59.4%

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

      1. pow1 59.4%

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

      2. associate-*r* 59.4%

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

      3. sqrt-div 63.7%

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

      4. sqrt-div 73.4%

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

      5. frac-times 73.4%

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

      6. add-sqr-sqrt 73.6%

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

      7. unpow-prod-down 73.6%

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

      8. metadata-eval 73.6%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 73.6%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 73.7%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 73.7%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 77.2%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 79.3%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 79.3%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 79.3%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 79.3%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 79.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 79.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 79.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 79.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 79.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 74.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 79.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 79.3%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   9. Taylor expanded in M around 0 68.6%

      \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h} \cdot \sqrt{\ell}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{-177}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+134}:\\ \;\;\;\;\frac{d \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{-178}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+130}:\\ \;\;\;\;\frac{d \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(M\_m \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
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 -2.5e-178)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (if (<= l -5e-310)
     (* d (pow (+ -1.0 (fma h l 1.0)) -0.5))
     (if (<= l 1.1e+130)
       (/
        (* d (+ 1.0 (* (/ h l) (* -0.125 (pow (* M_m (/ D_m d)) 2.0)))))
        (sqrt (* l h)))
       (* d (/ 1.0 (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
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 <= -2.5e-178) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else if (l <= -5e-310) {
		tmp = d * pow((-1.0 + fma(h, l, 1.0)), -0.5);
	} else if (l <= 1.1e+130) {
		tmp = (d * (1.0 + ((h / l) * (-0.125 * pow((M_m * (D_m / d)), 2.0))))) / sqrt((l * h));
	} else {
		tmp = d * (1.0 / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
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 <= -2.5e-178)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (l <= -5e-310)
		tmp = Float64(d * (Float64(-1.0 + fma(h, l, 1.0)) ^ -0.5));
	elseif (l <= 1.1e+130)
		tmp = Float64(Float64(d * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.125 * (Float64(M_m * Float64(D_m / d)) ^ 2.0))))) / sqrt(Float64(l * h)));
	else
		tmp = Float64(d * Float64(1.0 / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $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, -2.5e-178], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -5e-310], N[(d * N[Power[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.1e+130], N[(N[(d * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.125 * N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(1.0 / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.49999999999999988e-178

   1. Initial program 63.6%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in d around inf 6.7%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 46.5%

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

      4. mul-1-neg 46.5%

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

      5. *-commutative 46.5%

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

      6. associate-/r* 47.8%

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

   8. Simplified 47.8%

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

   if -2.49999999999999988e-178 < l < -4.999999999999985e-310

   1. Initial program 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in d around inf 39.9%

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

      1. pow1 39.9%

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

      2. metadata-eval 39.9%

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

      3. sqrt-pow1 14.9%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. sqrt-prod 14.9%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{h \cdot \ell}}} \]

      5. div-inv 14.9%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

      6. sqrt-div 15.0%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}} \]

      7. sqrt-pow1 35.2%

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

      8. metadata-eval 35.2%

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

      9. pow1 35.2%

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

   7. Applied egg-rr 35.2%

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

      1. div-inv 35.2%

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

      2. pow1/2 35.2%

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

      3. pow-flip 35.2%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5\right)}} \]

      4. *-commutative 35.2%

         \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{\left(-0.5\right)} \]

      5. metadata-eval 35.2%

         \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{\color{blue}{-0.5}} \]

   9. Applied egg-rr 35.2%

      \[\leadsto \color{blue}{d \cdot {\left(\ell \cdot h\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 35.2%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot h\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 62.7%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot h\right)} - 1\right)}}^{-0.5} \]

   11. Applied egg-rr 62.7%

       \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot h\right)} - 1\right)}}^{-0.5} \]
   12. Step-by-step derivation

       1. sub-neg 62.7%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot h\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 62.7%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(\ell \cdot h\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 62.7%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(\ell \cdot h\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 62.7%

          \[\leadsto d \cdot {\left(-1 + e^{\color{blue}{\log \left(1 + \ell \cdot h\right)}}\right)}^{-0.5} \]

       5. rem-exp-log 62.7%

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

       6. +-commutative 62.7%

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

       7. *-commutative 62.7%

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

       8. fma-define 62.7%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}\right)}^{-0.5} \]

   13. Simplified 62.7%

       \[\leadsto d \cdot {\color{blue}{\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}}^{-0.5} \]

   if -4.999999999999985e-310 < l < 1.09999999999999997e130

   1. Initial program 73.3%

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

   3. Simplified 72.2%

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

      1. pow1 72.2%

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

      2. associate-*r* 72.2%

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

      3. sqrt-div 80.6%

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

      4. sqrt-div 84.9%

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

      5. frac-times 84.8%

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

      6. add-sqr-sqrt 85.0%

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

      7. unpow-prod-down 83.9%

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

      8. metadata-eval 83.9%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 83.9%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 83.9%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 83.9%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 83.9%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 85.1%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 83.9%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 83.9%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 83.9%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 83.9%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 83.9%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 83.9%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 83.9%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 83.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 83.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 83.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 83.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 83.9%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 85.1%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. associate-*r/ 86.1%

         \[\leadsto \frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      3. sqrt-unprod 83.9%

         \[\leadsto \frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{M \cdot D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   10. Applied egg-rr 83.9%

       \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{M \cdot D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. fma-undefine 83.9%

          \[\leadsto \frac{d \cdot \color{blue}{\left(\left(-0.125 \cdot {\left(\frac{M \cdot D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell} + 1\right)}}{\sqrt{h \cdot \ell}} \]

       2. associate-/l* 82.8%

          \[\leadsto \frac{d \cdot \left(\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}\right) \cdot \frac{h}{\ell} + 1\right)}{\sqrt{h \cdot \ell}} \]

   12. Applied egg-rr 82.8%

       \[\leadsto \frac{d \cdot \color{blue}{\left(\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell} + 1\right)}}{\sqrt{h \cdot \ell}} \]

   if 1.09999999999999997e130 < l

   1. Initial program 56.0%

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

   3. Simplified 58.1%

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

      1. pow1 58.1%

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

      2. associate-*r* 58.1%

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

      3. sqrt-div 62.3%

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

      4. sqrt-div 71.8%

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

      5. frac-times 71.8%

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

      6. add-sqr-sqrt 72.0%

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

      7. unpow-prod-down 72.0%

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

      8. metadata-eval 72.0%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 72.0%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 72.0%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 72.0%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 72.0%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 75.5%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 77.6%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 77.6%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 77.6%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 77.6%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 77.6%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 77.6%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 77.6%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 77.6%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 77.5%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 73.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 77.5%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 77.5%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   9. Taylor expanded in M around 0 67.1%

      \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h} \cdot \sqrt{\ell}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{-178}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+130}:\\ \;\;\;\;\frac{d \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+134}:\\ \;\;\;\;\frac{d \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
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 -5e-310)
   (*
    (* d (sqrt (/ 1.0 (* l h))))
    (+ -1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
   (if (<= l 6e+134)
     (/
      (* d (- 1.0 (* h (* 0.125 (/ (pow (* D_m (/ M_m d)) 2.0) l)))))
      (sqrt (* l h)))
     (* d (/ 1.0 (* (sqrt h) (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
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 <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	} else if (l <= 6e+134) {
		tmp = (d * (1.0 - (h * (0.125 * (pow((D_m * (M_m / d)), 2.0) / l))))) / sqrt((l * h));
	} else {
		tmp = d * (1.0 / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
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 <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))))
    else if (l <= 6d+134) then
        tmp = (d * (1.0d0 - (h * (0.125d0 * (((d_m * (m_m / d)) ** 2.0d0) / l))))) / sqrt((l * h))
    else
        tmp = d * (1.0d0 / (sqrt(h) * sqrt(l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
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 <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	} else if (l <= 6e+134) {
		tmp = (d * (1.0 - (h * (0.125 * (Math.pow((D_m * (M_m / d)), 2.0) / l))))) / Math.sqrt((l * h));
	} else {
		tmp = d * (1.0 / (Math.sqrt(h) * Math.sqrt(l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[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 <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))))
	elif l <= 6e+134:
		tmp = (d * (1.0 - (h * (0.125 * (math.pow((D_m * (M_m / d)), 2.0) / l))))) / math.sqrt((l * h))
	else:
		tmp = d * (1.0 / (math.sqrt(h) * math.sqrt(l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
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 <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	elseif (l <= 6e+134)
		tmp = Float64(Float64(d * Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) / l))))) / sqrt(Float64(l * h)));
	else
		tmp = Float64(d * Float64(1.0 / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
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 <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))));
	elseif (l <= 6e+134)
		tmp = (d * (1.0 - (h * (0.125 * (((D_m * (M_m / d)) ^ 2.0) / l))))) / sqrt((l * h));
	else
		tmp = d * (1.0 / (sqrt(h) * sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $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, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6e+134], N[(N[(d * N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(1.0 / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 67.3%

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

   3. Simplified 66.5%

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

      1. sqrt-unprod 54.2%

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

      2. pow1/2 54.2%

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

      3. frac-times 36.5%

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

      4. pow2 36.5%

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

   6. Applied egg-rr 36.5%

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

      1. unpow1/2 36.5%

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

   8. Simplified 36.5%

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

   9. Taylor expanded in d around -inf 68.3%

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

   if -4.999999999999985e-310 < l < 5.99999999999999993e134

   1. Initial program 72.4%

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

   3. Simplified 71.4%

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

      1. pow1 71.4%

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

      2. associate-*r* 71.4%

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

      3. sqrt-div 79.7%

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

      4. sqrt-div 83.9%

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

      5. frac-times 83.9%

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

      6. add-sqr-sqrt 84.0%

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

      7. unpow-prod-down 82.9%

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

      8. metadata-eval 82.9%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 82.9%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 82.9%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 82.9%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 82.9%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 84.1%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 82.9%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 82.9%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 82.9%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 82.9%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 82.9%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 82.9%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 82.9%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 82.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 83.0%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 82.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 82.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 82.9%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 84.1%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. associate-*r/ 85.1%

         \[\leadsto \frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      3. sqrt-unprod 82.9%

         \[\leadsto \frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{M \cdot D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   10. Applied egg-rr 82.9%

       \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{M \cdot D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h \cdot \ell}}} \]

   11. Taylor expanded in h around -inf 59.4%

       \[\leadsto \frac{d \cdot \color{blue}{\left(-1 \cdot \left(h \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} - \frac{1}{h}\right)\right)\right)}}{\sqrt{h \cdot \ell}} \]
   12. Step-by-step derivation

       1. associate-*r* 59.4%

          \[\leadsto \frac{d \cdot \color{blue}{\left(\left(-1 \cdot h\right) \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} - \frac{1}{h}\right)\right)}}{\sqrt{h \cdot \ell}} \]

       2. neg-mul-1 59.4%

          \[\leadsto \frac{d \cdot \left(\color{blue}{\left(-h\right)} \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} - \frac{1}{h}\right)\right)}{\sqrt{h \cdot \ell}} \]

       3. sub-neg 59.4%

          \[\leadsto \frac{d \cdot \left(\left(-h\right) \cdot \color{blue}{\left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} + \left(-\frac{1}{h}\right)\right)}\right)}{\sqrt{h \cdot \ell}} \]

       4. distribute-lft-in 59.4%

          \[\leadsto \frac{d \cdot \color{blue}{\left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) + \left(-h\right) \cdot \left(-\frac{1}{h}\right)\right)}}{\sqrt{h \cdot \ell}} \]

   13. Simplified 86.5%

       \[\leadsto \frac{d \cdot \color{blue}{\left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) + 1\right)}}{\sqrt{h \cdot \ell}} \]

   if 5.99999999999999993e134 < l

   1. Initial program 57.2%

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

   3. Simplified 59.4%

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

      1. pow1 59.4%

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

      2. associate-*r* 59.4%

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

      3. sqrt-div 63.7%

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

      4. sqrt-div 73.4%

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

      5. frac-times 73.4%

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

      6. add-sqr-sqrt 73.6%

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

      7. unpow-prod-down 73.6%

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

      8. metadata-eval 73.6%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 73.6%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 73.7%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 73.7%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 77.2%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 79.3%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 79.3%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 79.3%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 79.3%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 79.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 79.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 79.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 79.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 79.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 74.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 79.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 79.3%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   9. Taylor expanded in M around 0 68.6%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+134}:\\ \;\;\;\;\frac{d \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{-178}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
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 -9e-178)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (if (<= l -5e-310)
     (* d (pow (+ -1.0 (fma h l 1.0)) -0.5))
     (/ d (* (sqrt h) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
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 <= -9e-178) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else if (l <= -5e-310) {
		tmp = d * pow((-1.0 + fma(h, l, 1.0)), -0.5);
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
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 <= -9e-178)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (l <= -5e-310)
		tmp = Float64(d * (Float64(-1.0 + fma(h, l, 1.0)) ^ -0.5));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $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, -9e-178], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -5e-310], N[(d * N[Power[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.99999999999999957e-178

   1. Initial program 63.6%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in d around inf 6.7%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 46.5%

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

      4. mul-1-neg 46.5%

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

      5. *-commutative 46.5%

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

      6. associate-/r* 47.8%

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

   8. Simplified 47.8%

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

   if -8.99999999999999957e-178 < l < -4.999999999999985e-310

   1. Initial program 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in d around inf 39.9%

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

      1. pow1 39.9%

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

      2. metadata-eval 39.9%

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

      3. sqrt-pow1 14.9%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. sqrt-prod 14.9%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{h \cdot \ell}}} \]

      5. div-inv 14.9%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

      6. sqrt-div 15.0%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}} \]

      7. sqrt-pow1 35.2%

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

      8. metadata-eval 35.2%

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

      9. pow1 35.2%

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

   7. Applied egg-rr 35.2%

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

      1. div-inv 35.2%

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

      2. pow1/2 35.2%

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

      3. pow-flip 35.2%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5\right)}} \]

      4. *-commutative 35.2%

         \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{\left(-0.5\right)} \]

      5. metadata-eval 35.2%

         \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{\color{blue}{-0.5}} \]

   9. Applied egg-rr 35.2%

      \[\leadsto \color{blue}{d \cdot {\left(\ell \cdot h\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 35.2%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot h\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 62.7%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot h\right)} - 1\right)}}^{-0.5} \]

   11. Applied egg-rr 62.7%

       \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot h\right)} - 1\right)}}^{-0.5} \]
   12. Step-by-step derivation

       1. sub-neg 62.7%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot h\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 62.7%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(\ell \cdot h\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 62.7%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(\ell \cdot h\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 62.7%

          \[\leadsto d \cdot {\left(-1 + e^{\color{blue}{\log \left(1 + \ell \cdot h\right)}}\right)}^{-0.5} \]

       5. rem-exp-log 62.7%

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

       6. +-commutative 62.7%

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

       7. *-commutative 62.7%

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

       8. fma-define 62.7%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}\right)}^{-0.5} \]

   13. Simplified 62.7%

       \[\leadsto d \cdot {\color{blue}{\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}}^{-0.5} \]

   if -4.999999999999985e-310 < l

   1. Initial program 67.3%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in d around inf 37.9%

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

      1. pow1 37.9%

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

      2. metadata-eval 37.9%

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

      3. sqrt-pow1 28.6%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. sqrt-prod 20.8%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{h \cdot \ell}}} \]

      5. div-inv 20.8%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

      6. sqrt-div 28.5%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}} \]

      7. sqrt-pow1 37.8%

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

      8. metadata-eval 37.8%

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

      9. pow1 37.8%

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

   7. Applied egg-rr 37.8%

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

      1. sqrt-prod 46.2%

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

   9. Applied egg-rr 46.2%

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

       1. *-commutative 46.2%

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

   11. Simplified 46.2%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{-178}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-187}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
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 -6e-187)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (if (<= l -5e-310)
     (/ d (cbrt (pow (* l h) 1.5)))
     (/ d (* (sqrt h) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
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 <= -6e-187) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else if (l <= -5e-310) {
		tmp = d / cbrt(pow((l * h), 1.5));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
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 <= -6e-187) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else if (l <= -5e-310) {
		tmp = d / Math.cbrt(Math.pow((l * h), 1.5));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
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 <= -6e-187)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (l <= -5e-310)
		tmp = Float64(d / cbrt((Float64(l * h) ^ 1.5)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $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, -6e-187], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -5e-310], N[(d / N[Power[N[Power[N[(l * h), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.00000000000000008e-187

   1. Initial program 64.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in d around inf 6.6%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 46.2%

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

      4. mul-1-neg 46.2%

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

      5. *-commutative 46.2%

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

      6. associate-/r* 47.5%

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

   8. Simplified 47.5%

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

   if -6.00000000000000008e-187 < l < -4.999999999999985e-310

   1. Initial program 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in d around inf 46.2%

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

      1. pow1 46.2%

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

      2. metadata-eval 46.2%

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

      3. sqrt-pow1 11.7%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. sqrt-prod 11.7%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{h \cdot \ell}}} \]

      5. div-inv 11.7%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

      6. sqrt-div 11.8%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}} \]

      7. sqrt-pow1 40.8%

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

      8. metadata-eval 40.8%

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

      9. pow1 40.8%

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

   7. Applied egg-rr 40.8%

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

      1. add-cbrt-cube 51.2%

         \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}}}} \]

      2. add-sqr-sqrt 51.2%

         \[\leadsto \frac{d}{\sqrt[3]{\color{blue}{\left(h \cdot \ell\right)} \cdot \sqrt{h \cdot \ell}}} \]

      3. pow1 51.2%

         \[\leadsto \frac{d}{\sqrt[3]{\color{blue}{{\left(h \cdot \ell\right)}^{1}} \cdot \sqrt{h \cdot \ell}}} \]

      4. pow1/2 51.2%

         \[\leadsto \frac{d}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1} \cdot \color{blue}{{\left(h \cdot \ell\right)}^{0.5}}}} \]

      5. pow-prod-up 51.2%

         \[\leadsto \frac{d}{\sqrt[3]{\color{blue}{{\left(h \cdot \ell\right)}^{\left(1 + 0.5\right)}}}} \]

      6. *-commutative 51.2%

         \[\leadsto \frac{d}{\sqrt[3]{{\color{blue}{\left(\ell \cdot h\right)}}^{\left(1 + 0.5\right)}}} \]

      7. metadata-eval 51.2%

         \[\leadsto \frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{\color{blue}{1.5}}}} \]

   9. Applied egg-rr 51.2%

      \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}} \]

   if -4.999999999999985e-310 < l

   1. Initial program 67.3%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in d around inf 37.9%

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

      1. pow1 37.9%

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

      2. metadata-eval 37.9%

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

      3. sqrt-pow1 28.6%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. sqrt-prod 20.8%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{h \cdot \ell}}} \]

      5. div-inv 20.8%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

      6. sqrt-div 28.5%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}} \]

      7. sqrt-pow1 37.8%

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

      8. metadata-eval 37.8%

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

      9. pow1 37.8%

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

   7. Applied egg-rr 37.8%

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

      1. sqrt-prod 46.2%

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

   9. Applied egg-rr 46.2%

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

       1. *-commutative 46.2%

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

   11. Simplified 46.2%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-187}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.12 \cdot 10^{-243}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 1.12e-243)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (/ d (* (sqrt h) (sqrt l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 1.12e-243) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
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 <= 1.12d-243) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
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 <= 1.12e-243) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[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 <= 1.12e-243:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 1.12e-243)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
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 <= 1.12e-243)
		tmp = d * -sqrt(((1.0 / l) / h));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 1.12e-243], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.12000000000000005e-243

   1. Initial program 68.2%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in d around inf 11.9%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 41.8%

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

      4. mul-1-neg 41.8%

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

      5. *-commutative 41.8%

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

      6. associate-/r* 42.8%

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

   8. Simplified 42.8%

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

   if 1.12000000000000005e-243 < l

   1. Initial program 66.2%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in d around inf 41.1%

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

      1. pow1 41.1%

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

      2. metadata-eval 41.1%

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

      3. sqrt-pow1 30.8%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. sqrt-prod 23.0%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{h \cdot \ell}}} \]

      5. div-inv 23.0%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

      6. sqrt-div 30.8%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}} \]

      7. sqrt-pow1 41.0%

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

      8. metadata-eval 41.0%

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

      9. pow1 41.0%

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

   7. Applied egg-rr 41.0%

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

      1. sqrt-prod 48.8%

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

   9. Applied egg-rr 48.8%

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

       1. *-commutative 48.8%

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

   11. Simplified 48.8%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.12 \cdot 10^{-243}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{-188}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell} \cdot \frac{1}{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.45e-188)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (* d (sqrt (* (/ 1.0 l) (/ 1.0 h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.45e-188) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else {
		tmp = d * sqrt(((1.0 / l) * (1.0 / h)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
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 <= (-1.45d-188)) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    else
        tmp = d * sqrt(((1.0d0 / l) * (1.0d0 / h)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
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 <= -1.45e-188) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) * (1.0 / h)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[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 <= -1.45e-188:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	else:
		tmp = d * math.sqrt(((1.0 / l) * (1.0 / h)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.45e-188)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) * Float64(1.0 / h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
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 <= -1.45e-188)
		tmp = d * -sqrt(((1.0 / l) / h));
	else
		tmp = d * sqrt(((1.0 / l) * (1.0 / h)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.45e-188], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * N[(1.0 / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.4500000000000001e-188

   1. Initial program 64.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in d around inf 6.6%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 46.2%

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

      4. mul-1-neg 46.2%

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

      5. *-commutative 46.2%

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

      6. associate-/r* 47.5%

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

   8. Simplified 47.5%

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

   if -1.4500000000000001e-188 < l

   1. Initial program 69.2%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in d around inf 38.9%

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

      1. inv-pow 38.9%

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

      2. *-commutative 38.9%

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

      3. unpow-prod-down 39.7%

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

      4. inv-pow 39.7%

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

      5. inv-pow 39.7%

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

   7. Applied egg-rr 39.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{1}{h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 42.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-188}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
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 (/ (/ 1.0 l) h))))
   (if (<= l -2.4e-188) (* d (- t_0)) (* d t_0))))

C

M_m = fabs(M);
D_m = fabs(D);
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(((1.0 / l) / h));
	double tmp;
	if (l <= -2.4e-188) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((1.0d0 / l) / h))
    if (l <= (-2.4d-188)) then
        tmp = d * -t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
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(((1.0 / l) / h));
	double tmp;
	if (l <= -2.4e-188) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[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(((1.0 / l) / h))
	tmp = 0
	if l <= -2.4e-188:
		tmp = d * -t_0
	else:
		tmp = d * t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
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(Float64(1.0 / l) / h))
	tmp = 0.0
	if (l <= -2.4e-188)
		tmp = Float64(d * Float64(-t_0));
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
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(((1.0 / l) / h));
	tmp = 0.0;
	if (l <= -2.4e-188)
		tmp = d * -t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $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[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.4e-188], N[(d * (-t$95$0)), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -2.4e-188

   1. Initial program 64.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in d around inf 6.6%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 46.2%

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

      4. mul-1-neg 46.2%

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

      5. *-commutative 46.2%

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

      6. associate-/r* 47.5%

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

   8. Simplified 47.5%

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

   if -2.4e-188 < l

   1. Initial program 69.2%

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

   3. Simplified 69.3%

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

      1. pow1 69.3%

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

      2. associate-*r* 69.3%

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

      3. sqrt-div 65.3%

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

      4. sqrt-div 70.6%

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

      5. frac-times 70.6%

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

      6. add-sqr-sqrt 70.7%

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

      7. unpow-prod-down 70.1%

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

      8. metadata-eval 70.1%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 70.1%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 70.1%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 70.1%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 70.1%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 71.8%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 71.8%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 71.8%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 71.8%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 71.8%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 71.8%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 71.8%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 71.8%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 71.8%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 71.8%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 70.4%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 71.8%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 71.8%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   9. Taylor expanded in M around 0 38.9%

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

       1. *-commutative 38.9%

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

       2. associate-/r* 39.7%

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

   11. Simplified 39.7%

       \[\leadsto d \cdot \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 42.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-187}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
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 -2.4e-187)
   (* d (- (pow (* l h) -0.5)))
   (* d (sqrt (/ (/ 1.0 l) h)))))

C

M_m = fabs(M);
D_m = fabs(D);
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 <= -2.4e-187) {
		tmp = d * -pow((l * h), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
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 <= (-2.4d-187)) then
        tmp = d * -((l * h) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
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 <= -2.4e-187) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[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 <= -2.4e-187:
		tmp = d * -math.pow((l * h), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
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 <= -2.4e-187)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
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 <= -2.4e-187)
		tmp = d * -((l * h) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $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, -2.4e-187], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.40000000000000013e-187

   1. Initial program 64.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in d around inf 6.6%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 46.2%

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

      4. mul-1-neg 46.2%

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

      5. unpow-1 46.2%

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

      6. metadata-eval 46.2%

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

      7. pow-sqr 46.2%

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

      8. rem-sqrt-square 46.5%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 46.2%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 46.2%

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

      11. rem-square-sqrt 46.5%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 46.5%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -2.40000000000000013e-187 < l

   1. Initial program 69.2%

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

   3. Simplified 69.3%

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

      1. pow1 69.3%

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

      2. associate-*r* 69.3%

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

      3. sqrt-div 65.3%

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

      4. sqrt-div 70.6%

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

      5. frac-times 70.6%

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

      6. add-sqr-sqrt 70.7%

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

      7. unpow-prod-down 70.1%

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

      8. metadata-eval 70.1%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 70.1%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 70.1%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 70.1%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 70.1%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 71.8%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 71.8%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 71.8%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 71.8%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 71.8%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 71.8%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 71.8%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 71.8%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 71.8%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 71.8%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 70.4%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 71.8%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 71.8%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   9. Taylor expanded in M around 0 38.9%

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

       1. *-commutative 38.9%

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

       2. associate-/r* 39.7%

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

   11. Simplified 39.7%

       \[\leadsto d \cdot \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-187}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 42.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -2.5 \cdot 10^{-209}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
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 (* l h) -0.5)))
   (if (<= l -2.5e-209) (* d (- t_0)) (* d t_0))))

C

M_m = fabs(M);
D_m = fabs(D);
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((l * h), -0.5);
	double tmp;
	if (l <= -2.5e-209) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
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) :: t_0
    real(8) :: tmp
    t_0 = (l * h) ** (-0.5d0)
    if (l <= (-2.5d-209)) then
        tmp = d * -t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
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.pow((l * h), -0.5);
	double tmp;
	if (l <= -2.5e-209) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[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.pow((l * h), -0.5)
	tmp = 0
	if l <= -2.5e-209:
		tmp = d * -t_0
	else:
		tmp = d * t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
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(l * h) ^ -0.5
	tmp = 0.0
	if (l <= -2.5e-209)
		tmp = Float64(d * Float64(-t_0));
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
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 = (l * h) ^ -0.5;
	tmp = 0.0;
	if (l <= -2.5e-209)
		tmp = d * -t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $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[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, -2.5e-209], N[(d * (-t$95$0)), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -2.5000000000000002e-209

   1. Initial program 65.6%

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

   3. Simplified 64.6%

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

   5. Taylor expanded in d around inf 7.3%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 45.8%

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

      4. mul-1-neg 45.8%

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

      5. unpow-1 45.8%

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

      6. metadata-eval 45.8%

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

      7. pow-sqr 45.8%

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

      8. rem-sqrt-square 46.1%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 45.9%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 45.9%

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

      11. rem-square-sqrt 46.1%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 46.1%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -2.5000000000000002e-209 < l

   1. Initial program 68.6%

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

   3. Simplified 68.7%

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

      1. pow1 68.7%

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

      2. associate-*r* 68.7%

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

      3. sqrt-div 66.6%

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

      4. sqrt-div 72.0%

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

      5. frac-times 72.0%

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

      6. add-sqr-sqrt 72.2%

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

      7. unpow-prod-down 71.5%

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

      8. metadata-eval 71.5%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      9. clear-num 71.5%

         \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      10. un-div-inv 71.5%

          \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 71.5%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 71.5%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

      2. associate-*l/ 73.3%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 73.3%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. fma-define 73.3%

         \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. associate-*r* 73.3%

         \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. fma-define 73.3%

         \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-commutative 73.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-*l* 73.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. metadata-eval 73.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. *-commutative 73.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. associate-/r/ 73.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. associate-*l/ 71.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      13. associate-/l* 73.2%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 73.2%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   9. Taylor expanded in M around 0 39.0%

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

       1. unpow-1 39.0%

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

       2. metadata-eval 39.0%

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

       3. pow-sqr 39.0%

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

       4. rem-sqrt-square 39.0%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 38.8%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 38.8%

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

       7. rem-square-sqrt 39.0%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 39.0%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{-209}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 26.0% accurate, 3.1× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
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 (pow (* l h) -0.5)))

C

M_m = fabs(M);
D_m = fabs(D);
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 * pow((l * h), -0.5);
}

Fortran

M_m = abs(m)
D_m = abs(d)
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 * ((l * h) ** (-0.5d0))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
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.pow((l * h), -0.5);
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[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.pow((l * h), -0.5)

Julia

M_m = abs(M)
D_m = abs(D)
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 * (Float64(l * h) ^ -0.5))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
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 * ((l * h) ^ -0.5);
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $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[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.3%

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

3. Simplified 66.9%

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

   1. pow1 66.9%

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

   2. associate-*r* 66.9%

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

   3. sqrt-div 37.7%

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

   4. sqrt-div 40.8%

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

   5. frac-times 40.8%

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

   6. add-sqr-sqrt 40.9%

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

   7. unpow-prod-down 40.5%

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

   8. metadata-eval 40.5%

      \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\color{blue}{0.25} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   9. clear-num 40.5%

      \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   10. un-div-inv 40.5%

       \[\leadsto {\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

6. Applied egg-rr 40.5%

   \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1}} \]
7. Step-by-step derivation

   1. unpow1 40.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   2. associate-*l/ 41.5%

      \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   3. associate-/l* 41.5%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   4. fma-define 41.5%

      \[\leadsto d \cdot \frac{\color{blue}{\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1}}{\sqrt{h} \cdot \sqrt{\ell}} \]

   5. associate-*r* 41.5%

      \[\leadsto d \cdot \frac{\color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \cdot \frac{h}{\ell}} + 1}{\sqrt{h} \cdot \sqrt{\ell}} \]

   6. fma-define 41.5%

      \[\leadsto d \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5, \frac{h}{\ell}, 1\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

   7. *-commutative 41.5%

      \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25\right)} \cdot -0.5, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. associate-*l* 41.5%

      \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \left(0.25 \cdot -0.5\right)}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   9. metadata-eval 41.5%

      \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot \color{blue}{-0.125}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. *-commutative 41.5%

       \[\leadsto d \cdot \frac{\mathsf{fma}\left(\color{blue}{-0.125 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   11. associate-/r/ 41.5%

       \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   12. associate-*l/ 40.7%

       \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   13. associate-/l* 41.5%

       \[\leadsto d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

8. Simplified 41.5%

   \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

9. Taylor expanded in M around 0 25.2%

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

    1. unpow-1 25.2%

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

    2. metadata-eval 25.2%

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

    3. pow-sqr 25.2%

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

    4. rem-sqrt-square 24.9%

       \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

    5. rem-square-sqrt 24.8%

       \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

    6. fabs-sqr 24.8%

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

    7. rem-square-sqrt 24.9%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

11. Simplified 24.9%

    \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

12. Final simplification 24.9%

    \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]
13. Add Preprocessing

Alternative 21: 26.0% accurate, 3.2× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
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

M_m = fabs(M);
D_m = fabs(D);
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

M_m = abs(m)
D_m = abs(d)
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

M_m = Math.abs(M);
D_m = Math.abs(D);
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

M_m = math.fabs(M)
D_m = math.fabs(D)
[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

M_m = abs(M)
D_m = abs(D)
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

M_m = abs(M);
D_m = abs(D);
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

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

Derivation

1. Initial program 67.3%

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

3. Simplified 66.9%

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

5. Taylor expanded in d around inf 25.2%

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

   1. pow1 25.2%

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

   2. metadata-eval 25.2%

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

   3. sqrt-pow1 26.1%

      \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   4. sqrt-prod 20.7%

      \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{h \cdot \ell}}} \]

   5. div-inv 20.7%

      \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

   6. sqrt-div 26.1%

      \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}} \]

   7. sqrt-pow1 24.8%

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

   8. metadata-eval 24.8%

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

   9. pow1 24.8%

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

7. Applied egg-rr 24.8%

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

8. Final simplification 24.8%

   \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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)))))